2.1 Shimura variety of orthogonal type

Let V be a nondegenerate quadratic even lattice over $\mathbb {Z}$ of signature $(2,n)$ , and let $G=\mathrm {SO}(V)$ be the corresponding special orthogonal group. The group $G(\mathbb {R})$ of real points of G is isomorphic to $\mathrm {SO} (2,n)$ . Fix a maximal compact subgroup of $K_{\mathbb {R}} \subset G(\mathbb {R})$ , and let $\mathrm {D}_V$ be the connected component of $G(\mathbb {R})/K_{\mathbb {R}}$ . It is a type IV Hermitian symmetric domain of complex dimension n.

Let $\hat {\mathbb {Z}}$ be the profinite completion of $\mathbb {Z}$ and $\mathbb {A}_f=\hat {\mathbb {Z}}\otimes \mathbb {Q}$ be the ring of finite adeles. Let $\mathbf {K}\subseteq G(\mathbb {A}_f)$ be the discriminant kernel: that is, the largest subgroup of $G(\hat {\mathbb {Z}})$ that acts trivially on the discriminant group $V^\vee /V$ . To any compact open subgroup $K\subseteq \mathbf {K}$ of $G(\mathbb {A}_f)$ , it corresponds a quotient space

$$ \begin{align*}\mathrm{Sh}_K(V) = G(\mathbb{Q})\backslash \widehat{\mathrm{D}}_V \times G (\mathbb{A}_f) /K.\end{align*} $$

We shall simply write $\mathrm {Sh}(V)$ when $K = \mathbf {K}$ .

If K is neat, such as when K is sufficiently small, then $\mathrm {Sh}_K(V)$ is a smooth quasi-projective variety. In general, $\mathrm {Sh}_K(V)$ is not connected. The connected component corresponding to the coset of $\mathrm {D}_V \times \{\mathrm {id} \}$ is an arithmetic quotient

$$ \begin{align*}X_{\Gamma}:=\Gamma \backslash \mathrm{D}_V,\end{align*} $$

where $\Gamma =G(\mathbb {Q})\cap K$ . If V contains a hyperbolic plane, then one has

$$ \begin{align*}\mathrm{Sh}(V)=\Gamma_V\backslash \mathrm{D}_V,\end{align*} $$

where $\Gamma _V=\mathbf {K}\cap G(\mathbb {Z})$ is the stable special orthogonal group.

2.2 Special cycles on $X_\Gamma $

Given a vector $\mathbf {v}\in (V^\vee )^r\subseteq V^r(\mathbb {Q})$ , we let $U=U(\mathbf {v})$ be the $\mathbb {Q}$ -subspace of $V(\mathbb {Q})$ spanned by the components of $\mathbf {v}$ . Let $\mathrm {D}_{\mathbf {v}}\subset \mathrm {D}_V$ be the subset consisting of $2$ -planes that lie in $U^\perp $ . The codimension r natural cycle $Z(\mathbf {v})$ on $X_\Gamma $ is defined to be the image of

(2.1) $$ \begin{align} \Gamma_{\mathbf{v}}\backslash \mathrm{D}_{\mathbf{v}}\rightarrow \Gamma\backslash \mathrm{D}_V, \end{align} $$

where $\Gamma _{\mathbf {v}}$ is the stabiliser of U in $\Gamma $ . For any $\beta \in \mathrm {Sym}_{r\times r} (\mathbb {Q})$ of rank $t(\beta )$ , we set

$$ \begin{align*}\Omega_\beta=\{\mathbf{v}\in V^r \; | \; \frac{1}{2} (\mathbf{v},\mathbf{v})=\beta,\dim U(\mathbf{v})=t(\beta)\},\end{align*} $$

which comes with a natural $\Gamma $ -action. For each function $\varphi \in \mathrm {Hom}((V^\vee /V)^r, \mathbb {C})$ , Kudla associates a special cycle

$$ \begin{align*}Z(\beta,\varphi;\Gamma)\in \mathrm{ CH}^r(X_\Gamma)\end{align*} $$

as some linear combinations of $\lambda ^{r-t(\beta )} \cdot Z(\mathbf {v})$ with $\mathbf {v}\in \Omega _\beta $ (compare [Reference Kudla27]). Here $\lambda $ is the Hodge line bundle on $X_\Gamma $ . They are compatible with natural pullback maps $\pi :X_{\Gamma '}\to X_\Gamma $ for $\Gamma '\subseteq \Gamma $ : that is,

$$ \begin{align*}\pi^\ast(Z(\beta,\varphi;\Gamma))=Z(\beta,\varphi;\Gamma').\end{align*} $$

Hence we may write $Z(\beta ,\varphi )=Z(\beta ,\varphi ;\Gamma )$ for simplicity. Let us denote by $Z(\beta )$ the map

$$ \begin{align*}\varphi \mapsto Z(\beta, \varphi ) \in \mathrm{Hom}((V^\vee/V)^r, \mathbb{C})^\vee \otimes \mathrm{CH}^r(X_\Gamma)_{\mathbb{C}} .\end{align*} $$

Let $\mathrm {Mp}_{2r}(\mathbb {Z})$ be the metaplectic double covering of $\mathrm {Sp}_{2r}(\mathbb {Z})$ and $\rho _r$ be the Weil representation of $\mathrm { Mp}_{2r}(\mathbb {Z})$ acting on $\mathrm {Hom}((V^\vee /V)^r, \mathbb {C})$ . A celebrated theorem for special cycles on $X_\Gamma $ is

Theorem 2.2.1 (Kudla’s modularity conjecture)

The generating series

(2.2) $$ \begin{align} \Theta(\tau)= \sum\limits_{\beta\geq0} Z(\beta) e^{2\pi i \mathrm{tr}(\beta\tau)},~ \tau\in \mathbb{H}_r \end{align} $$

with coefficients in $\mathrm {Hom}((V^\vee /V)^r, \mathbb {C})^\vee \otimes _{\mathbb {C}} \mathrm {CH}^r(X_\Gamma )_{\mathbb {C}}$ is a Siegel modular form of type $\rho _r$ , weight $\frac {n+2}{2}$ and genus r in with values in $\mathrm {CH}^r(X_\Gamma )_{\mathbb {C}}$ : that is,

$$ \begin{align*}\Theta(A\tau)=\det(c\tau +d)^{\frac{n+2}{2}}\rho_r (A, \pm \sqrt{\det(c\tau+d)})\Theta(\tau),\end{align*} $$

for all $A=\left (\begin {array}{ccc} a & b \\ c & d \end {array}\right )\in \mathrm {Sp}_{2r}(\mathbb {Z})$ .

Example 2.3. If V is unimodular, then $V^\vee /V$ is trivial, and the special cycle $Z(\beta )$ can be written as the quotient by $\Gamma $ of the union of hyperplanes

(2.3) $$ \begin{align} Z(\beta)= \Gamma\backslash \sum\limits_{\mathbf{v}\in \Omega_\beta} U(\mathbf{v})^\perp, \end{align} $$

which is a linear combination of $Z(\mathbf {v})$ for $\mathbf {v}\in \Gamma \backslash \Omega _\beta $ . In this case, Theorem 2.2.1 reads as the generating series $\Theta (\tau )$ is a classical Siegel modular form of weight $\frac {n+2}{2}$ and genus r.

Another important result concerning the special cycles is the following:

3.1 Moduli space of (quasi-)polarised K3 surfaces

A (quasi-)polarised K3 surface of genus $g\geq 2$ is a pair $(S,L)$ , where S is a K3 surface and $L\in \mathrm {Pic}(S)$ is an ample (respectively, nef) line bundle with $L^2=2g-2$ . The middle cohomology $\mathrm {H}^2(S,\mathbb {Z})$ equipped with the bilinear intersection form $q(-,-)$ is an even unimodular lattice that is isomorphic to the K3 lattice

$$ \begin{align*}\Lambda=U^{\oplus 3}\oplus E_8(-1)^{\oplus 2},\end{align*} $$

where U is the hyperbolic lattice of rank two and $E_8$ is the positive definite lattice associated to the Lie algebra of the same name. Let $(e_1,f_1)$ be a standard basis of the first hyperbolic lattice. The primitive part $\mathrm {H}^2_{\mathrm {prim}}(S,\mathbb {Z}):=c_1(L)^\perp $ has signature $(2,19)$ and it is isomorphic to the lattice

$$ \begin{align*}\Lambda_{g}:=\left<e_1-(g-1)f\right>\oplus U^{\oplus 2}\oplus E_8(-1)^{\oplus 2},\end{align*} $$

by identifying $c_1(L)$ with the vector $e_1+(g-1)f_1$ . The Hodge structures on $\mathrm {H}^2_{\mathrm {prim}}(S,\mathbb {Z})$ are parametrised by the period domain $\mathrm {D}_{\Lambda _g}$ associated to $\Lambda _g$ .

Let $\mathscr {F}_g$ be the moduli stack of primitively quasi-polarised complex K3 surfaces of genus g and $\mathscr {F}_{g}^p \subseteq \mathscr {F}_g$ the open substack of polarised K3 surfaces of genus g. Recall that for $g>2$ , we have also introduced the open subscheme $\mathscr {F}_g^\circ \subseteq \mathscr {F}_{g}^p$ of polarised K3 surfaces of genus g with only trivial automorphisms. The following result is well-known.

Theorem 3.1.1 (compare [Reference Huybrechts21, 5.1.3, 5.1.4 and 6.4.1])

The moduli stack $\mathscr {F}_g$ is a smooth Deligne-Mumford stack of finite type; it is coarsely represented by a quasi-projective variety $\mathcal {F}_g$ , and there is an isomorphism

$$ \begin{align*}\mathcal{F}_g\to \widetilde{\mathrm{O}}(\Lambda_g)\backslash \mathrm{D}_{\Lambda_g}.\end{align*} $$

Moreover, the open substack $\mathscr {F}_g^p$ is separated.

More generally, one can consider the moduli space of lattice polarised K3 surfaces, which behaves similarly to $\mathscr {F}_g$ .

Let $\mathcal {F}_{\Sigma }$ be the coarse moduli space of $\Sigma $ -quasi-polarised K3 surfaces. In [Reference Dolgachev11], Dolgachev proved that

Theorem 3.2.1 (compare [Reference Dolgachev11, Remark 3.4])

Fix a primitive embedding of $\Sigma $ into $\Lambda $ , and consider $\Sigma $ as a sublattice of $\Lambda $ via this embedding. Then there is an isomorphism

$$ \begin{align*}\mathcal{F}_\Sigma\cong \Gamma_{\Sigma}\backslash \mathrm{D}_{\Sigma^\perp},\end{align*} $$

where $\mathrm {D}_{\Sigma ^\perp }:= \{[\sigma ]\in \mathrm {D}_{\Lambda }|~q(\sigma , \Sigma )=0\}$ is the period domain of $\Sigma $ -(quasi-)polarised K3 surfaces and $\Gamma _{\Sigma }:=\{\gamma \in \mathrm {O}(\Lambda )|~\gamma |_{\Sigma }=id\}.$

3.3 Example: Moduli of elliptic K3 surfaces

Let us consider the U-quasi-polarised K3 surfaces. If S is U-quasi-polarised, we have $L_1,L_2\in \mathrm {Pic}(S)$ satisfying $L_i^2=0$ and $L_1L_2=1$ . We may assume that $L_1$ is effective without fixed component. Then the map

$$ \begin{align*}f:S\to |L_1|\cong \mathbb{P}^1\end{align*} $$

is an elliptic fibration, and the difference $L_1-L_2$ represents a section class of f. Conversely, if $f:S\to \mathbb {P}^1$ is an elliptic fibration with a section O, then the fibre class F and $F+O$ gives a U-quasi-polarisation. We may also call S a unigonal K3 surface. Let $\mathcal {F}_U$ be the coarse moduli space of U-quasi-polarised K3 surfaces. According to Theorem 3.2.1, we have

$$ \begin{align*}\mathcal{F}_U=\Gamma_{U} \backslash \mathrm{D}_{U^\perp}.\end{align*} $$

Note that the orthogonal complement $U^\perp \cong U^{\oplus 2}\oplus E_8^{\oplus 2}$ is a unimodular lattice.

Let us follow [Reference Miranda35, III.2] to describe the projective models of U-quasi-polarised K3 surfaces. Let $\pi :S\to \mathbb {P}^1$ be the elliptic fibration and C a section of $\pi $ . Then S has a unique Weierstrass model with at worst ADE singularities: that is, we can think of it as the closed subscheme of $\mathbb {P}(\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(-4)\oplus \mathcal {O}_{\mathbb {P}^1}(-6))$ defined by the equation

$$ \begin{align*}y^2z=x^3+axz^2+bz^3,\end{align*} $$

where $a\in \mathrm {H}^0(\mathbb {P}^1,\mathcal {O}_{\mathbb {P}^1}(8))$ and $b\in \mathrm {H}^0(\mathbb {P}^1,\mathcal {O}_{\mathbb {P}^1} (12))$ , satisfying certain conditions (compare [Reference Miranda34, Theorem 2.1]). As shown in [Reference Miranda34], such equations are naturally parametrised by an open subset of the weighted projective space $\mathbb {P}(2^{(9)},3^{(13)})$ equipped with a $\mathrm {SL}_2$ -action. Recently, Odaka-Oshima [Reference Odaka and Oshima38] have shown that there is actually an isomorphism

(3.1) $$ \begin{align} \mathbb{P}(2^{(9)},3^{(13)}) /\!\!/ \mathrm{SL}_2\cong \mathcal{F}_U^\ast \end{align} $$

between the GIT quotient and the Baily-Borel compactification of $\mathcal {F}_U$ .

There is another description of the projective model of $\pi :S\to \mathbb {P}^1$ : as

$$ \begin{align*}\pi_\ast \mathcal{O}_S(2C)\cong \mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(-4),\end{align*} $$

the natural surjection $\pi ^\ast \pi _\ast \mathcal {O}_S(2C) \to \mathcal {O}_S(2C)$ defines a double covering

$$ \begin{align*}S\to \mathbf{F}_4=:\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(-4)).\end{align*} $$

Its Weierstrass model is a double covering of $\mathbf {F}_4$ branched over the trisection $x^3+axz^2+bz^3=0$ and the section $z=0$ . Conversely, Let $A=\mathbb {P}(\mathcal {O}_{\mathbb {P}^1}(-4))$ and F a ruling of $\mathbf {F}_4$ . Choose $B\in |3A+12F|$ that is disjoint from A and smooth; then $(A+B)$ is a smooth $2$ -anticanonical curve on $\mathbf {F}_4$ , and the double covering of $\mathbf {F}_4$ branched over $(A+B)$ is a unigonal K3 surface.

3.4 Noether-Lefschetz loci on $\mathcal {F}_g$

The Noether-Lefschetz (NL) locus $\mathcal {N} \mathcal {L}^1(\mathcal {F}_g)\subset \mathcal {F}_g$ is the locus parametrising $K3$ surfaces in $\mathcal {F}_{g}$ with Picard number greater than $1$ . It is a union of countably many divisors defined as follows: given $d,n\in \mathbb {Z}$ , the NL-divisor $\mathcal {H}_{d,n}^{g}\subset \mathcal {F}_{g}$ is the locus of quasi-polarised K3 surfaces $(S,L)\in \mathcal {F}_g$ whose Picard lattice $\mathrm {Pic}(S)$ contains a rank two sublattice

(3.2) $$ \begin{align}\begin{array}{c|c|c} & L & \alpha \\\hline L & 2g-2 & d \\\hline \alpha & d & n\end{array} \end{align} $$

where $\alpha \in \mathrm {Pic}(S)$ . In general, $\mathcal {H}_{d,n}^g$ is not irreducible, and one can define the so-called irreducible NL-divisor $\mathcal {P}_{d,n}^g$ by requiring lattice (3.2) to be primitive. As divisors, $\mathcal {H}_{d,n}^g$ can be written as a linear combination of primitive NL divisors via the triangulated relation

(3.3) $$ \begin{align} \mathcal{H}^{g}_{d,n}=\sum\limits_{(d_i,n_i) } \mu_{d_i,n_i} \mathcal{P}^{g}_{d_i,n_i}, \end{align} $$

where $(d_i,n_i)$ runs over $\mathbb {Z}^2$ modulo the relation $(d,n)\sim (d', n')$ if

(3.4) $$ \begin{align} d^2-2n(g-1)=(d')^2-2n'(g-1)\quad and \quad d\equiv d'\mod 2g-2; \end{align} $$

the coefficient $\mu _{d_i,g_i}\in \{0,1,2\}$ is the number of integer solutions $(x,y)$ of the equations

$$ \begin{align*}(d^2_i-2n_i(g-1))x^2=d^2-2n(g-1); ~(2g-2)y=d-x n_i.\end{align*} $$

Let $\Delta _{d,g}$ be the determinant of matrix (3.2), and let $\varpi $ be the generator of the cyclic group $\Lambda _g^\vee /\Lambda _g$ . According to [Reference Maulik and Pandharipande32, Lemma 3], the projection $\overline {\alpha } $ of $\alpha $ to $\mathrm {H}^2_{prim}(S,\mathbb {Z})$ is a vector with norm, $\frac {\Delta _{d,g}}{4g-4}$ and the image of $\overline {\alpha }$ in $ \Lambda ^\vee _g/\Lambda _g$ is $d\varpi $ ; thus the NL divisor $\mathcal {H}^g_{d,n}$ can be identified as the special cycle of codimension one

(3.5) $$ \begin{align} Z(\frac{\Delta_{d,g}}{4g-4}, d):=\tilde{\mathrm{O}} (\Lambda_g)\backslash \sum_{\substack{ v^2=\frac{\Delta_{d,g}}{4g-4},\\ v\equiv d \varpi\ \mod \Lambda_g }} v^\perp. \end{align} $$

Note that $Z(\frac {\Delta _{d,g}}{4g-4}, d)$ has multiplicity two if $2d\varpi =0$ in $\Lambda _g^\vee /\Lambda _g$ because v and $-v$ will both occur in taking the sum.

Similarly, one can define the higher codimensional Noether-Lefschetz loci $\mathcal {N}\mathcal {L}^m(\mathcal {F}_g)$ parametrising quasi-polarised K3 surfaces in $\mathcal {F}_g$ with Picard number greater than m. To specify the irreducible components of $\mathcal {N}\mathcal {L}^m(\mathcal {F}_g)$ , we define the NL-cycles of codimension m on $\mathcal {F}_g$ as below.

Definition 3.5. Let M be a symmetric matrix of signature $(1,m)$ . We define

$$ \begin{align*}\mathcal{H}_{M}^g\subseteq \mathcal{N}\mathcal{L}^m(\mathcal{F}_g)\end{align*} $$

to be the locus of quasi-polarised K3 surfaces $(S,L)\in \mathcal {F}_g$ for which there exist classes

$$ \begin{align*}\alpha_1,\ldots, \alpha_m\in \mathrm{ Pic}(S)\end{align*} $$

such that the Gram matrix of ( $L, \alpha _1, \ldots , \alpha _m)$ is M.

Using the period map, the NL-cycle $\mathcal {H}_M^g$ can be identified as linear combinations of special cycles in the following way: assume $L\cdot \alpha _i=d_i\in \mathbb {Z}$ . Let $\overline {\alpha }_i$ be the projection of $\alpha _i$ in $\mathrm {H}^2_{\mathrm {prim}}(S,\mathbb {Z})$ , and let $\overline {M}$ be the Gram matrix of $\overline {\alpha _i}$ . Let $\Omega _{M}$ be the collection of vectors $\mathbf {v}=(v_1,\ldots ,v_m)\in \Lambda ^m_g$ such that

• ○ the Gram matrix of $(v_1,\ldots , v_m)$ is $\overline {M}$ ,

• ○ $v_i\equiv d_i \varpi \mod \Lambda _g$ .

Then $\mathcal {H}_M^g$ can be written as the quotient by $\widetilde {\mathrm {O}}(\Lambda _g)$ of the sum of subdomains

(3.6) $$ \begin{align} \widetilde{\mathrm{O}}(\Lambda_g)\backslash \sum\limits_{\substack{\{v_1,\ldots,v_m\} \\ \mathrm{s.t.} (v_1,\ldots, v_m)\in \Omega_M}} U(v_1,\ldots, v_m)^{\perp}, \end{align} $$

where the sum is taken over the m-ary subsets of $\Lambda _g$ consisting of components of vectors in $\Omega _M$ . One can easily see that the irreducible components of $\mathcal {H}^g_M$ are the natural cycles. A simple fact is that $\mathcal {H}_M^g$ is irreducible if and only if all the sublattices $\mathtt {M}\subseteq \Lambda _g$ containing $e_1+(g-1)f_1$ , having Gram matrix M with respect to $(e_1+(g-1)f_1, v_1, \ldots , v_m)$ for some $(v_1,\ldots ,v_m)\in \mathtt {M}^m$ , are lying in the same $\widetilde {\mathrm {O}}(\Lambda _g)$ -orbit. In the rest of this section, we will give some examples of NL-cycles and compute the irreducible components.

3.6 First example: locus of exceptional K3 surfaces

A typical example is the NL-locus of K3 surfaces in $\mathcal {F}_g$ with $(-2)$ -exceptional curves. To describe the irreducible components of this locus, we need the following result that characterises the $\widetilde {\mathrm {O}}(V)$ -orbits of primitive vectors for certain V.

Then we have

Proof. The irreducible components of $\mathcal {H}^g_{0,-2}$ correspond to the $\widetilde {\mathrm {O}}(\Lambda _g)$ -orbits of roots in $\Lambda _g$ . According to Proposition 3.7, any two roots x and y are lying in the same $\widetilde {\mathrm {O}}(\Lambda _g)$ -orbit if $x^\ast =y^\ast $ in $\Lambda _g^\vee /\Lambda _g$ . As the discriminant group

$$ \begin{align*}\Lambda_g^\vee/\Lambda_g\cong \mathbb{Z}/(2g-2)\mathbb{Z},\end{align*} $$

it is not difficult to show that $x^\ast $ is either $0$ or $g-1$ in $\Lambda _g^\vee /\Lambda _g$ . The latter situation occurs only when $g\equiv 2\mod 4$ . The assertion follows easily.

3.9 Second example: binodal and cuspidal loci

Let us describe some Noether-Lefschetz loci of codimension $2$ on $\mathcal {F}_g$ , which plays an important role in this paper. Let $\{t_1,\cdots , t_8\}$ be a standard basis of the $E_8$ lattice. Then we introduce two sublattices in $\Lambda $ : let $\mathtt {A_{1,1}}$ be the sublattice generated by $e_1+(g-1)f_1$ , $t_1$ and $t_3$ , and let $\mathtt {A}_{2}$ be the sublattice generated by $e_1+(g-1)f_1$ , $t_1$ and $t_2$ . The associated Gram matrices under $q(-,-)$ are

$$ \begin{align*} A_{1,1}=\left(\begin{array}{ccc} 2g-2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2\end{array}\right); A_{2}=\left(\begin{array}{ccc} 2g-2 & 0 & 0 \\ 0 & -2 & 1 \\ 0 & 1 & -2\end{array}\right). \end{align*} $$

We let $\mathcal {H}^g_{A_{1,1}}$ and $\mathcal {H}^g_{A_2}$ be the associated NL-cycles of codimension $2$ . Indeed, they parametrise quasi-polarised K3 surfaces in $\mathcal {F}_g$ with at least two exceptional $(-2)$ -curves. We may call $\mathcal {H}_{A_{1,1}}^g$ the binodal locus and $\mathcal {H}_{A_2}^g$ the cuspidal locus. This is because for K3 surfaces in $\mathcal {H}^g_{A_{1,1}}$ and $\mathcal {H}^g_{A_2}$ , after contracting the two exceptional $(-2)$ -curves, one can obtain singular K3 surfaces with two isolated $A_1$ -singularities or, respectively, an isolated $A_2$ -singularity.

Proof. Let us first consider the case $\mathcal {H}^g_{A_{1,1}}$ . Identify $\widetilde {\mathrm {O}}(\Lambda _g)$ as the subgroup of $\mathrm {O}(\Lambda )$ , fixing the vector $e_1+(g-1)f_1$ ; then we just need to classify the $\widetilde {\mathrm {O}}(\Lambda _g)$ -orbits of lattice $M\subset \Lambda $ satisfying $e_1+(g-1)f_1\in \mathtt {M}$ and $\mathtt {M}\cong \mathtt {A_{1,1}}$ via an isometry fixing the vector $e_1+(g-1)f_1$ . One should note that $\mathcal {P}^g_{0,-2}\cap \mathcal {P}^g_{g-1,\frac {g-2}{2}}=\emptyset $ by the Hodge index theorem. According to Proposition 3.8, $ \mathcal {H}^g_{A_{1,1}}$ is contained in the primitive NL-locus $\mathcal {P}^g_{0,-2}$ . Hence, up to a $\widetilde {\mathrm {O}}(\Lambda _g)$ -action, we can assume that $\mathtt {M}$ is spanned by the vectors $e_1+(g-1)f_1, t_1$ and v for some root $v\in \Lambda $ orthogonal to $e_1+(g-1)f_1$ and $t_1$ .

Denote by

$$ \begin{align*}\Lambda_{A_1}=\left<e_1-(g-1)f_1\right>\oplus U^{\oplus 2}\oplus E_8\oplus W_7\end{align*} $$

the orthogonal complement of $e_1+(g-1)f_1$ and $t_1$ , where $W_7=t_1^\perp \subset E_8$ is the orthogonal complement of $t_1$ in $E_8$ . We are reduced to consider the $\widetilde {\mathrm {O}}(\Lambda _{A_1})$ -orbits of v. Due to Proposition 3.7, we know that the $\widetilde {\mathrm {O}}(\Lambda _{A_1})$ -orbits of v are determined by $v^\ast $ in $\Lambda _{A_1}^\vee /\Lambda _{A_1}$ . Let $div(v)$ be the divisibility of v. Note that $div(v)$ is at most $2$ as $v^2=-2$ . The discriminant group $\Lambda _{A_1}^\vee /\Lambda _{A_1}$ is isomorphic to $\mathbb {Z}/(2g-2)\mathbb {Z}\times \mathbb {Z}/2\mathbb {Z}$ , which is generated by $\frac {(e_1-(g-1)f_1)}{2g-2}$ and $\frac {(t_1+2t_2)}{2}$ . This gives three possibilities:

1. i) $div(v)=1$ , and $v^\ast =0$ in $\Lambda _{A_1}^\vee /\Lambda _{A_1}$ .

2. ii) $div(v)=2$ , and $v^\ast =\frac {1}{2}(e_1-(g-1)f_1)$ in $\Lambda _{A_1}^\vee /\Lambda _{A_1}$ . By computing the norm of $v^\ast $ , we get $g\equiv 2 \mod 4$ . In this case, one can take v to be the vector

   $$ \begin{align*}e_1-(g-1)f_1+2(e_2+\frac{g-2}{4}f_2),\end{align*} $$
   where $(e_2,f_2)$ is the standard basis of the second hyperbolic lattice.

3. iii) $div(v)=2$ , and $v^\ast =\frac {1}{2}(e_1-(g-1) f_1+t_1+2t_2)$ in $\Lambda _{A_1}^\vee /\Lambda _{A_1}$ . The normal computation shows that $g\equiv 3\mod 4$ . In this case, one can take v to be the vector

   $$ \begin{align*}e_1-(g-1) f_1+t_1+2t_2+2(e_2+\frac{g+1}{4} f_2).\end{align*} $$

4. iiv) $div(v)=2$ , and $v^\ast =\frac {1}{2}(t_1+2t_2) $ in $\Lambda _{A_1}^\vee /\Lambda _{A_1}$ . This is impossible because the norm of $v^\ast $ is $-\frac {1}{2}$ , while $ \frac {1}{2}(t_1+2t_2)$ is $-\frac {3}{2}$ .

For $\mathcal {H}^g_{A_2}$ , the computation is similar. We just need to classify the $\widetilde {\mathrm {O}}(\Lambda _g)$ -orbits of lattices spanned by $e_1+(g-1)f_1$ , $t_1$ and v for some v orthogonal to $e_1+(g-1)f_1$ , $v^2=-2$ and $t_1\cdot v=1$ . The vector $t_1+2v$ lies in $\Lambda _{A_1}$ with norm $-6$ . One can see that the divisibility of $t_1+2v$ in $\Lambda _{A_1}$ is either $2$ or $6$ . Among all cases, there is only one possibility:

$$ \begin{align*}(t_1+2v)^\ast= \frac{t_1+2t_2}{2}~in \Lambda_{A_1}^\vee/\Lambda_{A_1}.\end{align*} $$

One can take $v=t_2$ in this case. The rest of the assertion follows easily.

Furthermore, one can consider the binodal locus and cuspidal locus on the moduli space of lattice polarised K3 surfaces. The following result is straightforward.

Proof. According to Example 3.3, $U^\perp \cong U^{\oplus 2} \oplus E_8^{\oplus 2} $ is unimodular. Using the period map, we can write $\mathcal {H}_{U,A_2}$ and $\mathcal {H}_{U,A_{1,1}}$ as a multiple of special cycles via equation (2.3) and equation (3.6):

(3.7) $$ \begin{align} \mathcal{H}_{U,A_2}=\frac{1}{2}Z(\left(\begin{array}{ccc} -2 & 1 \\ 1 & -2 \end{array}\right) ) ~\mathrm{and}~\mathcal{H}_{U,A_{1,1}}=\frac{1}{2}Z(\left(\begin{array}{ccc} -2 & 0 \\ 0 & -2 \end{array}\right) ). \end{align} $$

The multiplicity $\frac {1}{2}$ comes from the fact that if $\mathbf {v}=(v_1,v_2)\in \Omega _{A_2}$ (respectively, $\Omega _{A_{1,1}}$ ), then the vector $\mathbf {v}'=(v_2,v_1)$ also occurs in $\Omega _{A_2}$ (respectively, $\Omega _{A_{1,1}}$ ). The NL-cycle takes the sum over the collection of the set consisting of components $\{v_1,v_2\}$ , while the special cycle takes the sum over the collection of vectors $(v_1,v_2)$ .

The irreducibility of them follows from Nikulin’s result [Reference Nikulin36, Theorem 1.14.4], which shows that the primitive embeddings of $\left (\begin {array}{ccc} -2 & 1 \\ 1 & -2 \end {array}\right ) $ and $\left (\begin {array}{ccc} -2 & 0 \\ 0 & -2 \end {array}\right ) $ into $U^{\oplus 2} \oplus E_8^{\oplus 2}$ are unique.

4.1 DB cohomology of smooth varieties and arbitrary varieties

Definition 4.2 [Reference Beĭlinson5, Reference Esnault and Viehweg13]

Let X be a smooth complex algebraic variety and $j:X\hookrightarrow \overline {X}$ be a good compactification with boundary divisor Z. The Deligne-Beilinson cohomology of X is defined as the analytic hypercohomology

$$ \begin{align*}\mathrm{H}^q_{\mathrm{DB}}(X,A(p))= \mathrm{H}^q (\overline{X}_{\mathrm{an}},A(p)_{\mathrm{DB}, (\overline{X}, X)}),\end{align*} $$

where

$$ \begin{align*}A(p)_{\mathrm{DB}, (\overline{X}, X)}= \mathrm{Cone} (R j_* A(p) \oplus \Omega_{\overline{X}}^{\geq p}(\log Z)\xrightarrow{\epsilon-\iota} R j_* \Omega_{X}^\bullet)[-1]\end{align*} $$

is a complex of analytic sheaves on $\overline {X}$ called the Deligne-Beilinson complex of the pair $(\overline {X}, X)$ . Here, A is a subring of $\mathbb {R}$ and $A(p)$ is the constant sheaf with value $(2\pi i)^p A$ ; the complex $\Omega ^\bullet _X$ is the de Rham complex of holomorphic forms on X, and $\Omega ^{\geq p}_{\overline {X}}(\log Z)$ is the brutal truncation of the complex of meromorphic forms on X with at most logarithmic poles along Z; $\epsilon $ is the composite $R j_* A(p)\to R j_* \mathbb {C} \to R j_*\Omega _{X}^\bullet $ induced by $A(p)\subseteq \mathbb {C}\subseteq \Omega _{X}^\bullet $ , and $\iota $ is the composite $\Omega ^{\geq p}_{\overline {X}}(\log Z)\subseteq \Omega _{\overline {X}}^\bullet (\log Z)\subseteq j_*\Omega _{X}^\bullet =R j_*\Omega _{X}^\bullet $ .

Moreover, there is a natural map $\mathrm {H}^{2p}_{\mathrm {DB}}(X,A(p))\to \mathrm {H}^{2p}(X,A(p))$ and a DB-cycle class map

$$ \begin{align*}\mathrm{cl}_{\mathrm{DB}}:\mathrm{CH}^p(X)\to \mathrm{H}^{2p}_{\mathrm{DB}}(X,\mathbb{Z}(p))\end{align*} $$

defined by lifting the Betti fundamental classes. When X is smooth and projective, they are compatible with the Abel-Jacobi map

(4.1)

The notion of DB cohomology can be extended to the case of arbitrary varieties and, even more generally, arbitrary (separated) simplicial schemes via the theory of cohomological descent (compare [Reference Esnault and Viehweg13, §5]). We briefly recall the constructions. For a separated simplicial scheme $X_{\bullet }$ , one can find a diagram

(4.2)

where p is (a proper hypercovering, hence) of cohomological descent: that is, the cohomologies of $X_{\bullet }$ and of $U_{\bullet }$ are identified via p, and j is a good compactification [Reference Deligne10, 8.3.2]. Then the Deligne-Beilinson cohomology of $X_{\bullet }$ is defined to be the analytic hypercohomology

$$ \begin{align*}\mathrm{H}^q_{\mathrm{DB}}(X_{\bullet},A(p))= \mathrm{H}^q (\overline{U}_{\bullet, \mathrm{an}}, A(p)_{\mathrm{DB}, (\overline{U}_{\bullet}, U_{\bullet})}),\end{align*} $$

where $A(p)_{\mathrm {DB}, (\overline {U}_{\bullet }, U_{\bullet })}$ is the Deligne-Beilinson complex of $(\overline {U}_{\bullet }, U_{\bullet })$ .

Many properties still hold in such a general case (see [Reference Esnault and Viehweg13, 5.4]). Among them, we state a long exact sequence involving the DB cohomology and Betti cohomology of $X_{\bullet }$ , which is the most important for our purpose. Recall [Reference Deligne10, Definition 8.3.4] that the mixed Hodge structure on $\mathrm {H}^{q}(X_{\bullet }, \mathbb {Z})$ is defined by transport from that on $\mathrm {H}^{q}(U_{\bullet }, \mathbb {Z})$ via the isomorphism $p^*$ (for any choice of diagram (4.2)). We denote by $\{F^p\mathrm {H}^q(X_{\bullet },\mathbb {C})\}$ the Hodge filtration on $\mathrm {H}^q(X_{\bullet },\mathbb {C})$ . Then we have

Proposition 4.3 ([Reference Esnault and Viehweg13, 2.10(b) and 5.4])

Let $X_{\bullet }$ be a (separated) simplicial scheme; then there exists a long exact sequence

(4.3) $$ \begin{align} \cdots \to \mathrm{H}^q_{\mathrm{DB}}(X_{\bullet}, A(p)) \to \mathrm{H}^q(X_{\bullet}, A(p)) \to \mathrm{H}^q(X_{\bullet}, \mathbb{C})/F^p\mathrm{H}^q(X_{\bullet}, \mathbb{C}) \to \cdots \end{align} $$

and this sequence is contravariantly functorial for any morphism $X^{\prime }_{\bullet }\to X_{\bullet }$ .

4.4 Cohomology on Deligne-Mumford stacks

Let $\mathscr {X}$ be a Deligne-Mumford stack of finite type over $\mathbb {C}$ . We denote by $\mathrm {\acute {E}t}(\mathscr {X})$ the étale site of $\mathscr {X}$ , whose objects are étale morphisms $(U\to \mathscr {X})$ ; the morphisms from $(U'\to \mathscr {X})$ to $(U\to \mathscr {X})$ are morphisms of schemes $(U'\to U)$ such that the corresponding diagram with base $\mathscr {X}$ is (2-)commutative, and the coverings of $(U\to \mathscr {X})$ are those $((U_i\to \mathscr {X})\to (U\to \mathscr {X}))_i$ for which $(U_i\to U)_i$ is an étale covering of U. The resulting topos is denoted by $\mathscr {X}_{\mathrm {\acute {e}t}}$ , and the sheaves are called étale sheaves on $\mathscr {X}$ .

Let $\mathcal {F}$ be an étale sheaf on $\mathscr {X}$ . Then one can define the localised topos $\mathscr {X}_{\mathrm {\acute {e}t}}/{\mathcal {F}}$ , which is simply the category $\mathscr {X}_{\mathrm {\acute {e}t}}$ over $\mathcal {F}$ . In the case where $\mathcal {F}$ is given by an étale morphism $U\to \mathscr {X}$ , it is also denoted by $\mathscr {X}_{\mathrm {\acute {e}t}}/U$ , and the fact that it is indeed a topos follows from the following description: let $\mathrm { \acute {E}t}(\mathscr {X})/U$ be the localised site; then the topos associated to $\mathrm {\acute {E}t}(\mathscr {X})/U$ is equivalent to $\mathscr {X}_{\mathrm { \acute {e}t}}/U$ . Moreover, it is not hard to see that $\mathscr {X}_{\mathrm {\acute {e}t}}/U$ is equivalent to $U_{\mathrm {\acute {e}t}}$ , the étale topos of U.

We now review the method to ‘compute’ the cohomology using an atlas. Let $\pi :X\to \mathscr {X}$ be an étale atlas of $\mathscr {X}$ : that is, X is a scheme and $\pi $ is an étale surjective morphism. To any such $\pi $ , we can associate its 0-coskeleton

$$ \begin{align*}X_{\bullet}:=\mathrm{cosk}_0(\pi),\end{align*} $$

which is a simplicial scheme, and an augmentation $e:X_{\bullet }\to \mathscr {X}$ ([Reference Deligne10, 5.1.4], [Reference Laumon and Moret-Bailly29, 13.5]). Note that replacing X by $\amalg _i U_i$ , where $(U_i)_i$ is a Zariski open covering of X consisting of affine schemes (and finite in number), we may always assume that X is separated, and $X_{\bullet }$ is then a separated simplicial scheme.

Since $X_{\bullet }$ is a simplicial étale sheaf on $\mathscr {X}$ , there is a localised topos $\mathscr {X}_{\mathrm {\acute {e}t}}/X_{\bullet }$ and a morphism of topoi

$$ \begin{align*}e_{\mathrm{\acute{e}t}}:\mathscr{X}_{\mathrm{\acute{e}t}}/X_{\bullet}\to \mathscr{X}_{\mathrm{\acute{e}t}}\end{align*} $$

(compare [Reference Olsson40, §2.4.5 and §2.4.11]). It follows from definition that the localised topos $\mathscr {X}_{\mathrm {\acute {e}t}}/X_{\bullet }$ is indeed equivalent to the étale topos $X_{\bullet , \mathrm { \acute {e}t}}$ of $X_{\bullet }$ defined by Deligne (compare [Reference Laumon and Moret-Bailly29, 12.4]). Recall that sheaves in $X_{\bullet , \mathrm {\acute {e}t}}$ are those

$$ \begin{align*}((\mathcal{F}_n)_{n\geq 0}, (\theta_{\delta})_{\delta:[n']\to [n]}),\end{align*} $$

where for each n, $\mathcal {F}_n$ is an étale sheaf on $X_n$ , and for each $\delta :[n']\to [n]$ , $\theta _{\delta }: \mathcal {F}_{n'}\to \delta _*\mathcal {F}_n$ is a morphism of étale sheaves on $X_{n'}$ (here we also denote by $\delta $ the morphism $X_n\to X_{n'}$ associated to $\delta :[n']\to [n]$ ), satisfying suitable compatibility conditions on composition of transition maps.

Since $\pi $ is étale, surjective and of finite type, and hence a covering in the canonical topology, $e_{\mathrm {\acute {e}t}}:X_{\bullet , \mathrm {\acute {e}t}} \to \mathscr {X}_{\mathrm {\acute {e}t}}$ is of cohomological descent (compare [Reference Olsson40, 2.4.16]): that is, let $D^+(\mathscr {X}_{\mathrm {\acute {e}t}}, \mathbb {Z})$ be the derived category of bounded below complexes of $\mathbb {Z}$ -modules in $\mathscr {X}_{\mathrm {\acute {e}t}}$ . Then for any $\mathcal {F}\in D^+(\mathscr {X}_{\mathrm {\acute {e}t}}, \mathbb {Z})$ , the morphism

$$ \begin{align*} \mathcal{F} \to Re_{\mathrm{\acute{e}t},*}e^{-1}_{\mathrm{\acute{e}t}} \mathcal{F} \end{align*} $$

in $D^+(\mathscr {X}_{\mathrm {\acute {e}t}}, \mathbb {Z})$ is an isomorphism. Note that $e^{-1}_{\mathrm {\acute {e}t}}(\mathcal {F})$ has the following concrete description: it is the complex of sheaves in $X_{\bullet , \mathrm {\acute {e}t}}$ , where the degree n component is simply the restriction of $\mathcal {F}$ to $X_{n, \mathrm {\acute {e}t}}$ , and the transition maps are those induced by $X_{\bullet }$ . Under these notations, we therefore have an isomorphism

(4.4) $$ \begin{align} \mathrm{H}^q(\mathscr{X}_{\mathrm{\acute{e}t}}, \mathcal{F})\to \mathrm{H}^q(X_{\bullet, \mathrm{\acute{e}t}}, e^{-1}_{\mathrm{\acute{e}t}}(\mathcal{F})). \end{align} $$

There is a similar picture on the analytic side. For an étale atlas $\pi :X\to \mathscr {X}$ , the analytification $\pi ^{\mathrm {an}}:X^{\mathrm {an}}\to \mathscr {X}^{\mathrm {an}}$ is also a covering in the canonical topology (of the analytic topos $\mathscr {X}_{\mathrm {an}}$ ). Therefore for $\mathcal {F}\in D^+(\mathscr {X}_{\mathrm {an}}, \mathbb {Z})$ , we have also an isomorphism

(4.5) $$ \begin{align} \mathrm{H}^q(\mathscr{X}_{\mathrm{an}}, \mathcal{F})\to \mathrm{H}^q(X_{\bullet, \mathrm{an}}, e^{-1}_{\mathrm{an}}(\mathcal{F})). \end{align} $$

We now give the construction of mixed Hodge structures on the cohomology groups of $\mathscr {X}$ using an atlas.

Proof. Let $X_3$ be the (2-)fibre product $X_1\times _{\mathscr {X}}X_2$ . We have the following pull-back diagram

where $\widetilde {\pi _1}$ and $\widetilde {\pi _2}$ are étale surjective. In the associated commutative diagram

$\widetilde {e_1}$ and $\widetilde {e_2}$ (the morphisms between $0$ -coskeletons induced by $\widetilde {\pi _1}$ and $\widetilde {\pi _2}$ , respectively) are then étale hypercoverings, hence of cohomological descent. Therefore, in the following commutative diagram of cohomology groups

(4.6)

$(\widetilde {e_1})^*$ and $(\widetilde {e_2})^*$ are (morphisms of mixed Hodge structures and isomorphisms of the underlying $\mathbb {Z}$ -modules, hence) isomorphisms of mixed Hodge structures; so for $i=1,2$ , the transports of mixed Hodge structures from those of $\mathrm {H}^{q}((X_i)_{\bullet }, \mathbb {Z})$ via $e_i^*$ to $\mathrm {H}^{q}(\mathscr {X}, \mathbb {Z})$ are both equal to the transport from that of $\mathrm {H}^{q}((X_3)_{\bullet }, \mathbb {Z})$ .

From this, a mixed Hodge structure on $\mathrm {H}^{q}(\mathscr {X}, \mathbb {Z})$ is defined as the transport from that of $\mathrm {H}^{q}(X_{\bullet }, \mathbb {Z})$ via $e^*$ using any étale atlas. This gives in particular the Hodge filtration $\{F^p\mathrm {H}^{q}(\mathscr {X}, \mathbb {C})\}$ of $\mathrm {H}^{q}(\mathscr {X}, \mathbb {Z})$ , and (4.6) becomes a commutative diagram of isomorphisms of mixed Hodge structures.

4.8 DB cohomology of Deligne-Mumford stacks: via atlases

As before, we let $\mathscr {X}$ be a Deligne-Mumford stack of finite type over $\mathbb {C}$ . The first result is

Lemma 4.9. With the same notations as in Lemma 4.7, there is an isomorphism

(4.7) $$ \begin{align} \mathrm{H}^q_{\mathrm{DB}}((X_1)_{\bullet}, A(p))\cong \mathrm{H}^q_{\mathrm{DB}}((X_2)_{\bullet}, A(p)) \end{align} $$

for any q.

Proof. The proof is similar to that of [Reference Esnault and Viehweg13, 5.2]. Keep notations the same as in the proof of Lemma 4.7. The morphism $\widetilde {e_2}$ induces a morphism of two long exact sequences (4.3):

Since $(\widetilde {e_2})^*$ is an isomorphism of mixed Hodge structures, $\overline {(\widetilde {e_2})^*}$ is an isomorphism, and hence $(\widetilde {e_2})^*_{\mathrm {DB}}$ is also an isomorphism. Similarly, the induced map

$$ \begin{align*}(\widetilde{e_1})^*_{\mathrm{DB}}: \mathrm{H}^q_{\mathrm{DB}}((X_2)_{\bullet}, A(p))\to \mathrm{H}^q_{\mathrm{DB}}((X_3)_{\bullet}, A(p))\end{align*} $$

is an isomorphism as well. We conclude that $((\widetilde {e_1})^*_{\mathrm {DB}})^{-1} \circ (\widetilde {e_2})^*_{\mathrm {DB}}$ gives the desired isomorphism.

Remark 4.10. Suppose that there is a morphism $f:X_1\to X_2$ such that the following diagram is commutative:

Let $f_{\bullet }:(X_1)_{\bullet }\to (X_2)_{\bullet }$ be the morphism induced by f. Then the same argument shows that

$$ \begin{align*}((f_{\bullet})^*_{\mathrm{DB}})^{-1}: \mathrm{H}^q_{\mathrm{DB}}((X_1)_{\bullet}, A(p))\to \mathrm{H}^q_{\mathrm{DB}}((X_2)_{\bullet}, A(p))\end{align*} $$

is an isomorphism, denoted by $\eta _{f}$ .

Definition 4.11. Let $\mathscr {X}$ be a Deligne-Mumford stack of finite type over $\mathbb {C}$ . Fix any étale atlas $\pi :X\to \mathscr {X}$ (with X separated); we define the Deligne-Beilinson cohomology of $\mathscr {X}$ to be

(4.8) $$ \begin{align} \mathrm{H}^q_{\mathrm{DB}}(\mathscr{X}, A(p)):=\mathrm{H}^q_{\mathrm{DB}}(X_{\bullet}, A(p)), \end{align} $$

where $X_{\bullet }$ is the $0$ -coskeleton of $\pi $ .

Moreover, the construction above is functorial. Let us fix for any $\mathscr {X}$ an arbitrary étale atlas $\pi :X\to \mathscr {X}$ (with X separated). Let $f:\mathscr {X}'\to \mathscr {X}$ be a morphism. Take the étale atlas $\pi ^{\prime }:X^{\prime }\to \mathscr {X}'$ of $\mathscr {X}'$ we have fixed, and let $\widetilde {\pi }: X^{\prime }_{(X)}\to X^{\prime }$ be the pull-back of $\pi $ along $f\circ \pi ^{\prime }$ , which fits into the following commutative diagram:

Then we have

Proposition 4.12. Set

(4.9) $$ \begin{align} f^*_{\mathrm{DB}}:=\eta_{\widetilde{\pi}} \circ (\widetilde{f_{\bullet}})^*_{\mathrm{DB}}: \mathrm{H}^q_{\mathrm{DB}}(\mathscr{X}, A(p))\to \mathrm{H}^q_{\mathrm{DB}}(\mathscr{X}', A(p)), \end{align} $$

where $\eta _{\widetilde {\pi }}$ is defined in Remark 4.10 and $\widetilde {f_{\bullet }}$ is the morphism of simplicial schemes induced by $\widetilde {f}$ . Then the assignment $\mathscr {X}\to \mathrm {H}^q_{\mathrm {DB}}(\mathscr {X}, A(p))$ , $f\to f^*_{\mathrm {DB}}$ defined above is contravariantly functorial for any morphism $\mathscr {X}'\to \mathscr {X}$ .

Proof. Suppose that we are also given $f': \mathscr {X}"\to \mathscr {X}'$ . Let $\pi ":X"\to \mathscr {X}"$ be the étale atlas of $\mathscr {X}"$ we have fixed. Consider the following diagram

where all squares are pull-backs. Denote the arrow $X^{\prime \prime }_{(X)}\times _{X^{\prime \prime }}X^{\prime \prime }_{(X^{\prime })}\to X^{\prime \prime }$ by g and the arrow $X^{\prime \prime }_{(X)}\times _{X^{\prime \prime }}X^{\prime \prime }_{(X^{\prime })}\to X$ by h. It follows that $(f\circ f^{\prime })^*_{\mathrm {DB}}$ and $(f^{\prime })^*_{\mathrm {DB}}\circ (f)^*_{\mathrm {DB}}$ are both naturally isomorphic to $\eta _{g}\circ (h_{\bullet })^*_{\mathrm {DB}}$ .

Proposition 4.13. Let $\mathscr {X}$ be a Deligne-Mumford stack of finite type over $\mathbb {C}$ . Then there is an exact sequence

(4.10)

Proof. Take $\pi : X\to \mathscr {X}$ as in Definition 4.11; then the assertion follows from the construction of $\mathrm {H}^q_{\mathrm {DB}}(\mathscr {X}, A(p))$ and the exact sequence in equation (4.3) for the 0-coskeleton $X_{\bullet }$

4.14 DB cohomology for smooth Deligne-Mumford stacks: via DB complexes

In this subsection, we give an alternative definition of DB cohomology for smooth Deligne-Mumford stacks, which will be used to define the DB-cycle class maps.

Let us assume now that $\mathscr {X}$ is smooth. In particular, for every étale morphism $U\to \mathscr {X}$ , U must be smooth. Consequently, if $X\to \mathscr {X}$ is an étale atlas with X separated (and smooth), its 0-coskeleton is a smooth separated simplicial scheme.

Let $Sm/\mathbb {C}$ be the category of smooth separated schemes. In [Reference Beĭlinson5], Beilinson showed that there is a complex of (big) sheaves $\mathbb {Q}(p)_{\mathrm {DB} , \mathrm {Zar}}$ on $(Sm/\mathbb {C})_{\mathrm {Zar}}$ , which computes DB cohomology. In particular, for all smooth separated simplicial scheme $X_{\bullet }$ , we have an isomorphism

$$ \begin{align*} \mathrm{H}^q_{\mathrm{DB}}(X_{\bullet}, \mathbb{Q}(p)) \cong \mathrm{H}^q(X_{\bullet, \mathrm{Zar}}, \mathbb{Q}(p)_{\mathrm{DB}, \mathrm{Zar}}|_{X_{\bullet, \mathrm{ Zar}}}) \end{align*} $$

(see also [Reference Esnault and Viehweg13, §5.5]).

Let $\mathbb {Q}(p)_{\mathrm {DB}, \mathrm {\acute {e}t}}$ be the sheafification of $\mathbb {Q}(p)_{\mathrm {DB} , Zar}$ in the étale topology. It has been proved that the complex of étale sheaves $\mathbb {Q}(p)_{\mathrm {DB}, \mathrm {\acute {e}t}}$ also computes the DB cohomology. In particular, for all smooth separated simplicial schemes $X_{\bullet }$ , we have

(4.11) $$ \begin{align} \mathrm{H}^q_{\mathrm{DB}}(X_{\bullet}, \mathbb{Q}(p)) \cong \mathrm{H}^q(X_{\bullet, \mathrm{\acute{e}t}}, \mathbb{Q}(p)_{\mathrm{DB}, \mathrm{ \acute{e}t}}|_{X_{\bullet, \mathrm{\acute{e}t}}}) \end{align} $$

(compare [Reference Kohrita26, Thm. 2.5 (i)]). This holds even in $\mathbb {Z}$ -coefficients.

Now since $\mathscr {X}$ is smooth, we can restrict $\mathbb {Q}(p)_{\mathrm {DB}, \mathrm {\acute {e}t}}$ to $\mathscr {X}_{\mathrm {\acute {e}t}}$ (see Remark 4.5, and note that the objects of $\mathrm {\acute {E}}\mathrm {t}^{\mathrm {sep}}(\mathscr {X})$ are all smooth).

Proposition 4.15. Let $\mathscr {X}$ be a smooth Deligne-Mumford stack. Then there is an isomorphism

(4.12) $$ \begin{align} \mathrm{H}^q_{\mathrm{DB}}(\mathscr{X}, \mathbb{Q}(p)) \cong \mathrm{H}^q(\mathscr{X}_{\mathrm{\acute{e}t}}, \mathbb{Q}(p)_{\mathrm{DB}, \mathrm{ \acute{e}t}}|_{\mathscr{X}_{\mathrm{\acute{e}t}}}). \end{align} $$

4.16 DB-cycle class maps for smooth Deligne-Mumford stacks

Now we proceed to define the DB-cycle class map from the Chow group of a smooth stack to the DB cohomology group.

Let $\mathscr {X}$ be a smooth Deligne-Mumford stack. For each integer p, let $\mathrm {CH}^p(\mathscr {X})_{\mathbb {Q}}$ be the pth rational Chow group of $\mathscr {X}$ defined in [Reference Gillet18, Reference Vistoli42]. Let $\mathbb {Q}(p)$ denote the motivic complex of weight p with rational coefficients (see [Reference Mazza, Voevodsky and Weibel33, Definition 3.1]), which is a complex of sheaves on $(Sm/\mathbb {C})_{\mathrm {\acute {e}t}}$ . For a smooth separated scheme X, $\mathbb {Q}(p)$ computes higher Chow groups of X (see [Reference Mazza, Voevodsky and Weibel33, Theorem 19.1]). In particular, both the $2p$ th Zariski and étale hypercohomology on X of $\mathbb {Q}(p)$ can be identified with $\mathrm {CH}^p(X)_{\mathbb {Q}}$ (see [Reference Mazza, Voevodsky and Weibel33, Corollary 19.2 and Theorem 14.24]). For stacks, we also have the following:

Proposition 4.17 (compare [Reference Joshua23, Theorem 2 (i)])

There is a canonical isomorphism

$$ \begin{align*}\mathrm{CH}^p(\mathscr{X})_{\mathbb{Q}}\cong \mathrm{H}^{2p}(\mathscr{X}_{\mathrm{\acute{e}t}}, \mathbb{Q}(p)|_{\mathscr{X}_{\mathrm{\acute{e}t}}}).\end{align*} $$

According to [Reference Kohrita26, Theorem 5.2 (i)] (and see also [Reference Mazza, Voevodsky and Weibel33, Theorem 16.7]), there is a map

$$ \begin{align*}\mathbb{Q}(p)\to \mathbb{Q}(p)_{\mathrm{DB}, \mathrm{Zar}},\end{align*} $$

such that the induced map of the $2p$ th Zariski hypercohomologies, restricting to $X_{\mathrm {Zar}}$ for any smooth separated scheme X, can be identified with the classical DB-cycle class map

$$ \begin{align*}\mathrm{cl}_{\mathrm{DB}}:\mathrm{CH}^p(X)_{\mathbb{Q}}\to \mathrm{H}^{2p}_{\mathrm{DB}}(X,\mathbb{Q}(p)).\end{align*} $$

Sheafifying the above map in the étale topology (note that $\mathbb {Q}(p)$ is a complex of sheaves in the étale topology), we get a map

(4.13) $$ \begin{align} \mathbb{Q}(p)\to \mathbb{Q}(p)_{\mathrm{DB}, \mathrm{\acute{e}t}}. \end{align} $$

This yields

Definition 4.18. Let $\mathscr {X}$ be a smooth Deligne-Mumford stack of finite type over $\mathbb {C}$ . The DB-cycle class map for $\mathscr {X}$ with $\mathbb {Q}$ -coefficients

$$ \begin{align*}\mathrm{cl}_{\mathrm{DB}}:\mathrm{CH}^p(\mathscr{X})_{\mathbb{Q}}\to \mathrm{H}^{2p}_{\mathrm{DB}}(\mathscr{X},\mathbb{Q}(p))\end{align*} $$

is defined by taking the $2p$ th étale hypercohomology of the map (4.13) restricting to $\mathscr {X}_{\mathrm {\acute {e}t}}$ .

We have

5.1 Oriented K3 surfaces

Following [Reference Gritsenko, Hulek and Sankaran20], we first introduce the spin structures on a K3 surface.

Definition 5.2. A spin structure on a quasi-polarised K3 surface $(S, L)$ of genus g is a choice $\omega $ of the two generators of

$$ \begin{align*}\mathrm{ Hom}_{\mathbb{Z}}(\det(\mathrm{H}^2_{\mathrm{prim}}(S, \mathbb{Z})), \det \Lambda_g).\end{align*} $$

The triple $(S, L, \omega )$ is called an oriented quasi-polarised K3 surface of genus g. A morphism between two oriented quasi-polarised K3 surfaces $(S, L, \omega )$ and $(S', L', \omega ')$ is a morphism $f:S'\to S$ satisfying $f^*(L)=L'$ and $\omega '=\omega \circ \det (f^*)$ .

Let $\widetilde {\mathscr {F}_g}$ be the moduli stack of quasi-polarised oriented K3 surfaces of genus g. The natural forgetful functor

$$ \begin{align*}u:\widetilde{\mathscr{F}_g}\to \mathscr{F}_{g}\end{align*} $$

is étale of degree $2$ . It follows that $\widetilde {\mathscr {F}}_g$ is a smooth Deligne-Mumford stack. Let $\widetilde {\mathcal {F}_g}$ be the coarse moduli space of $\widetilde {\mathscr {F}}_g$ . Then it is isomorphic to a Shimura variety of orthogonal type.

The stack $\widetilde {\mathscr {F}}_{g}$ remains nonseparated, but it behaves slightly better than $\mathscr {F}_g$ in the sense that it admits a larger separated locus.

5.4 Separated locus of $\widetilde {\mathscr {F}}_g$

Let us consider the open substack $\widetilde {\mathscr {F}}_g^s$ of oriented quasi-polarised K3 surfaces of genus g containing at most one exceptional $(-2)$ -curve. A key observation is

Proof. Let R be a discrete valuation ring over $\mathbb {C}$ and K its fraction field. To prove the separatedness, it suffices to show that for any two families $(\mathcal {X}, \mathcal {L}, \omega ) \to \mathrm {Spec~} R$ and $(\mathcal {X}',\mathcal {L}',\omega ')\to \mathrm {Spec~} R$ of quasi-polarised oriented K3 surfaces of genus g with at most one exceptional $(-2)$ -curve, if there is an isomorphism

(5.1) $$ \begin{align} \phi_K: (\mathcal{X}_K, \mathcal{L}_K, \omega_K)\cong (\mathcal{X}'_K,\mathcal{L}'_K, \omega'_K) \end{align} $$

of quasi-polarised oriented K3 surfaces over the generic point, then $\phi _K$ can be uniquely extended to an isomorphism over R. There are three possibilities:

(1) If $\mathcal {L}$ is relative ample, one can apply [Reference Matsusaka and Mumford31, Theorem 1] to the pairs $(\mathcal {X},\mathcal {L})$ and $(\mathcal {X},\mathcal {L}')$ . It is not hard to see that such extension must preserve the orientation.

(2) If $\mathcal {L}_K$ is not ample, then by our assumption, there exist unique families of exceptional $(-2)$ -curves on $\mathcal {X}\to \mathrm {Spec~} R$ and $\mathcal {X}'\to \mathrm {Spec~} R$ , denoted by $\mathcal {C}\to \mathrm {Spec~} R$ and $\mathcal {C}'\to \mathrm {Spec~} R$ , respectively. One can apply [Reference Matsusaka and Mumford31, Theorem 1] to the pairs $(\mathcal {X},\mathcal {L}(-\mathcal {C}))$ or $(\mathcal {X}', \mathcal {L}'(-\mathcal {C}'))$ to obtain the extension.

(3) If $\mathcal {L}_K$ is ample, but the restriction $\mathcal {L}_0=\mathcal {L}|_{\mathcal {X}_0}$ is not ample, then $\mathcal {X}_0$ contains an exceptional $(-2)$ -curve C. If the isomorphism of the pairs $(\mathcal {X}_K,\mathcal {L}_K)\cong (\mathcal {X}^{\prime }_K,\mathcal {L}_K^{\prime })$ cannot be extended to $\mathrm {Spec~} R$ , then the birational map $(\mathcal {X},\mathcal {L})\dashrightarrow (\mathcal {X}', \mathcal {L}') $ is an elementary transform associated to C (compare [Reference Burns and Rapoport7, Paragraph 7, Theorem 2]). However, the monodromy group of a single elementary transform acts on $\mathrm {H}^2(\mathcal {X}_\eta ,\mathbb {Z})$ as a reflection, which does not preserve the orientation of the generic fibre. This contradicts our assumption.

Let $\widetilde {\pi }: \widetilde {\mathscr {S}}_g^s\to \widetilde {\mathscr {F}}_g^s$ be the universal family. Then $\widetilde {\mathscr {S}}_g^s$ is also a separated Deligne-Mumford stack (non quasi-projective), and we denote by $\mathrm {CH}^2(\widetilde {\mathscr {S}}_g^s)$ the rational Chow group. We can state a stacky version of O’Grady’s generalised Franchetta conjecture (GFC) for universal oriented K3 surfaces.

Certainly, Conjecture 5.6 will automatically imply Conjecture 1.1 via applying the push-forward map. The cohomological version of Conjecture 5.6 has also been confirmed in [Reference Bergeron and Li4].

6.1 Vanishing of odd cohomology groups

Based on the vanishing result in Theorem 2.3.1, let us first give a sufficient condition for the vanishing of $\mathrm {H}^3(\widetilde {\mathscr {S}}_g^s,\mathbb {C})$ .

Proof. Write $Y=\widetilde {\mathcal {F}}_g-\widetilde {\mathcal {F}}^s_g=\mathcal {H}^g_{A_{1,1}}\cup \mathcal {H}^g_{A_2}$ . We first compute $\mathrm {H}^3(\widetilde {\mathcal {F}}_g^s,\mathbb {C})$ . Note that there is a topological commutative diagram of exact sequences

(6.1)

Here, the first row is the classical sequence of cohomology (and with supports in Y) associated to the pair $(\widetilde {\mathcal {F}}_g,Y)$ , the second row is the standard exact sequence of Borel-Moore homology of the complement $\widetilde {\mathcal {F}}_g^s=\widetilde {\mathcal {F}}_g-Y$ , and the vertical arrows are given by cap products with the fundamental classes.

Note that $\widetilde {\mathcal {F}}_g$ and $\widetilde {\mathcal {F}}_g^s$ are finite quotient of smooth quasi-projective varieties; the cap products $\cap ~[\widetilde {\mathcal {F}}_g]$ and $\cap ~[\widetilde {\mathcal {F}}_g^s]$ are isomorphisms on all degrees. Hence the map

$$ \begin{align*}\cap~[Y]: \mathrm{H}^4_{Y}(\widetilde{\mathcal{F}}_g,\mathbb{C})\to \mathrm{H}_{34}(Y,\mathbb{C})\end{align*} $$

is also an isomorphism. According to Proposition 5.3 and Theorem 2.3.1, we have $\mathrm {H}^3(\widetilde {\mathcal {F}}_g,\mathbb {C})=0$ . Since the space $\mathrm {H}^4_Y(\widetilde {\mathcal {F}}_g,\mathbb {C})$ is spanned by the fundamental classes of irreducible components of Y, it follows from our assumption that $\rho $ is injective. This implies $\mathrm {H}^3(\widetilde {\mathcal {F}}_g^s,\mathbb {C})=0$ . A similar argument shows that $\mathrm {H}^1(\widetilde {\mathcal {F}}_g, \underline {\Lambda _g}|_{\widetilde {\mathcal {F}}_g^s})=0$ .

We will see later that the conclusion of the above lemma leads quickly to our main result. The rest of this section is therefore devoted to the proof of Proposition 6.9.

6.3 Families of K3 surfaces in $\mathcal {F}_g$

Regard $\mathcal {F}_g$ as the coarse moduli space of polarised K3 surfaces with at worst isolated ADE singularities (compare [Reference Huybrechts21, Remark 6.4.5]). Let us construct two-dimensional proper families of polarised K3 surfaces in $\mathcal {F}_g$ for all g.

Hyperelliptic family for $g>1$ odd

Let $\Sigma _1=\mathbb {P}^2$ . Let us take $\mathcal {C}_1\subseteq \Sigma _1 \times \mathbb {P}^1\times \mathbb {P}^1$ a general member in $| \mathcal {O}_{\mathbb {P}^2}(2)\boxtimes \mathcal {O}_{\mathbb {P}^1\times \mathbb {P}^1}(-2K_{\mathbb {P}^1\times \mathbb {P}^1})|$ . Each fibre $\mathcal {C}_{1,t}$ of the projection $\mathcal {C}_1\to \Sigma _1$ over a point $t\in \Sigma _1$ is a curve in $|\mathcal {O}_{\mathbb {P}^1\times \mathbb {P}^1}(-2K_{\mathbb {P}^1\times \mathbb {P}^1})|$ with at worst a cusp or two nodes. Consider the double covering

$$ \begin{align*}\mathcal{X}_1\to \Sigma_1 \times \mathbb{P}^1\times \mathbb{P}^1\end{align*} $$

branched over $\mathcal {C}_1$ , equipped with natural projections to the three factors. The first projection gives a family $\phi _1:\mathcal {X}_1\to \Sigma _1$ of surfaces over $\Sigma _1$ . For any $t\in \Sigma _1$ , the fibre $\mathcal {X}_{1,t}=\phi _1^{-1}(t)$ is a double covering of $\mathbb {P}^1\times \mathbb {P}^1$ branched over the curve $\mathcal {C}_{1,t}$ . It is a hyperelliptic K3 surface and is smooth (respectively, with $A_1$ or $A_2$ singularity) if and only if the branching curve $\mathcal {C}_{1,t}$ is smooth (respectively, nodal or cuspidal). Moreover, the family $\mathcal {X}_1\to \Sigma _1$ has a relative polarisation

$$ \begin{align*}\mathcal{L}_1=\phi_1^\ast\mathcal{O}_{\mathbb{P}^2}(1)+\pi_2^\ast \mathcal{O}_{\mathbb{P}^1}(1)+ \frac{g-1}{2}\pi_3^\ast \mathcal{O}_{\mathbb{P}^1}(1),\end{align*} $$

of odd genus g, which induces a map $\Sigma _1\to \mathcal {F}_g$ factoring through the natural map $\Sigma _1\to |-2K_{\mathbb {P}^1\times \mathbb {P}^1}|$ . Conversely, a hyperelliptic K3 surface in $\mathcal {F}_g$ ( $g>1$ is odd) with at worst ADE singularities arises as a double covering of $\mathbb {P}^1\times \mathbb {P}^1$ branching over an integral curve in $|-2K_{\mathbb {P}^1\times \mathbb {P}^1}|$ .

Hyperelliptic family for $g>2$ even

Let $\Sigma _2=\mathbb {P}^2$ . Consider the Hirzebruch surface $\mathbf {F}_1=\mathbb {P}(\mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(-1))$ ; we denote by s the section of $\mathbf {F}_1\to \mathbb {P}^1$ and h the fibre class. Take $\mathcal {C}_2\subseteq \mathbb {P}^2\times \mathbf {F}_1$ , a general member in $|\mathcal {O}_{\mathbb {P}^2}(2)\boxtimes \mathcal {O}_{\mathbb {F}_1}(-2K_{\mathbf {F}_1})|$ , and let $\mathcal {X}_2\to \Sigma _2 \times \mathbf {F}_1$ be the double covering branched over $\mathcal {C}_2$ . The projection

$$ \begin{align*}\phi_2:\mathcal{X}_2\to \Sigma_2\end{align*} $$

to the first factor gives a family of hyperelliptic K3 surfaces. The fibre $\mathcal {X}_{2,t}=\phi _2^{-1}(t)$ is a double covering of $\mathbf {F}_1$ branched over the curve $\mathcal {C}_{2,t}$ . Similarly as before, this is a family of hyperelliptic K3 surfaces that carries a relative polarisation

$$ \begin{align*}\mathcal{L}_2=(\pi_1')^\ast \mathcal{O}_{\mathbb{P}^2}(1)+ (\pi_2')^\ast \mathcal{O}_{\mathbf{F}_1}(s+\frac{g}{2}h)\end{align*} $$

of even genus g. Similarly, we have a moduli map $\Sigma _2\to \mathcal {F}_g$ factoring through $\Sigma _2\to |-2K_{\mathbf {F}_1}|$ for a generic choice of $\mathcal {C}_2$ .

Unigonal family

The construction of the family of unigonal K3 surfaces needs more attention. To get a two-dimensional proper family of unigonal K3 surfaces, we follow the construction in [Reference Laza and O’Grady30, §4.2.2] to proceed as follows:

Let $\Sigma _3=(\mathbb {P}^2)^*$ be the dual projective plane, $\Phi =\{([L], p)\in (\mathbb {P}^2)^*\times \mathbb {P}^2|~p\in L\}$ the universal line, and $\pi _1: \Phi \to (\mathbb {P}^2)^*$ and $\pi _2: \Phi \to \mathbb {P}^2$ the projections.

Consider the fourfold

$$ \begin{align*}\mathcal{Y}=\mathbb{P}_{\Phi}(\mathcal{O}_{\Phi}\oplus \mathcal{O}_{\Phi}(-4)),\end{align*} $$

where $\mathcal {O}_{\Phi }(1)$ is the pull-back of $\mathcal {O}_{\mathbb {P}^2}(1)$ along $\pi _2: \Phi \to \mathbb {P}^2$ . Let $\mathcal {A}=\mathbb {P}_{\Phi }(\mathcal {O}_{\Phi }(-4))$ and $\mathcal {F}$ the pull-back of a line in $\mathbb {P}^2$ along $\mathcal {Y}\to \Phi \xrightarrow {\pi _2} \mathbb {P}^2$ . Take a general member

$$ \begin{align*}\mathcal{B}\in |3\mathcal{A}+12\mathcal{F}|\end{align*} $$

such that it is smooth and disjoint from $\mathcal {A}$ , and form the double covering $\mathcal {X}_3\to \mathcal {Y}$ branched over $(\mathcal {A}+\mathcal {B})$ . Then the composite

$$ \begin{align*}\mathcal{X}_3\to \mathcal{Y} \to \Phi \xrightarrow{\pi_1} (\mathbb{P}^2)^*\end{align*} $$

gives a family of unigonal K3 surfaces, denoted by $\phi _3: \mathcal {X}_3\to \Sigma _3$ .

6.4 The NL-number computation

For the three families $\phi _i$ ( $i=1, 2, 3$ ) constructed above, we are going to compute the intersection numbers

(6.2) $$ \begin{align} \Sigma_i\cdot \mathcal{H}_{A_{1,1}}^g~\mathrm{and}~\Sigma_i\cdot \mathcal{H}_{A_2}^g \end{align} $$

in $\mathcal {F}_g$ .

Hyperelliptic family

We first consider the hyperelliptic families. By taking $\mathcal {C}_1$ and $\mathcal {C}_2$ generically, we can further assume that all the fibres of $\mathcal {C}_i\to \Sigma _i$ have only trivial automorphisms. Then we have embeddings

$$ \begin{align*}\psi_1:\Sigma_1\to |-2K_{\mathbb{P}^1\times \mathbb{P}^1}|~\mathrm{and}~ \psi_2:\Sigma_2\to |-2K_{\mathbf{F}_1}|.\end{align*} $$

Let us first consider the case $i=1$ , and similar arguments will apply to the case $i=2$ . Denote by

$$ \begin{align*} \mathcal{A}_{1,1}\subseteq |-2K_{\mathbb{P}^1\times \mathbb{P}^1}|\ \mathrm{and}\ \mathcal{A}_2\subseteq |-2K_{\mathbb{P}^1\times \mathbb{P}^1}| \end{align*} $$

the Zariski closure of the binodal locus and, respectively, the cuspidal locus in $| -2K_{\mathbb {P}^1\times \mathbb {P}^1}|\cong \mathbb {P}^{24}$ that parametrise curves in $|-2K_{\mathbb {P}^1\times \mathbb {P}^1}|$ with at worst two nodes (respectively, a cusp). It is known that $\mathcal {A}_{1,1}$ and $\mathcal {A}_{2}$ are both of codimension two. Then we have

(6.3) $$ \begin{align} \begin{aligned} \Sigma_1\cdot \mathcal{H}_{A_{1,1}}^g &=\Sigma_1\cdot \mathcal{A}_{1,1}\\[6pt] \Sigma_1\cdot \mathcal{H}_{A_{2}}^g &=\Sigma_1\cdot \mathcal{A}_{2}, \end{aligned} \end{align} $$

where the right-hand side is taking intersections in $|-2K_{\mathbb {P}^1\times \mathbb {P}^1}|$ , since by construction, the intersection numbers are the numbers of binodal and cuspidal members in the two-dimensional family of the branching curves $\mathcal {C}_1\to \Sigma _1$ , respectively. It suffices to compute the degree of $\mathcal {A}_{1,1}$ and $\mathcal {A}_{2}$ in $|-2K_{\mathbb {P}^1\times \mathbb {P}^1}|$ . This becomes a classical enumeration problem for singular members in a family of curves, which has been studied extensively in [Reference Kazarian24], [Reference Kazarian25], and so on.

By taking a general two-dimensional linear subspace $B=\mathbb {P}^2\subseteq |-2K_{\mathbb {P}^1\times \mathbb {P}^1} |$ , we obtain a family $f:\mathcal {C}\to B$ of curves in $|-2K_{\mathbb {P}^1\times \mathbb {P}^1}|$ called a net of curves in $|-2K_{\mathbb {P}^1\times \mathbb {P}^1}|$ . Let $\Delta _f \subseteq B$ be the discriminant curve of f: that is, the locus in B over which the fibres are singular. Suppose that f has at worst cuspidal or binodal fibres, and let $a_2(f)$ and $a_{1,1}(f)$ be the numbers of cuspidal and binodal fibres of f, respectively. In this case, we have

$$ \begin{align*} \deg(\mathcal{A}_2)=a_2(f)\ \mathrm{and}\ \deg(\mathcal{A}_{1,1})=a_{1,1}(f). \end{align*} $$

By our assumption, the discriminant curve $\Delta _f$ has only nodes and cusps as its singularities, and $a_2(f)$ is equal to the number of cusps of $\Delta _f$ , while $a_{1,1}(f)$ is equal to the number of nodes of $\Delta _f$ . Then we have the following explicit formulae that compute $a_{1,1}(f)$ and $a_2(f)$ .

Proposition 6.5 (compare [Reference Eisenbud and Harris12, §11.4.4])

Let S be a smooth projective surface and L a very ample line bundle on S. Let $f:\mathcal {C}\to \mathbb {P}^2$ be a general net of curves in $|L|$ whose fibres are at worst cuspidal or binodal. Then the numbers of cuspidal fibres and binodal fibres of f are given by

(6.4) $$ \begin{align} a_2(f)&=2g-d+2(e-1) \end{align} $$

(6.5) $$ \begin{align} a_{1,1}(f)&=d-3g-e(e-1)+\frac{3}{2}(e-1)(e-2), \end{align} $$

respectively, where $c_i=c_i(\Omega _S^1)$ , $\alpha =c_1(\mathcal {L})$ , $g=\frac {1}{2}(9\alpha ^2+9\alpha c_1+2c_1^2)+1$ , $d=3\alpha ^2+\alpha c_1$ , and $e=3\alpha ^2+2\alpha c_1+c_2$ .

Note that for $(S,L)= (\mathbb {P}^1\times \mathbb {P}^1, |-2K_{\mathbb {P}^1\times \mathbb {P}^1}|)$ or $(\mathbf {F}_1, |-2K_{\mathbf {F}_1}|)$ , we always have

$$ \begin{align*}\alpha^2=32,~c_1^2=8,~\alpha c_1=-16,~c_2=4.\end{align*} $$

By applying Proposition 6.5 to a general net of curves in $|-2K_{\mathbb {P}^1\times \mathbb {P}^1}| $ or $|-2K_{\mathbf {F}_1}|$ , one can compute that in both cases the numbers of cuspidal and binodal members in such a net are

(6.6) $$ \begin{align} a_2(f)&=216 \end{align} $$

(6.7) $$ \begin{align} a_{1,1}(f)&=1914, \end{align} $$

respectively. Since the surface $\Sigma _i$ has degree $4$ in the linear system, it follows that

Unigonal family

Now we consider the unigonal family $\phi _3:\mathcal {X}_3\to \Sigma _3=(\mathbb {P}^2)^*$ . Let

$$ \begin{align*}\psi_3:\Sigma_3\to \mathcal{F}_{g}\end{align*} $$

be the induced moduli map. As above, the intersection numbers $\deg (\psi _3^\ast (\mathcal {H}^g_{A_2}))$ and $\deg (\psi _3^\ast (\mathcal {H}^g_{A_2}))$ are, by construction, the numbers of binodal and cuspidal members in the two-dimensional family of the branching curves $\mathcal {B}\to \Sigma _3$ .

Let M be the locus in $\mathcal {B}$ of the singularities of all singular fibres of $\mathcal {B}\to \Sigma _3$ , which is smooth by genericity of $\mathcal {B}$ . They fit into the following commutative diagram:

Note that the family $\pi $ in which the family $\mathcal {B}\to \Sigma _3$ of curves is embedded is no longer a trivial family of surfaces.

The image $\Delta _h=f(M)\subseteq \Sigma _3$ is exactly the discriminant curve of h. By genericity of $\mathcal {B}$ , we may assume that the singular fibres of $h:\mathcal {B}\to \Sigma _3$ are at worst cuspidal or binodal. As before, the intersection number $\psi _3^\ast (\mathcal {H}^g_{A_2})=a_2(h)$ is equal to the number of cusps of $\Delta _h$ , while $\psi _3^\ast (\mathcal {H}^{g}_{A_{1,1}})= a_{1,1}(h)$ is equal to the number of nodes of $\Delta _h$ . They satisfy the natural relation

(6.8) $$ \begin{align} 2(a_2(h)+a_{1,1}(h))=2(p_a(\Delta_h)-p_g(\Delta_h))=\deg \mathbb{D}(f), \end{align} $$

where $\mathbb {D}(f)\in \mathrm {CH}^1(\mathcal {M})$ is the double point class of f given by ([Reference Fulton17, Theorem 9.3])

(6.9) $$ \begin{align} \mathbb{D}(f)=f^*f_*[M]-(c_1(f^*\mathcal{T}_{\Sigma_3})-c_1(\mathcal{T}_{M}))\cap [M]. \end{align} $$

According to [Reference Kazarian24, §8-10], we have the following formula for $a_2(h)$ :

(6.10) $$ \begin{align} a_2(h)=\deg ((c_1(\Omega_{\pi}\otimes\mathcal{O}_{\mathcal{Y}}(\mathcal{B}))-c_1(\mathcal{T}_{\pi}))\cap [M]). \end{align} $$

This gives

Proposition 6.7. The intersection numbers are

$$ \begin{align*} a_2(h)&=816\\ a_{1,1}(h)&=33480. \end{align*} $$

Proof. We need to compute the class of M. Let $dh:\mathcal {T}_{\mathcal {B}}\to h^*\mathcal {T}_{\Sigma _3}$ be the differential of h. Consider the projective bundle $p:\mathcal {P}=\mathbb {P}_{\mathcal {B}}(h^*\mathcal {T}_{\Sigma _3})\to \mathcal {B}$ . The composition

$$ \begin{align*}p^*\mathcal{T}_{\mathcal{X}_3}\xrightarrow{p^*dh} p^*h^*\mathcal{T}_{\Sigma_3}\to \mathcal{O}_{\mathcal{P}}(1)\end{align*} $$

gives a section $s\in \mathrm {H}^0(\mathcal {P},\mathcal {O}_{\mathcal {P}}(1)\otimes p^*\Omega _{\mathcal {B}})$ . Let $\widetilde {M}$ be the zero locus of s. According to [Reference Kazarian25, Lemma 2.3.1], the restriction of p to $\widetilde {M}$ is an isomorphism onto M. In particular, we have

(6.11) $$ \begin{align} \begin{aligned} i_*[M]&=p_*c_{\mathrm{top}}(\mathcal{O}_{\mathcal{P}}(1)\otimes p^*\Omega_{\mathcal{B}})\\[4pt] \deg c_1(\mathcal{T}_M)&=\deg c_1(\mathcal{T}_{\widetilde{M}}). \end{aligned} \end{align} $$

Set

$$ \begin{align*} \alpha_i&=c_i(\Omega_{\mathcal{X}_3}),~ \beta_j=c_j(\mathcal{T}_{\Sigma_3}),~ \delta=h_*(\alpha_1)\beta_1+h_*(\alpha_2) \end{align*} $$

for $i=1,2,3$ and $j=1,2$ . Combining equations (6.8), (6.9), (6.10) and (6.11), we get

(6.12) $$ \begin{align} \begin{aligned} a_2(h)=&~4h_*(\alpha_1)\beta_1^2+2h_*(\alpha_1^2+\alpha_2)\beta_1-2h_*(\alpha_1)\beta_2+2h_*(\alpha_1\alpha_2)\\[3pt] 2(a_2(h)+a_{1,1}(h))=&~\delta^2-\beta_1\delta+(-2h_*(\alpha_1\alpha_2)-h_*(\alpha_3) -2h_*(\alpha_1^2+\alpha_2)\beta_1\\[3pt] & +h_*(\alpha_1)(-4\beta_1^2+3\beta_2)). \end{aligned} \end{align} $$

Set $\zeta =c_1(\mathcal {O}_{\Sigma _3}(1))$ . Then the classes of $h_\ast (\alpha _i)$ and $h_\ast (\alpha _1\alpha _2)$ and $\delta $ are given in the following table:

$$ \begin{align*} \begin{array}{c|c|c|c|c|c} h_*(\alpha_1) & h_*(\alpha_2) & h_*(\alpha_3) & h_*(\alpha_1^2) & h_*(\alpha_1\alpha_2) & \delta\\ \hline 18 & 210\zeta & -450\zeta^2 & 36\zeta & -600\zeta^2 & 264\zeta\\ \end{array} \end{align*} $$

One can get the numbers by substituting these data into (6.12).

Now we are ready to derive the following:

Proof. As the covering map $\widetilde {\mathcal {F}}_g\to \mathcal {F}_g$ is branched over $\mathcal {H}^g_{0,-2}$ containing $\mathcal {H}^g_{A_{1, 1}}\cup \mathcal {H}^g_{A_2}$ , we know that the preimages of the irreducible components of $\mathcal {H}^g_{A_{1, 1}}$ or $\mathcal {H}^g_{A_2}$ remain irreducible. It suffices to show that the fundamental classes of irreducible components of $\mathcal {H}^g_{A_{1, 1}}$ and $\mathcal {H}^g_{A_2}$ are linearly independent in $\mathrm {H}^4(\mathcal {F}_g, \mathbb {Q})$ . According to Proposition 3.10, the irreducible components of $\mathcal {H}^g_{A_{1,1}}$ and $\mathcal {H}^g_{A_{2}}$ are described as follows:

1. i) If $g\equiv 0 ~\mathrm{or} ~1\ \mod 4$ , $\mathcal {H}^g_{A_{1,1}}$ and $\mathcal {H}^g_{A_{2}}$ are irreducible.

2. ii) If $g\equiv 2\ \mod 4$ , $\mathcal {H}^g_{A_{2}}$ is irreducible and $\mathcal {H}^g_{A_{1,1}}=\mathcal {H}_{A_{1,1}}^{g'}\cup \mathcal {H}_{A_{1,1}}^{g"}$ .

3. iii) If $g\equiv 3\ \mod 4$ , $\mathcal {H}^g_{A_{2}}$ is irreducible and $\mathcal {H}^g_{A_{1,1}}=\mathcal {H}_{A_{1,1}}^{g'}\cup \mathcal {H}_{A_{1,1}}^{g"'}$ .

In case i), we take the testing families $\phi _i:\mathcal {X}_i\to \Sigma _i, i=1,2,3$ depending on the parity of g. We have the following intersection matrix

$$ \begin{align*} \begin{array}{c|cc} & \mathcal{H}_{A_{1,1}} & \mathcal{H}_{A_2} \\\hline \Sigma_i & 7656 & 864 \\ \Sigma_3 & 33480 & 816 \end{array} \end{align*} $$

for $i=1$ or $2$ . As the matrix is nondegenerate, we can conclude our assertion easily.

In case ii), one can take a general two-dimensional proper family $\phi _4:\mathcal {X}_4\to \Sigma _4$ of K3 surfaces in $\mathscr {F}_g$ such that there exist finitely many fibres of $\phi _4$ lying in $\mathcal {H}^{g"}_{A_{1,1}}$ . This is valid because the boundary of the Baily-Borel compactification of $\mathcal {F}_g$ is only $1$ -dimensional. Next, note that $\Sigma _2$ and $\Sigma _3$ do not intersect with the NL-locus $\mathcal {H}^g_{g-1,\frac {g-2}{2}}$ , which contains $\mathcal {H}^{g"}_{A_{1,1}}$ . The corresponding intersection matrix of $\Sigma _i$ with $\mathcal {H}^g_{A_{1,1}}$ and $\mathcal {H}^g_{A_2}$ is given by

$$ \begin{align*} \begin{array}{c|ccc} & \mathcal{H}^{g'}_{A_{1,1}} & \mathcal{H}_{A_{1,1}}^{g"} & \mathcal{H}^g_{A_2} \\\hline \Sigma_2 & 7656 & 0 & 864 \\ \Sigma_3 & 33480 & 0 & 816 \\ \Sigma_4 & a & b\neq 0 & c \end{array} \end{align*} $$

which is clearly nondegenerate. Therefore, the fundamental classes of $\mathcal {H}_{A_{1,1}}^{g'}, \mathcal {H}_{A_{1,1}}^{g"}$ and $\mathcal {H}^g_{A_2}$ are linearly independent.

In case iii), as above, one can just take a general two-dimensional proper family $\phi _5:\mathcal {X}_5\to \Sigma _5$ of K3 surfaces in $\mathscr {F}_g$ such that there exist finitely many fibres of $\phi _5$ lying in $\mathcal {H}^{g"'}_{A_{1,1}}$ . Then we can obtain an intersection matrix

$$ \begin{align*} \begin{array}{c|ccc} & \mathcal{H}^{g'}_{A_{1,1}} & \mathcal{H}_{A_{1,1}}^{g"'} & \mathcal{H}^g_{A_2} \\\hline \Sigma_1 & 7656 & 0 & 864 \\ \Sigma_3 & 33480 & 0 & 816 \\ \Sigma_5 & a' & b'\neq 0 & c' \end{array} \end{align*} $$

and the assertion follows similarly.

6.10 NL-numbers via Kudla’s modularity conjecture

We will provide an alternative way to compute the binodal and cuspidal fibres on a family of unigonal K3 surfaces that uses less classical enumerative geometry. For any two-dimensional nontrivial family of unigonal K3 surfaces (with only ADE singularities)

$$ \begin{align*}f:\mathcal{X}\to \Sigma\end{align*} $$

over a smooth projective surface $\Sigma $ , it induces a map $\psi :\Sigma \to \mathcal {F}_U\cong \Gamma _U \backslash \mathrm {D}_{U^\perp }$ . To compute the Noether-Lefschetz numbers with respect to f, we can apply Theorem 2.2.1 (see also Example 2.3 and Example 3.3) to $\mathcal {F}_U$ and get that the generating series

(6.13) $$ \begin{align} \begin{aligned} \Theta_\psi(\tau): &=\sum\limits_{\beta\geq 0} (\deg(\psi^\ast Z(\beta))) e^{2\pi i \mathrm{tr} (\beta \tau )} \\ &=\sum\limits_{\substack{k,l,m\in \mathbb{Z},\\ k, m\geq 0,\\ l^2-4km\leq 0}} N_{k,l,m}\tilde{q}^kp^lq^m \end{aligned} \end{align} $$

is a classical scalar-valued Siegel modular form of weight $10$ and genus $2$ , where we write $\tau =\left (\begin {array}{ccc} z_1 & z_2 \\ z_2 & z_3 \end {array}\right )\in \mathbb {H}_2$ , $\tilde {q}=e^{2\pi i z_1}$ , $p=e^{2\pi i z_2}$ and $q=e^{2\pi i z_3}$ for $z_1,z_3\in \mathbb {H}, z_2\in \mathbb {C}$ . Then we see that all the Noether-Lefschetz numbers can be read off from the Fourier coefficients of some Siegel modular form. The geometric meaning of some coefficients $N_{k,l,m} $ are given as below (see also Proposition 3.11):

• ○ The constant term $N_{0,0,0}=(R^2f_\ast \mathcal {O}_{\mathcal {X}})^2 $ is the self-intersection of the Hodge line bundle on $\Sigma $ .

• ○ $N_{0,0,1}=\deg ((\psi ^\ast \lambda )|_\Delta )$ is the degree of the Hodge line bundle on the discriminant curve $\Delta \subseteq \Sigma $ .

• ○ $N_{1,1,1}=\deg (\psi ^\ast Z(\left (\begin {array}{ccc} -2 & 1 \\ 1 & -2 \end {array}\right ) ))$ is twice the number of cuspidal fibres.

• ○ $N_{1,0,1}=\deg (\psi ^\ast (Z(\left (\begin {array}{ccc} -2 & 0 \\ 0 & -2 \end {array}\right ) )))$ is twice the number of binodal fibres.

Next, due to Igusa’s work, a Siegel modular form of weight $10$ must be a linear combination of $E_4^{(2)}E_6^{(2)}$ and $\chi _{10}$ . Here, $E_{n}^{(2)}$ is the Siegel Eisenstein series of weight n and genus $2$ defined by

$$ \begin{align*} E_n^{(2)}(\tau)=\frac{1}{2\zeta(n)} \sum\limits_{(c,d)} \det(c\tau+d)^{-n}, \tau\in \mathbb{H}_2, \end{align*} $$

where summation runs over the bottom row of a complete system of representatives $(c,d)$ of elements in $\{ \left (\begin {array}{ccc} \ast & \ast \\ 0 & \ast \end {array}\right ) \in \mathrm {Sp}_4(\mathbb {Z})\}\backslash \mathrm {Sp}_{4}(\mathbb {Z})$ ; $\chi _{10}$ is the Igusa cusp form defined by the infinite product

(6.14) $$ \begin{align} \begin{aligned} \chi_{10}(\tau)& =\tilde{q} p q\prod\limits_{(r,s,t)> 0} (1-\tilde{q}^rp^s q^t)^{c(4rt-s^2)} \\ &=\tilde{q}pq-2\tilde{q}q-16 \tilde{q}pq^2+\cdots, \end{aligned} \end{align} $$

where $(r,s,t)> 0$ means either $r>0, t\geq 0$ or $r\geq 0,t>0$ or $r=t=0, s<0$ ; $c(m)$ is the Fourier coefficient of a modular form of weight $-\frac {1}{2}$ given by

(6.15) $$ \begin{align} \sum\limits_{m} c(m)q^m=2q^{-1}+20-128q^3+216q^4-1026q^7+1618q^8+\cdots. \end{align} $$

See also [Reference Oberdieck and Pixton37, §0.2] for the explicit construction of equation (6.15). This provides a way to construct many testing surfaces that implies Proposition 6.9.

Proof. Note that

(6.16) $$ \begin{align} \begin{aligned} E_4^{(2)}E_6^{(2)}=~&(1+240\tilde{q}+240q+\cdots+13440\tilde{q}pq+ 30240\tilde{q}q+\cdots) \times \\&(1-504\tilde{q}-504q +\cdots+44352\tilde{q}p q+166320\tilde{q}q+\cdots)\\ =~&1-264\tilde{q}-264q +\cdots+57792 \tilde{q}pq-45360 \tilde{q}q+\cdots. \end{aligned} \end{align} $$

Kudla’s modularity theorem shows that $\Theta _\psi $ must have the form

$$ \begin{align*}\Theta_\psi=aE_4^{(2)}E_6^{(2)}+b\chi_{10}\end{align*} $$

for some $a,b\in \mathbb {Q}$ . As the coefficients $N_{0,0,0}$ and $N_{1,1,1}$ are integers, we know that a and b are integers as well. We can write

$$ \begin{align*}\deg (\psi^{\ast} Z(\left(\begin{array}{ccc} -2 & 1 \\ 1 & -2 \end{array}\right)))= 57792a+b;~\deg \psi^\ast (Z(\left(\begin{array}{ccc} -2 & 0 \\ 0 & -2 \end{array}\right)))=-45360a -2b.\end{align*} $$

The assumption then implies that $\frac {57792a+b}{-45360a -2b}\neq \frac {864}{7656}$ , which proves Proposition 6.9.

For the last statement, we need to construct such a family. The idea is to make use of Miranda’s construction. As in Section 3.3, we set $W=\mathbb {P}(2^{(9)}, 3^{(13)})$ for convenience, and one can regard $\mathcal {F}_U$ as an open subset of $W/\!\!/ \mathrm {SL}_2$ via the isomorphism in equation (3.1). Embed W into $|\mathcal {O}_{W}(6\cdot 2^n)|$ with n sufficiently large. One can take general hyperplanes $H_1,\ldots ,H_{16}$ in $|\mathcal {O}_{W}(6\cdot 2^n)|$ , such that they meet transversally with W and their intersection $\Sigma =W\cap H_1\cap \ldots \cap H_{16}$ is contained in the open locus $W^\circ $ consisting of Weierstrass K3 surfaces with no extra automorphisms. This is valid because the complement of $W^\circ $ in W has codimension $>2$ . Let $\psi :\Sigma \to \mathcal {F}_U$ be the map induced by the quotient map $W^\circ \to W^\circ / \mathrm {SL}_2\subseteq \mathcal {F}_U$ . According to the computation in [Reference Laza and O’Grady30, Proposition 4.1.1], $\mathcal {O}_{W^\circ }(1)$ descends to the Hodge line bundle $\lambda $ on $W^\circ /\mathrm {SL}_2$ . One can compute that the degree of $(\psi ^\ast \lambda )^2$ equal to the degree of S in $|\mathcal {O}_{W}(6\cdot 2^n)|$ , which is only divisible by $2$ and $3$ . Thus we can conclude our assertion.

As an example, one can verify our computations for NL-numbers on $\psi _3$ .

Corollary 6.12.

(6.17) $$ \begin{align} \Theta_{\psi_3}=E_4^{(2)}E_6^{(2)}-56160\chi_{10}. \end{align} $$

6.13 Proof of the main theorem

Using the vanishing result, we can immediately get

Lemma 6.14. The natural map

(6.18) $$ \begin{align} \mathrm{H}^{4}_{\mathrm{DB}}(\widetilde{\mathscr{S}}^s_g,\mathbb{Q}(2))\rightarrow \mathrm{H}^{4}(\widetilde{\mathscr{S}}^s_g,\mathbb{Q}) \end{align} $$

is injective.

Proof. Consider the smooth and proper map $\widetilde {\pi }: \widetilde {\mathscr {S}}_g^s\to \widetilde {\mathscr {F}}_g^s$ , we can apply Deligne’s decomposition theorem [Reference Deligne9] to get

(6.19) $$ \begin{align} \mathrm{H}^k(\widetilde{\mathscr{S}}^s_g,\mathbb{C})\cong \bigoplus\limits_{i+j=k} \mathrm{H}^i(\widetilde{\mathscr{F}}^s_g; R^{j}\widetilde{\pi}_\ast (\mathbb{C})). \end{align} $$

Since $\widetilde {\mathscr {F}}_g^s$ is separated, the terms on the right-hand side can be identified with cohomology groups (with values in automorphic local systems) of the coarse moduli space $\widetilde {\mathcal {F}}_g^s$ (compare [Reference Behrend3, Proposition 36]). According to Proposition 6.2, we have

$$ \begin{align*}\mathrm{H}^{3}(\widetilde{\mathscr{S}}_{g}^s,\mathbb{Q})=0.\end{align*} $$

The assertion follows from the long exact sequence in equation (4.10).

For each $\alpha \in \mathrm {CH}^2(\widetilde {\mathscr {S}}^s_g)$ , according to Theorem 2.3.1(2) and the decomposition in equation (6.19), there exists $m\in \mathbb {Q}$ such that

(6.20) $$ \begin{align} \mathrm{cl}_{\mathrm{B}}(\alpha-mc_{2} (\mathcal{T}_{\widetilde{\pi}}))\in \mathrm{H}^4(\widetilde{\mathscr{S}}^s_g, \mathbb{Q}) \end{align} $$

can be written as a linear combination of universal line bundles over some moduli spaces of lattice polarised oriented K3 surfaces in $\mathrm {H}^4(\widetilde {\mathscr {S}}^s_g,\mathbb {Q})$ (see also the proof of [Reference Bergeron and Li4, Theorem 8.1.1]). Then Lemma 6.14 implies that this also holds in the Deligne-Beilinson cohomology group. Thus we obtain

Theorem 6.14.1. Let $\mathcal {T}_{\widetilde {\pi }}$ be the relative tangent bundle of $\widetilde {\pi }:\widetilde {\mathscr {S}_g^s}\to \widetilde {\mathscr {F}_g^s}$ . Then for any $\widetilde {\alpha } \in \mathrm {CH}^2(\widetilde {\mathscr {S}_{g}^s)}$ , there exists a rational number $m\in \mathbb {Q}$ such that the class

(6.21) $$ \begin{align} \mathrm{cl}_{\mathrm{DB}}(\widetilde{\alpha}-m c_2(\mathcal{T}_{\widetilde{\pi}})) \in \mathrm{H}^4_{\mathrm{DB}}(\widetilde{\mathscr{S}_g^s}, \mathbb{Q}(2)) \end{align} $$

is supported on a proper closed subset of $\widetilde {\mathscr {F}_{g}^s}$ : that is, the restriction of $ \mathrm {cl}_{\mathrm {DB}}(\widetilde {\alpha }-m c_2(\mathcal {T}_{\widetilde {\pi }})) $ is zero on $\widetilde {\pi }^{-1}(W)$ for some open substack W in $\widetilde {\mathscr {F}}_g^s$ .

It remains to prove Theorem 1.1.1. We can assume $g>2$ so that $\mathscr {S}_g^\circ \to \mathscr {F}_g^\circ $ is a smooth morphism between smooth varieties. Recall that we have the following commutative diagram:

(6.22)

We let $\widetilde {\pi ^\circ }:\widetilde {\mathscr {S}_{g}^\circ }\to \widetilde {\mathscr {F}}_g^\circ $ be the pull-back of $\pi $ via equation (6.22) and set $v:\widetilde {\mathscr {S}_{g}^\circ }\to \mathscr {S}_{g}^\circ $ to be the covering map.

For any class $\alpha \in \mathrm {CH}^2(\mathscr {S}_{g}^\circ )$ , take a lifting $\overline {\alpha }$ of $v^\ast (\alpha )$ in $ \mathrm { CH}^2(\widetilde {\mathscr {S}_{g}^{s}})$ . According to Theorem 6.14.1, there exists a rational number $m\in \mathbb {Q}$ such that $\mathrm { cl}_{\mathrm {DB}}(\overline {\alpha }-m c_2(\mathcal {T}_{\widetilde {\pi }}))$ is supported on a proper closed subset of $\widetilde {\mathscr {F}_{g}^s}$ . Restricting to $\widetilde {\mathscr {S}_{g}^\circ }$ , we see that $\mathrm {cl}_{\mathrm {DB}}(v^\ast (\alpha )-m c_2(\mathcal {T}_{\widetilde {\pi ^{\circ }}}))$ is also supported on a proper closed subset of $\widetilde {\mathscr {F}_{g}^\circ }$ .

Recall that there exists also the theory of Deligne-Beilinson homology $\mathrm {H}^{\mathrm {DB}}_q(X, A(p))$ for any separated algebraic scheme X, such that $\mathrm {H}^{\mathrm {DB}}$ and $\mathrm {H}_{\mathrm {DB}}$ form a twisted Poincaré duality theory (see [Reference Jannsen22, Theorem 1.19]). In particular, we have the push-forward map

$$ \begin{align*}f_{!}: \mathrm{H}^{\mathrm{DB}}_q(X', A(p))\to \mathrm{H}^{\mathrm{DB}}_q(X, A(p))\end{align*} $$

for a proper morphism $f:X'\to X$ (see [Reference Jannsen22, Theorem 1.18(a)]), and the twisted Poincaré duality

$$ \begin{align*}\mathrm{H}_{\mathrm{DB}}^q(X, A(p))\cong \mathrm{H}^{\mathrm{DB}}_{2\dim X-q}(X, A(\dim X-p))\end{align*} $$

for a smooth X (see [Reference Jannsen22, Theorem 1.15]). Combining these, we therefore have the push-forward map

$$ \begin{align*}\mathrm{H}_{\mathrm{DB}}^q(X', A(p))\to \mathrm{H}_{\mathrm{DB}}^{q-2(\dim X'-\dim X)}(X, A(p-(\dim X'-\dim X)))\end{align*} $$

in cohomology for any proper morphism $f:X'\to X$ between smooth separated algebraic schemes. By abuse of notations, we may still denote it by $f_{!}$ .

Applying the push-forward map

$$ \begin{align*}v_{!}: \mathrm{H}^4_{\mathrm{DB}}(\widetilde{\mathscr{S}_{g}^\circ}, \mathbb{Q}(2)) \to \mathrm{H}^4_{\mathrm{DB}}(\mathscr{S}_{g}^\circ, \mathbb{Q}(2))\end{align*} $$

to $\mathrm {cl}_{\mathrm {DB}}(v^\ast (\alpha )-m c_2(\mathcal {T}_{\widetilde {\pi ^{\circ }}}))$ , we find that $\mathrm {cl}_{\mathrm {DB}}(\alpha -m c_2(\mathcal {T}_{\pi }))$ is likewise supported on a proper closed subset of $\mathscr {F}_{g}^\circ $ . This finishes the proof of Theorem 1.1.1.