2.1 Covering families of rational curves in $\operatorname {\mathrm {SU}}_C(2)$

Rational curves in the moduli spaces $\operatorname {\mathrm {SU}}_C(r,d)$ were extensively studied; see, for example, [Reference Narasimhan and RamananNR75, Reference Oxbury, Pauly and PreviatoOPP98, Reference HwangHwa00, Reference Hwang and RamananHR04, Reference SunSun05, Reference Mok and SunMS09, Reference PalPal16]. Restricting to $g=3$ , $r=2$ and $d=0$ , the results of [Reference Mustopa and Teixidor i BigasMTiB20] show that there exist two different families of covering lines, that is, families of rational curves of degree one with respect to the Theta embedding

$$ \begin{align*}{\mathcal{C}}:=\operatorname{\mathrm{SU}}_2(C)\hookrightarrow |2\Theta|={\mathbb{P}}(V_8),\end{align*} $$

passing through a general point of the moduli space. We will denote these two families by ${\mathcal {F}}_H$ and ${\mathcal {F}}_R$ and consider them as subvarieties of the Grassmannain $G(2,V_8)$ . In the sequel, we describe these two covering families in some detail. They are both of dimension six but behave very differently; we will illustrate this by showing how different are the corresponding variety of minimal rational tangents(VMRTs), which in our case, since we deal with lines, are just the spaces of lines through a fixed general point.

2.2 Hecke lines

Before going through the body of this section, we need to recall the following

Definition 2.1. A vector bundle V on C is called $(k,\ell )$ -semistable (resp. $(k,\ell )$ -stable) if for any proper subbundle $W\subset V$ we have

$$ \begin{align*}\frac{\deg(W)+k}{\operatorname{\mathrm{rank}}(W)}\leq (resp. <) \frac{\deg(V)+k-\ell }{\operatorname{\mathrm{rank}}(V)}.\end{align*} $$

A generic Hecke line can be described by choosing a point $c\in C$ and a rank two vector bundle F on C with determinant $\det (F)={\mathcal {O}}_C(c)$ . Then the bundles E that fit into an exact sequence

$$ \begin{align*}0{\longrightarrow} E{\longrightarrow} F{\longrightarrow} {\mathcal{O}}_c{\longrightarrow} 0\end{align*} $$

are parametrized by ${\mathbb {P}} (F_c^\vee )\simeq {\mathbb {P}}^1$ . They have trivial determinant and are all stable when F is $(1,0)$ -semistable in the sense of [Reference Mok and SunMS09, Definition 2.5]. For vector bundles of rank two and degree one, this condition is equivalent to stability, hence also to semistability. The resulting curve in $\operatorname {\mathrm {SU}}_C(2)$ is a line and such lines are called Hecke lines. Note that dualizing, we get an exact sequence

$$ \begin{align*}0{\longrightarrow} F^\vee{\longrightarrow} E^\vee{\longrightarrow} {\mathcal{O}}_c{\longrightarrow} 0,\end{align*} $$

so a Hecke line parametrizes all the possible extensions of ${\mathcal {O}}_c$ by $F^\vee $ .

By [Reference PalPal16, Remark 5.3], a general Hecke line defines a vector bundle ${\mathcal {E}}$ of rank $2$ over $C\times {\mathbb {P}}^1$ fitting into an exact sequence

$$ \begin{align*}0{\longrightarrow} p_1^*F^\vee\otimes p_2^*{\mathcal{O}}_{{\mathbb{P}}^1}(-1){\longrightarrow} {\mathcal{E}}^\vee{\longrightarrow} p_1^*{\mathcal{O}}_c{\longrightarrow} 0,\end{align*} $$

where $p_1$ and $p_2$ are the projections of $C \times {\mathbb {P}}^1$ onto the two factors C and ${\mathbb {P}}^1$ . An easy consequence is that, since ${\mathcal {E}}^\vee $ admits a unique jumping line at c, this point can be uniquely recovered from the Hecke line. (Beware this is only true for general Hecke lines.)

We will denote by ${\mathcal {F}}_H$ the family of Hecke lines in $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}_C)$ , considered as a subvariety of the space $G(2,V_8)$ of lines in ${\mathbb {P}}(V_8)$ .

Conversely, Hecke lines passing through a general point $[E]$ of $\operatorname {\mathrm {SU}}_C(2,K_C)$ (we make this choice of determinant just for convenience) are obtained by choosing a projection $E{\rightarrow } E_c{\rightarrow }{\mathcal {O}}_c$ , where $E_c$ denotes the fiber of the vector bundle E at the point $c\in C$ . So they are parametrized by (the image in ${\mathcal {F}}_H$ of) the total space of the projective bundle ${\mathbb {P}}(E^\vee )$ over C. The tangent map of this morphism sends ${\mathbb {P}}(E^\vee )$ to the tangent space of the moduli space at $[E]$ , which is the projectivization of

$$ \begin{align*}H^1(C,\mathcal{E}nd_0(E))\simeq H^0(C,K_C\otimes \mathcal{E}nd_0(E))^\vee\simeq H^0(C,S^2E)^\vee\end{align*} $$

since $K_C\simeq \det (E)$ . Here, $ \mathcal {E}nd_0(E)$ denotes the vector bundle of traceless endomorphisms of E. This implies (see [Reference HwangHwa00, Reference Hwang and RamananHR04] for more general statements):

Equivalently, the latter claim means that a general Hecke line is not contained in any larger linear space contained in $\operatorname {\mathrm {SU}}_C(2)$ , although such larger linear spaces do exist.

2.3 Lines in the ruling

For each line bundle $L\in \operatorname {\mathrm {Pic}}^1(C)$ , consider the rank two vector bundles E obtained as extensions of the form

$$ \begin{align*}0{\longrightarrow} L{\longrightarrow} E{\longrightarrow} K_C\otimes L^\vee{\longrightarrow} 0.\end{align*} $$

Such extensions are parametrized by ${\mathbb {P}}_L:={\mathbb {P}} (\operatorname {\mathrm {Ext}}^1(K_C\otimes L^\vee ,L))\simeq {\mathbb {P}}^3$ . Hence, we obtain a ruling of $\operatorname {\mathrm {SU}}_C(2)$ by a family of ${\mathbb {P}}^3$ s parametrized by $\operatorname {\mathrm {Pic}}^1(C)$ , which we denote by ${\mathbb {P}}({\mathcal {R}}){\rightarrow } \operatorname {\mathrm {Pic}}^1(C)$ .

Note that ${\mathbb {P}}_L$ intersects the Kummer threefold along a copy $C_L$ of C [Reference Oxbury, Pauly and PreviatoOPP98, 1.1]. According to [Reference Oxbury, Pauly and PreviatoOPP98, Theorem 1.3], a line in ${\mathbb {P}}_L$ is a Hecke line if and only if it meets $C_L$ .

Moreover, by [Reference Oxbury, Pauly and PreviatoOPP98, Proposition 1.2], two spaces ${\mathbb {P}}_L$ and ${\mathbb {P}}_M$ are always distinct for $L\ne M$ and, for sufficiently general choices of L and M, they are disjoint. When they meet, their intersection is a single point, or a line; the latter case happens exactly when $K_C-L-M$ is effective. In particular, if a line is contained in ${\mathbb {P}}_L\cap {\mathbb {P}}_M$ , it must be a bisecant to both $C_L$ and $C_M$ .

Now, consider the family ${\mathcal {F}}_R$ of lines contained in the ${\mathbb {P}}^3$ s of the ruling. By what we have just recalled, ${\mathcal {F}}_R$ is the birational image in $G(2,V_8)$ of the quadric bundle $G(2,{\mathcal {R}})$ over $\operatorname {\mathrm {Pic}}^1(C)$ .

For future use, we record the following easy consequence.

3.1 A simple construction of the Coble quartic

In this section, we recall some results from [Reference Gruson, Sam and WeymanGSW13]. The starting point is a general four-form in eight variables, $v\in \wedge ^4V_8\simeq \wedge ^4V_8^\vee $ , where $V_8$ denotes a complex eight-dimensional vector space. Recall that this is a theta-representation, being part of a ${\mathbb {Z}}_2$ -grading of the exceptional Lie algebra

$$ \begin{align*}{\mathfrak e}_7\simeq \mathfrak{sl}(V_8)\oplus \wedge^4V_8.\end{align*} $$

The action of the so-called theta-group, which here is $\operatorname {\mathrm {SL}}(V_8)$ , behaves very much as the action of the adjoint group on a simple complex Lie algebra. In particular, one has Jordan decompositions, and the GIT-quotient

$$ \begin{align*}\wedge^4V_8 /\hspace{-1mm}/ \operatorname{\mathrm{SL}}(V_8) \simeq {\mathfrak h}/W\end{align*} $$

for some finite complex reflection group W acting on what is called a Cartan subspace ${\mathfrak h}$ of the theta-representation. We will make this Cartan subspace explicit later on. For now, we just need to know that it coincides with the seven-dimensional representation of the Weyl group of $E_7$ . As a consequence, the choice of v determines uniquely a nonhyperelliptic curve C of genus three (a plane quartic) with a marked flex point [Reference Gruson, Sam and WeymanGSW13, Remark 6.1].

We will construct from our general $v\in \wedge ^4V_8$ a collection of geometric objects defined as orbital degeneracy loci. The main point of this approach is that it allows us to reduce to simpler representations. Typically, the Borel–Weil theorem gives an isomorphism

$$ \begin{align*}\wedge^4V_8\simeq H^0({\mathbb{P}}(V_8),\wedge^4{\mathcal{Q}}) \simeq H^0({\mathbb{P}}(V_8),\wedge^3{\mathcal{Q}}^\vee(1)),\end{align*} $$

where ${\mathcal {Q}}$ denotes the rank seven quotient vector bundle on ${\mathbb {P}}(V_8)$ . At the price of passing to a relative setting over ${\mathbb {P}}(V_8)$ , this reduces the study of $\wedge ^4V_8$ to that of three-forms in seven variables.

But then the situation is much simpler because if $V_7$ is a seven-dimensional complex vector space, $\wedge ^3V_7^\vee \cong \wedge ^4 V_7$ has finitely many orbits under the action of $\operatorname {\mathrm {GL}}(V_7)$ . Each orbit closure Y allows to associate to $v\in \wedge ^4V_8$ the locus $D_Y(v)\subset {\mathbb {P}}(V_8)$ of points x where the image of v lies in the corresponding $Y_x\subset \wedge ^3{\mathcal {Q}}^\vee (1)_x$ (this is exactly how orbital degeneracy loci are defined). By the general results of [Reference Benedetti, Filippini, Manivel and TanturriBFMT20a], for v general the main properties of Y will be transferred to $D_Y(v)$ , starting from its codimension. We can therefore focus on the orbit closures in $\wedge ^3V_7^\vee $ of codimension at most seven. Remarkably, there are only three such orbit closures (not counting the whole space), that we can index by their codimension: $Y_1$ is a hypersurface of degree $7$ , $Y_4$ is its singular locus, $Y_7$ is the singular locus of $Y_4$ . The corresponding orbital degeneracy loci have been described in [Reference Gruson, Sam and WeymanGSW13, 6.1, 6.2].

Since the Coble quartic can be characterized as the unique quartic hypersurface that is singular along the Kummer threefold [Reference BeauvilleBea91, Proposition 3.1], we can immediately deduce that it coincides with $D_{Y_1}(v)$ .

3.2 Kempf collapsings

A nice feature of our orbital degeneracy loci is the following. It turns out that the orbit closures they are associated to, although singular, admit nice resolutions of singularities by Kempf collapsings, which are birational contractions from total spaces of homogeneous vector bundles on flag manifolds. These homogeneous vector bundles are typically nonsemisimple, making them more difficult to handle. Nevertheless, these collapsings allow to describe the corresponding orbital degeneracy loci in terms of zero loci of sections of vector bundles.

In the cases we are interested in, we obtain the following descriptions, where $U_k$ stands for a k-dimensional subspace of $V_8$ . For $A,B$ subspaces of a vector space V, we will denote by $(\wedge ^p A) \wedge (\wedge ^q B) \subset \wedge ^{p+q}V$ the linear subspace spanned by the elements of the form $a_1\wedge \cdots \wedge a_p \wedge b_1 \wedge \cdots \wedge b_q$ with $a_1,\ldots ,a_p \in A$ and $b_1,\ldots ,b_q\in B$ . For vector subbundles $\mathcal {A},\mathcal {B}$ of a the trivial bundle $V \otimes {\mathcal {O}}$ , we use the same convention to define $(\wedge ^p \mathcal {A}) \wedge (\wedge ^q \mathcal {B})$ in $\wedge ^{p+q} V \otimes {\mathcal {O}}$ .

Proposition 3.2. Let $v\in \wedge ^4 V_8$ be a generic element. The Coble quartic ${\mathcal {C}}$ associated to v can be described as

$$\begin{align*}\Big\{ [U_1]\in {\mathbb{P}}(V_8) \mid \exists U_4\supset U_1, \; v\in (\wedge^2 U_4) \wedge (\wedge^2 V_8) +\wedge^3 V_8\wedge U_1\Big\}. \end{align*}$$

The corresponding Kummer threefold ${\mathrm {Kum}_C}$ is

$$\begin{align*}\Big\{ [U_1]\in {\mathbb{P}}(V_8) \mid \exists U_6\supset U_2\supset U_1, \; v\in \wedge^4 U_6+ \wedge^2 U_6\wedge U_2\wedge V_8 + \wedge^3 V_8\wedge U_1\Big\}. \end{align*}$$

The singular locus ${\mathrm {Kum}_C}[2]$ of ${\mathrm {Kum}_C}$ is

$$\begin{align*}\Big\{ [U_1]\in {\mathbb{P}}(V_8) \mid \exists U_7\supset U_4\supset U_1, \; v\in \wedge^3 U_4\wedge V_8 + (\wedge^2 U_4)\wedge (\wedge^2 U_7) +\wedge^3 V_8\wedge U_1\Big\}. \end{align*}$$

These results follow from a combination of [Reference Gruson, Sam and WeymanGSW13, Section 6] and [Reference Kraskiewicz and WeymanKW13, Section 3]. Let us clarify the statement for instance for ${\mathrm {Kum}_C}$ , the explanations for the other loci being similar. In [Reference Gruson, Sam and WeymanGSW13, Section 6], it is shown that ${\mathrm {Kum}_C}:=D_{Y_4}(v)$ is the Kummer threefold. In [Reference Kraskiewicz and WeymanKW13, Section 3] it is proved that $Y_4\subset \wedge ^4 V_7 $ is desingularized by a the total space of the vector bundle

$$ \begin{align*}{\mathcal{W}}:=\wedge^4 {\mathcal{U}}_5+ \wedge^2 {\mathcal{U}}_5\wedge {\mathcal{U}}_1 \wedge V_7\end{align*} $$

over the flag variety $Fl(1,5,V_7)$ . Here, we denoted by ${\mathcal {U}}_1$ and ${\mathcal {U}}_5$ , respectively, the rank one and rank five tautological vector bundles on $Fl(1,5,V_7)$ . The projection from the total space of ${\mathcal {W}}$ to $Y_4\subset \wedge ^4 V_7$ is given by the composition of the inclusion of ${\mathcal {W}}$ inside $\wedge ^4 V_7 \otimes {\mathcal {O}}_{Fl(1,5,V_7)}$ with the projection to $\wedge ^4 V_7$ . For v general, this desingularization $\operatorname {\mathrm {Tot}}({\mathcal {W}})\to Y_4$ of $Y_4$ can be relativized to obtain a desingularization of $D_{Y_4}(v)$ , as explained in [Reference Benedetti, Filippini, Manivel and TanturriBFMT20a, Section 2]. For this, we simply consider the flag bundle $Fl(1,5,{\mathcal {Q}})$ : By the previous discussion, any point of $x=[U_1]\in D_{Y_4}(v)$ must be the image of a flag $\bar {U}_1 \subset \bar {U}_5\subset {\mathcal {Q}}_x=V_8/U_1$ such that $v \; \mathrm {mod} \; U_1$ belongs to $\wedge ^4 \bar {U}_5+ \wedge ^2 \bar {U}_5\wedge \bar {U}_1 \wedge {\mathcal {Q}}_x\subset \wedge ^4{\mathcal {Q}}_x$ . This flag originates from a flag $(U_1\subset U_2\subset U_6\subset V_8)$ (such that $\bar {U}_1=U_2/U_1$ , etc.), and we can rewrite the previous condition as asking that $x=[U_1]$ belongs to the projection of

$$ \begin{align*}Z(v):=\Big\{ (U_1\subset U_2\subset U_6)\in Fl(1,2,6,V_8), \; v\in \wedge^4 U_6+ \wedge^2 U_6\wedge U_2\wedge V_8 + \wedge^3 V_8\wedge U_1\Big\}.\end{align*} $$

This is the zero locus of a global section of a globally generated bundle, obtained as a quotient of the trivial bundle with fiber $\wedge ^4V_8$ . For v general, this section is general, so $Z(v)$ is smooth. Moreover, the projection $Z(v)\to D_{Y_4}(v)\subset {\mathbb {P}}(V_8)$ , obtained by just forgetting $U_2$ and $U_6$ , is birational.

3.3 Self-duality of the Coble quartic

Because of the natural isomorphism $\wedge ^4V_8\simeq \wedge ^4V_8^\vee $ (defined up to scalar or, more precisely, up to the choice of a volume form on $V_8$ ), the same constructions can be performed in the dual projective space ${\mathbb {P}}(V_8^\vee )$ . This is related to the remarkable fact that the Coble quartic is projectively self-dual [Reference PaulyPau02]. Let us show how this duality statement easily follows from our approach in terms of orbital degeneracy loci.

First, consider a general point $[U_1]$ of ${\mathcal {C}}=D_{Y_1}(v)$ . As we have seen in the previous section, there exists (a unique) $U_4\supset U_1$ such that v belongs to $(\wedge ^2 U_4) \wedge (\wedge ^2 V_8) +\wedge ^3 V_8\wedge U_1$ . Reducing modulo $(\wedge ^2 U_4) \wedge (\wedge ^2 V_8)$ , we get

$$ \begin{align*}\bar{v}\in \wedge^3(V_8/U_4)\otimes U_1\simeq (V_8/U_4)^\vee.\end{align*} $$

In general, $\bar {v}$ is nonzero and defines a hyperplane in $V_8/U_4$ , that is, a hyperplane $U_7$ of $V_8$ , containing $U_4$ . Note that this exactly means that

(3.1) $$ \begin{align} v\in (\wedge^2 U_4) \wedge (\wedge^2 V_8) +\wedge^3 U_7\wedge U_1. \end{align} $$

Proof. Let $\tilde {{\mathcal {C}}}$ denote the variety of flags $(U_1\subset U_4)$ such that v belongs to

$$ \begin{align*}{\mathcal{F}}(U_1,U_4):=(\wedge^2 U_4) \wedge (\wedge^2 V_8) +\wedge^3 V_8\wedge U_1.\end{align*} $$

We know that the projection $\tilde {{\mathcal {C}}}{\longrightarrow }{\mathcal {C}}$ is birational. Moreover, as a subvariety of the flag manifold $Fl(1,4,V_8)$ , $\tilde {{\mathcal {C}}}$ is the zero-locus of the section of the vector bundle $\wedge ^4V_8/{\mathcal {F}} (U_1,U_4)$ defined by v since such a section vanishes exactly when $v\in {\mathcal {F}}(U_1,U_4)\subset \wedge ^4 V_8$ . Let $\mathfrak {p}(U_1,U_4)$ denote the stabilizer of the flag $(U_1\subset U_4)$ inside $\mathfrak {gl}(V_8)$ . The tangent space to $Fl(1,4,V_8)$ at the corresponding point is the quotient $\mathfrak {gl}(V_8)/\mathfrak {p}(U_1,U_4)$ , and the tangent space to $\tilde {{\mathcal {C}}}$ is the image, in this quotient, of the space of endomorphisms $X\in \mathfrak {gl}(V_8)$ such that $X(v)$ belongs to ${\mathcal {F}}(U_1,U_4)$ , as follows from the normal exact sequence

$$ \begin{align*}0\to T_{\tilde{{\mathcal{C}}}}\to T_{Fl(1,4,V_8)}|_{\tilde{{\mathcal{C}}}} \to (\wedge^4V_8/{\mathcal{F}} (U_1,U_4))|_{\tilde{{\mathcal{C}}}} \to 0. \end{align*} $$

The tangent space to ${\mathcal {C}}$ is then the image of this space inside $\mathfrak {gl}(V_8)/\mathfrak {p}(U_1)\simeq \operatorname {\mathrm {Hom}}(U_1,V_8/U_1)$ , where $\mathfrak {p}(U_1)$ denotes the stabilizer of the line $U_1$ .

So our claim will follow, if we can check that any $X\in \mathfrak {gl}(V_8)$ such that $X(v)$ belongs to ${\mathcal {F}}(U_1,U_4)$ , must send $U_1$ into the hyperplane $U_7$ . But (3.1) implies, once we apply X, that

$$ \begin{align*}X(v)\in U_4 \wedge (\wedge^3 V_8) +\wedge^3 U_7\wedge X(U_1).\end{align*} $$

If $X(v)$ belongs to ${\mathcal {F}}(U_1,U_4)$ , it has to vanish modulo $U_4$ . So $\wedge ^3 U_7\wedge X(U_1)$ must also vanish modulo $U_4$ , which is the case only if $X(U_1)\subset U_7$ .

Recall that once we fix a volume form on $V_8$ , we get an isomorphism of $\wedge ^4V_8$ with $\wedge ^4V_8^\vee $ . We will denote by $v^\vee $ the image of v. (Strictly speaking, it is uniquely defined only up to scalar, but this is irrelevant in our constructions.) To make things clearer, we will denote by ${\mathcal {C}}(v)$ the Coble quartic defined by v in ${\mathbb {P}}(V_8)$ and by ${\mathcal {C}}(v^\vee )$ the Coble quartic defined by $v^\vee $ in ${\mathbb {P}}(V_8^\vee )$ .

Proof. For $[U_1]$ a general point of ${\mathcal {C}}$ , we have a flag $(U_1\subset U_4\subset U_7)$ such that v belongs to $(\wedge ^2 U_4) \wedge (\wedge ^2 V_8) +\wedge ^3 U_7\wedge U_1$ . Choose an adapted basis $e_1,\ldots , e_8$ so that $e_1$ generates $U_1$ , etc. The condition means that v is a linear combination of elementary tensors $e_i\wedge e_j\wedge e_k\wedge e_\ell $ with $i,j\le 4$ , and of $e_5\wedge e_6\wedge e_7\wedge e_1$ .

Now, recall that if the chosen volume form on $V_8$ is $e_1\wedge \cdots \wedge e_8$ , and $e_1^\vee ,\ldots , e_8^\vee $ is the dual basis of $e_1,\ldots , e_8$ , then the isomorphism of $\wedge ^4V_8$ with $\wedge ^4V_8^\vee $ sends the elementary tensor $e_i\wedge e_j\wedge e_k\wedge e_\ell $ to $\pm e_p^\vee \wedge e_q^\vee \wedge e_r^\vee \wedge e_s^\vee $ , where $\{i,j,k,l\} \cap \{p,q,r,s\}=\emptyset $ .

As a consequence, $v^\vee $ will be a linear combination of elementary tensors $e_p^\vee \wedge e_q^\vee \wedge e_r^\vee \wedge e_s^\vee $ with $p,q\ge 5$ , and $e_2^\vee \wedge e_3^\vee \wedge e_4^\vee \wedge e_8^\vee $ . In other words,

$$ \begin{align*}v^\vee \in (\wedge^2 U_4^\perp) \wedge (\wedge^2 V_8^\vee) +\wedge^3 U_1^\perp\wedge U_7^\perp.\end{align*} $$

This is exactly the condition that ensures that $[U_7^\perp ]$ belongs to ${\mathcal {C}}(v^\vee )$ . Thanks to the previous lemma, we deduce that ${\mathcal {C}}(v)^\vee \subset {\mathcal {C}}(v^\vee )$ . Moreover, the symmetry between $U_1$ and $U_7^\perp $ implies that in general, $U_1$ can be recovered from $U_7$ exactly as $U_7$ is constructed from $U_1$ , which means that ${\mathcal {C}}(v)^\vee $ is birationally equivalent to ${\mathcal {C}}(v)$ . Finally, since ${\mathcal {C}}(v)^\vee $ and ${\mathcal {C}}(v^\vee )$ are both irreducible hypersurfaces, they must be equal.

The previous discussion shows that it is natural to define the variety ${\mathcal {C}}(v,v^\vee )\subset Fl(1,4,7,V_8)$ parametrizing the flags $(U_1\subset U_4\subset U_7\subset V_8) $ satisfying condition (3.1). This is a smooth variety dominating birationally both ${\mathcal {C}}(v)$ and ${\mathcal {C}}(v^\vee )$ ; there is a diagram

One recovers that way the constructions explained is [Reference PaulyPau02, section 3.3]. We used the suggestive notation $d{\mathcal {C}}$ for the Gauss map, which sends a smooth point of ${\mathcal {C}}(v)$ to its tangent hyperplane, given by the differential of the cubic’s equation.

3.4 The Cartan subspace

Recall that a Cartan subspace for the ${\mathbb {Z}}_2$ -graded Lie algebra ${\mathfrak e}_7=\mathfrak {sl}(V_8)\oplus \wedge ^4V_8$ is a maximal subspace of $\wedge ^4V_8$ , made of elements of ${\mathfrak e}_6$ which are semisimple and commute [Reference VinbergVin76]. Among other nice properties, a general element of $\wedge ^4V_8$ is $\operatorname {\mathrm {SL}}(V_8)$ -conjugate to (finitely many) elements of any given Cartan subspace.

An explicit Cartan subspace of $\wedge ^4V_8$ is worked out in [Reference OedingOed22, (3.1)]. It coincides with the space of Heisenberg invariants provided in [Reference Ren, Sam, Schrader and SturmfelsRSSS13, Remark 4.2]. Here is a list of seven generators, for a given basis $e_1,\ldots , e_8$ of $V_8$ :

$$ \begin{align*}\begin{array}{rcl} h_1 & = & e_1\wedge e_2\wedge e_3\wedge e_4 +e_5\wedge e_6\wedge e_7\wedge e_8 ,\\ h_2 & = & e_1\wedge e_3\wedge e_5\wedge e_7 +e_6\wedge e_8\wedge e_2\wedge e_4 ,\\ h_3 & = & e_1\wedge e_5\wedge e_6\wedge e_2 +e_8\wedge e_4\wedge e_3\wedge e_7 ,\\ h_4 & = & e_1\wedge e_6\wedge e_8\wedge e_3 +e_4\wedge e_5\wedge e_7\wedge e_2 ,\\ h_5 & = & e_1\wedge e_8\wedge e_4\wedge e_5 +e_7\wedge e_2\wedge e_6\wedge e_3 ,\\ h_6 & = & e_1\wedge e_4\wedge e_7\wedge e_6 +e_2\wedge e_3\wedge e_8\wedge e_5 ,\\ h_7 & = & e_1\wedge e_7\wedge e_2\wedge e_8 +e_3\wedge e_5\wedge e_4\wedge e_6. \end{array}\end{align*} $$

Combinatorially, each of these generators is given by a pair of complementary fourtuples of indices in $\{1,\ldots ,8\}$ . Each of these $14$ fourtuples shares a pair of indices with any other distinct, not complementary fourtuple. This is the property that ensures the commutation in ${\mathfrak e}_6$ , since the Lie bracket of ${\mathfrak e}_6$ , restricted to $\wedge ^4V_8$ , is given by the unique (up to scalar) $\mathfrak {sl}_8$ -equivariant morphism

$$ \begin{align*}\wedge^2(\wedge^4V_8) {\longrightarrow} \wedge^4V_8\otimes \wedge^4V_8{\longrightarrow} S_{21111110}V_8\simeq\mathfrak{sl}_8,\end{align*} $$

where $S_{21111110}$ denotes the Schur functor corresponding to the partition $(2,1,1,1,1,1,1,0)$ . If we start with two elementary tensors given by fourtuples with a common pair of indices, we can include them into $\wedge ^4U_6$ for some codimension two subspace $U_6\subset V_8$ . But then the Lie bracket factors through $S_{21111110}U_6=\{0\}$ , so it has to vanish.

Each pair of indices in $\{1,\ldots ,8\}$ belongs to three of the $14$ fourtuples. For any triple $(ijk)$ among

$$ \begin{align*}(124), \; (137),\; (156),\; (235),\; (267),\; (346), \; (457),\end{align*} $$

$h_i$ , $h_j$ and $h_k$ share four disjoint pairs (for example $h_1$ , $h_2$ and $h_4$ share $(13), (24), (57), (68)$ ). These seven triples always meet in exactly one index, so they are in correspondence with the lines in a Fano plane. One can find more on this in [Reference ManivelMan06, Section 4].

A nice consequence of this description is the following

3.5 The abelian threefold

Remarkably, one can construct the abelian threefold whose Kummer variety is ${\mathrm {Kum}_C}$ by considering another orbital degeneracy locus. The idea is to use the flag variety $Fl(1,7,V_8)$ , the incidence correspondence in ${\mathbb {P}}(V_8)\times {\mathbb {P}}(V_8^\vee )$ parametrizing flags $(U_1\subset U_7)$ . The rank six quotient bundle ${\mathcal {N}}={\mathcal {U}}_7/{\mathcal {U}}_1$ allows to realize the space of four-forms as

$$ \begin{align*}\wedge^4V_8^\vee=H^0(Fl(1,7,V_8), p_1^*{\mathcal{O}}(1)\otimes\wedge^3{\mathcal{N}}^\vee).\end{align*} $$

Exactly as before, this allows to associate to any $\operatorname {\mathrm {GL}}(V_6)$ -orbit closure Y in $\wedge ^3V_6$ an orbital degeneracy locus $D_Y(v)\subset Fl(1,7,V_8)$ . Here, $V_6$ is a six-dimensional vector space. In particular, the cone $Y_{10}$ over the Grassmannian $G(3,V_6)$ yields, for v generic, a smooth threefold ${\mathrm {A}_C}:=D_{Y_{10}}(v)$ . In similar terms as for the other orbital degeneracy loci, this threefold is

(3.2) $$ \begin{align} {\mathrm{A}_C}=\Big\{ [U_1\subset U_7]\in Fl(1,7,V_8) \mid \exists U_1\subset U_4\subset U_7,\; v\in \wedge^3 U_4\wedge V_8 +\wedge^4 U_7 +\wedge^3 V_8\wedge U_1\Big\}. \end{align} $$

Over a point of ${\mathrm {A}_C}$ , given by a flag $U_1\subset U_7$ , the four-form v defines a decomposable tensor in $\wedge ^3(U_7/U_1)$ . This tensor is never zero if v is general and therefore defines a four-dimensional space $U_4$ such that $U_1\subset U_4\subset U_7$ . Hence, a rank-four vector bundle ${\mathcal {U}}_4$ on ${\mathrm {A}_C}$ , a subbundle of the trivial bundle $V_8\otimes {\mathcal {O}}_{\mathrm {A}_C}$ .

4.1 The ruling and its lines

Recall the definition of the abelian threefold ${\mathrm {A}_C}$ from Equation (3.2). Our next result relates it to the ruling described in section 2.3.

Proof. Consider a point of ${\mathrm {A}_C}$ and the associated flag $U_1\subset U_4\subset U_7$ . By the very definition of ${\mathrm {A}_C}$ , this means we can write

$$ \begin{align*}v=e_1\wedge w+v'+e_2\wedge e_3\wedge e_4\wedge e_8\end{align*} $$

for some vectors $e_1\in U_1$ and $e_2,e_3,e_4\in U_4$ , with $w\in \wedge ^3V_8$ , $e_8\in V_8$ and $v'\in \wedge ^4U_7$ . Under the generality hypothesis we can suppose that $U_4=\langle e_1,e_2,e_3,e_4\rangle $ , and it suffices to check that $U^{\prime }_1={\mathbb {C}} e_2$ defines a point of ${\mathcal {C}}$ .

Modulo $e_1$ and $e_2$ , the tensor w is a three-form in six variables. Since the secants of the Grassmannian $G(3,6)$ in its Plücker embedding fill up the ambient projective space, generically we can write $w=a\wedge b\wedge c+d\wedge e\wedge f$ modulo $e_1$ and $e_2$ , for some vectors $a,b,c,d,e,f$ . Modulo $e_1$ and $e_2$ again, $v'$ is a four-form in only five variables, so it defines a hyperplane that will cut the three-dimensional space $\langle a,b,c\rangle $ in codimension one, say along $\langle a,b\rangle $ , and similarly it will cut $\langle d,e,f\rangle $ in codimension one, say along $\langle d,e\rangle $ . In other words, we may suppose that modulo $e_1$ and $e_2$ , $v'=a\wedge b\wedge d\wedge e$ . But then, modulo $e_2$ we get

$$ \begin{align*}v=e_1\wedge (a\wedge b\wedge c+d\wedge e\wedge f)+a\wedge b\wedge d\wedge e.\end{align*} $$

So v belongs to $(\wedge ^2U^{\prime }_4)\wedge (\wedge ^2V_8)+\wedge ^3V_8\wedge U^{\prime }_1$ if $U^{\prime }_4=\langle e_1,e_2,a,d\rangle .$ The existence of such a space $U^{\prime }_4\supset U^{\prime }_1$ is precisely the required condition for $U^{\prime }_1$ to belong to ${\mathcal {C}}$ , so we are done.

Proof. This can be done by a Chern class computation, being equivalent to the fact that the degree of ${\mathbb {P}}({\mathcal {U}}_4)$ with respect to the relative hyperplane class does not vanish. Notice that by Equation (3.2), ${\mathrm {A}_C}$ can be considered as a subvariety of $Fl(1,4,7,V_8)$ . Even more, it is the zero locus in the flag manifold of the section $\overline {v}$ of the rank $19$ vector bundle

$$ \begin{align*}{\mathcal{G}}:=\wedge^4 V_8 /(\wedge^3 {\mathcal{U}}_4\wedge V_8 + \wedge^4 {\mathcal{U}}_7 + \wedge^3 V_8\wedge {\mathcal{U}}_1)\end{align*} $$

over $Fl(1,4,7,V_8)$ defined by v. Since this section is general, the class of ${\mathrm {A}_C}$ in the Chow ring of the flag manifold is the top Chern class of ${\mathcal {G}}$ . So the degree we are looking for is

$$ \begin{align*}\int_{{\mathbb{P}}({\mathcal{U}}_4)}c_1({\mathcal{U}}_1^\vee)^6=\int_{{\mathrm{A}_C}}s_3({\mathcal{U}}_4^\vee)=\int_{Fl(1,4,7,V_8)}c_{19}({\mathcal{G}})s_3({\mathcal{U}}_4^\vee)=32,\end{align*} $$

as can be computed using [Reference Grayson and StillmanGS, Schubert2 package]. This implies the claim.

The previous statement allows to reconstruct the curve C purely in terms of the four-form and its associated orbital degeneracy loci. Indeed, we have recalled that a ${\mathbb {P}}^3$ of the ruling meets the Kummer threefold along a copy of the curve.

And of course we also recover the family of lines in the ruling as a quadric bundle. Indeed, the same arguments as in section 2.3 yield:

4.2 Hecke lines from alternating forms

In the previous section, we have defined some ODL $D_{Y_i}(v)$ from orbits inside the space of three-forms in seven variables (i.e., in the notation of the previous sections, inside $\wedge ^3 V_7$ ). We will use a similar construction to obtain ODL inside the Grassmannian $G(2,V_8)$ . The Borel–Weil theorem gives an isomorphism

$$ \begin{align*}\wedge^4V_8\simeq H^0(G(2,V_8),\wedge^4{\mathcal{Q}})=H^0(G(2,V_8),\wedge^2{\mathcal{Q}}^\vee(1)),\end{align*} $$

where ${\mathcal {Q}}$ denotes the rank six quotient vector bundle on $G(2,V_8)$ . Thus, in this case, we need to look at two-forms in six variables.

If $V_6$ is as before a six-dimensional complex vector space, $\wedge ^4 V_6^\vee \simeq \wedge ^2 V_6$ has only two proper $\operatorname {\mathrm {GL}}(V_6)$ -orbit closures, that we will index by their codimension: the Pfaffian cubic hypersurface $Z_1$ and its singular locus $Z_6$ , that is the cone over the Grassmannian $G(2,V_6)$ . These allow us to construct inside $G(2,V_8)$ the two orbital degeneracy loci $D_{Z_1}(v)$ and $D_{Z_6}(v)$ .

Let us first consider $D_{Z_6}(v)$ , which can also be defined by

$$\begin{align*}D_{Z_6}(v):=\Big\{[U_2]\in G(2,V_8)\mid \exists U_6 \supset U_2, \;\; v \in \wedge^3 V_8\wedge U_2+\wedge^4 U_6\Big\}. \end{align*}$$

Proof. By definition, $D_{Z_6}(v)$ is the projection in $G(2,V_8)$ of the locus $Z_6(v)$ in $Fl(2,6,V_8)$ parametrizing flags $(U_2\subset U_6\subset V_8)$ such that v belongs to the $56$ -dimensional space $\wedge ^3 V_8\wedge U_2+\wedge ^4 U_6$ . Taking the quotient of $\wedge ^4V_8$ by the latter, we get a rank $14$ vector bundle ${\mathcal {P}}$ on $Fl(2,6,V_8)$ , generated by global sections since it is a quotient of a trivial bundle. Moreover, v defines a generic section of this bundle, and $Z_6(v)$ is the zero-locus of this section, hence it is smooth. Since $Fl(2,6,V_8)$ has dimension $20$ , $Z_6(v)$ has dimension $6$ and its canonical bundle is given by the adjunction formula. A straightforward computation yields

$$ \begin{align*}K_{Z_6(v)}=\det({\mathcal{U}}_2)^{-3}\otimes \det({\mathcal{U}}_6)^{5}.\end{align*} $$

On the other hand, for any $[U_2]\in G(2,V_8)$ , the quotient of $\wedge ^4V_8$ by $\wedge ^3 V_8\wedge U_2$ is isomorphic to $\wedge ^4(V_8/U_2)\simeq \wedge ^2(V_8/U_2)^\vee \otimes \det (V_8/U_2)$ . This is a space of skew-symmetric forms in six dimensions, and the existence of $U_6$ exactly means that v defines a skew-symmetric form in $\wedge ^2(V_8/U_2)^\vee \otimes \det (V_8/U_2)$ whose rank is at most two. In fact, the rank must be exactly two, since for v generic, a simple dimension count shows that the rank can never be zero. In particular, the projection of $Z_6(v)$ to $D_{Z_6}(v)$ is an isomorphism.

More than that, the kernel of our two-form on $V_8/U_2$ is $U_6/U_2$ , so we get a nondegenerate skew-symmetric form on the quotient $V_8/U_6$ , which is therefore identified with its dual. To be precise, since the skew-symmetric form has values in $\det (V_8/U_2)$ , we get an isomorphism $V_8/U_6\simeq (V_8/U_6)^\vee \otimes \det (V_8/U_2)$ . Taking determinants, we deduce that $\det (U_2)^2\simeq \det (U_6)^2$ ; in other words, the line bundle ${\mathcal {L}}=\det ({\mathcal {U}}_6)\otimes \det ({\mathcal {U}}_2)^\vee $ is $2$ -torsion on $Z_6(v)$ .

But then we can rewrite the canonical bundle as

$$ \begin{align*}K_{Z_6(v)}=\det({\mathcal{U}}_2)\otimes \det({\mathcal{U}}_6)\otimes {\mathcal{L}}^{\otimes 4}. \end{align*} $$

Note that $\det ({\mathcal {U}}_2)^\vee \otimes \det ({\mathcal {U}}_6)^\vee $ is very ample on $Fl(2,6,V_8)$ since it defines its canonical Plücker type embedding. Since ${\mathcal {L}}$ is torsion, we deduce that $Z_6(v)$ is Fano. But then its Picard group is torsion free, so ${\mathcal {L}}$ is actually trivial. So finally $K_{Z_6(v)}=\det ({\mathcal {U}}_2)^2$ , hence the index is even.

The previous discussion shows that $D_{Z_6}(v)$ is a Pfaffian locus defined by a skew-symmetric map $\psi _v : {\mathcal {Q}}{\rightarrow } {\mathcal {Q}}^\vee (1)$ associated with v. The restriction of $\psi _v$ to $D_{Z_6}(v)$ has constant rank, hence its Kernel $\operatorname {\mathrm {Ker}} (\psi _v)$ is a rank four vector bundle on $D_{Z_6}(v)$ , that coincides with $({\mathcal {U}}_6/{\mathcal {U}}_2)|_{D_{Z_6}(v)}$ and fits into an exact sequence

(4.1) $$ \begin{align} 0{\rightarrow} \operatorname{\mathrm{Ker}} (\psi_v){\longrightarrow} {\mathcal{Q}}|_{D_{Z_6}(v)}{\longrightarrow} {\mathcal{Q}}^\vee (1)|_{D_{Z_6}(v)}{\longrightarrow} \operatorname{\mathrm{Ker}} (\psi_v)^\vee (1){\longrightarrow} 0. \end{align} $$

Let us set once and for all the more compact notation $D:=D_{Z_6}(v)$ and $G:=G(2,V_8)$ . The exact sequence (4.1) allows to describe the normal bundle ${\mathcal {N}}_{D/G}$ as follows.

We will use this information later on. Our next goal is to prove the following theorem.

Let us begin by showing that $D_{Z_6}(v)$ defines a six-dimensional family of Hecke lines.

Proof. Let $[U_1]\in {\mathbb {P}}(U_2)$ be a point in the line. By definition of $D_{Z_6}(v)$ , one can write

$$ \begin{align*}(v\quad \mod U_1)=u_2\wedge v'+ a\wedge b\wedge c\wedge d\end{align*} $$

for some $u_2\in U_2$ , some trivector $v'$ and some vectors $a,b,c,d$ . The trivector $v'$ is a trivector in six variables; therefore, it can in general be written as $e\wedge f\wedge g + h\wedge i\wedge l$ for some vectors $e,f,g,h,i,l$ since the secant variety of $G(3,6)$ is the whole Plücker space. Now, modulo $U_2$ , $\dim (\langle a,b,c,d \rangle \cap \langle e,f,g \rangle )\geq 1$ and $\dim (\langle a,b,c,d \rangle \cap \langle h,i,l \rangle )\geq 1$ . Thus, we can suppose that $a=e$ and $b=h$ . But then if we let $U_4=\langle U_2,a,b \rangle ,$ it is straightforward to check that $( v\ \mod U_1) \in (\wedge ^2 U_4)\wedge (\wedge ^2 V_8)$ . This ensures that $[U_1]$ belongs to ${\mathcal {C}}$ .

The point-line incidence variety of the family of lines parametrized by $D_{Z_6}(v)$ is given by the projective bundle ${\mathbb {P}}({\mathcal {U}}_2)\to D_{Z_6}(v)$ .

5.1 The relative Pfaffian

As we have seen, the fact that $D_{Z_6}(v)$ is defined as a Pfaffian locus in $G(2,V_8)$ implies that it is the singular locus of a Pfaffian hypersurface, defined as the first degeneracy locus $D_{Z_1}(v)$ of the skew-symmetric morphism ${\mathcal {Q}}{\longrightarrow } {\mathcal {Q}}^\vee (1)$ defined by v.

5.1.1 Structure of the proof of Theorem 5.1

Recall that D stands for $D_{Z_6}(v)$ and G for $G(2,V_8)$ . That $D_{Z_1}(v)$ is a quadratic section of G follows from the fact that it is defined by a rank six Pfaffian, obtained as the image of v by the cubic morphism

$$ \begin{align*}S^3(\wedge^2{\mathcal{Q}}^\vee(1)){\rightarrow} \wedge^6{\mathcal{Q}}^\vee(3)= {\mathcal{O}}_G(2).\end{align*} $$

In order to prove that this is the only quadratic section that is singular along D, recall that the conormal bundle of D in the Grassmannian G is the quotient of the ideal sheaf ${\mathcal {I}}_D$ by its square ${\mathcal {I}}^2_D$ . Twisting by ${\mathcal {O}}_G(2)$ and taking cohomology, we get an exact sequence

$$ \begin{align*}0{\longrightarrow} H^0(G,{\mathcal{I}}^2_D(2)){\longrightarrow} H^0(G,{\mathcal{I}}_D(2)){\longrightarrow} H^0(D,{\mathcal{N}}_{D/G}^\vee(2)){\longrightarrow} H^1(G,{\mathcal{I}}^2_D(2)).\end{align*} $$

Observe that $H^0(G,{\mathcal {I}}_D(2))$ parametrizes quadratic sections of G (up to scalar) that contain D, while, since D is smooth, $H^0(G,{\mathcal {I}}^2_D(2))$ parametrizes quadratic sections that are singular along D. Our claim is that the latter space is one-dimensional. This will be proved in three steps: First, compute the dimension of the space of quadrics containing D; second, bound $H^0(D,{\mathcal {N}}_{D/G}^\vee (2))$ from below; third, prove that $H^1(G,{\mathcal {I}}^2_D(2))$ vanishes. These results are contained in Lemmas 5.3, 5.6, 5.11. From the fact that $H^1({\mathcal {I}}_D^2(2))=0$ , the exact sequence

$$ \begin{align*}0{\longrightarrow} H^0({\mathcal{I}}_D^2(2)){\longrightarrow} H^0({\mathcal{I}}_D(2)) {\longrightarrow} H^0({\mathcal{N}}_{D/G}^\vee(2))= H^0({\mathcal{N}}_{D/G}){\longrightarrow} 0,\end{align*} $$

knowing that $h^0({\mathcal {I}}_D(2))=71$ and $h^0({\mathcal {N}}_{D/G})\ge 70$ , will allow us to conclude that $h^0({\mathcal {I}}_D^2(2))\le 1$ and the proof will be complete.

5.1.2 Quadrics containing the moduli space

Let us count the quadric sections of $G=G(2,V_8)$ that contain the moduli space $D\simeq \operatorname {\mathrm {SU}}_C(2,L)$ .

Lemma 5.3. $h^0(G,{\mathcal {I}}_D(2))=71.$

Proof. Let us first recall the classical minimal resolution of the ideal I generated by submaximal Pfaffians of a generic skew-symmetric matrix of size $6$ ; in other words, the ideal of the cone over the Grassmannian $G(2,V_6)$ inside $\wedge ^2V_6$ . As usual, we will use the notation $S_\lambda $ , for $\lambda $ a partition, in order to indicate the corresponding Schur functor. Letting $S={\mathbb {C}}[\wedge ^2V_6]$ , this resolution is the following [Reference WeymanWey03, (6.4.6)]:

Since D is a Pfaffian locus of the expected dimension, given by a skew-symmetric map ${\mathcal {Q}}{\rightarrow }{\mathcal {Q}}^\vee (1)$ , we deduce the following free resolution of its twisted ideal sheaf (we used identifications like $S_{332222}{\mathcal {Q}}^\vee =\wedge ^2{\mathcal {Q}}^\vee (-2)$ ):

Note that this resolution is self-dual, up to twist. Moreover, using the Bott–Borel–Weil theorem one can check that all the factors are acyclic homogeneous vector bundles, with two exceptions: $\wedge ^4{\mathcal {Q}}$ has a nonzero space of sections, isomorphic to $\wedge ^4V_8$ ; and $S^2{\mathcal {Q}}^\vee (-1)$ , which is one of the two irreducible factors of $\Omega _G^2$ , has a one dimensional cohomology group in degree two. We end up with a canonical exact sequence

(5.1) $$ \begin{align} 0{\longrightarrow} \wedge^4V_8{\longrightarrow} H^0(G,{\mathcal{I}}_D(2)){\longrightarrow} {\mathbb{C}}{\longrightarrow} 0, \end{align} $$

and our claim follows.

Remark 5.4. Being defined by a cubic Pfaffian, the equation of the hypersurface $D_{Z_1}(v)$ must be a cubic $\operatorname {\mathrm {SL}}(V_8)$ -covariant of v in $\wedge ^4V_8$ , taking values in $H^0({\mathcal {O}}_G(2))\simeq S_{22}V_8$ . In fact, it is a $\operatorname {\mathrm {GL}}(V_8)$ -covariant, that by homogeneity with respect to $V_8$ , must take its values in $S_{22}V_8\otimes \det (V_8)$ . One can check that the latter module has multiplicity one inside $S^3(\wedge ^4V_8)$ , so this covariant is unique up to scalar. For example, it can be obtained as the composition

$$ \begin{align*} S^3(\wedge^4V_8)\hookrightarrow &S^3(\wedge^2V_8\otimes \wedge^2V_8)\to S^3(\wedge^2V_8)\otimes S^3(\wedge^2V_8)\to S^3(\wedge^2V_8)\otimes \wedge^6V_8 \to \\ &\to S^3(\wedge^2V_8)\otimes \wedge^2V_8^\vee\otimes\det(V_8) \to S^2(\wedge^2V_8)\otimes\det(V_8) \to S_{22}V_8\otimes\det(V_8). \end{align*} $$

Following the natural morphisms involved in these arrows, this would allow to give an explicit formula for an equation of the quadratic hypersurface $D_{Z_1}(v)$ in terms of the coefficients of v (this was done in [Reference Ren, Sam, Schrader and SturmfelsRSSS13] for the Coble quartic itself). It would suffice to do this when v belongs to our preferred Cartan subspace; this is in principle a straightforward computation but the resulting formulas would be huge.

Remark 5.5. The embedding of $\wedge ^4V_8$ inside $H^0(G,{\mathcal {I}}_D(2))$ in Equation (5.1) is given by the derivatives of $D_{Z_1}(v)$ with respect to v, that is, can be obtained by polarizing the cubic morphism discussed in the previous remark. On the other hand, modulo these derivatives, (4) shows that there is a uniquely defined ‘non-Pfaffian’ quadric vanishing on D. This non-Pfaffian quadric comes from the contribution of $S^2{\mathcal {Q}}^\vee (-1)$ in the resolution of ${\mathcal {I}}_D(2)$ . Since in this resolution, these two terms are connected one to the other through three morphisms having respective degree two, one and two with respect to v, the non-Pfaffian quadric must be given by a quintic covariant in v. And indeed, a computation with Lie [Reference van Leeuwen, Cohen and LisservLCL92] shows that

$$ \begin{align*}\operatorname{\mathrm{Hom}}(S^5(\wedge^4 V_8), S_{22}V_8\otimes (\det (V_8))^2)^{\operatorname{\mathrm{GL}}(V_8)}\simeq{\mathbb{C}}^2.\end{align*} $$

A special line in this space of covariants is generated by the cubic covariant defining the Pfaffian quadric, twisted by the invariant quadratic form (defined by the wedge product). The quotient is our non-Pfaffian quadric. As before, we could in principle compute it explicitly by constructing a specific covariant. One way to construct such a covariant is to observe that

$$ \begin{align*}S^2(\wedge^4 V_8)\twoheadrightarrow S_{221111}V_8\subset \wedge^2 V_8\otimes \wedge^6 V_8= \wedge^2 V_8\otimes \wedge^2 V_8^\vee\otimes\det (V_8).\end{align*} $$

Taking the square of the resulting morphism, we can define a quartic covariant

$$ \begin{align*}S^4(\wedge^4 V_8)\to S^2(\wedge^2 V_8)\otimes S^2(\wedge^2 V_8^\vee)\otimes\det (V_8)^2\to S_{22}V_8\otimes \wedge^4 V_8^\vee\otimes\det (V_8)^2,\end{align*} $$

hence the desired quintic covariant.

5.1.3 The normal bundle of D in $G(2,V_8)$

Let us now bound from below the dimension of $H^0(D,{\mathcal {N}}_{D/G}^\vee (2))$ . By Lemma 4.7, this space is isomorphic with $H^0({\mathcal {N}}_{D/G})$ , which parametrizes infinitesimal deformations of D inside G. Some of these deformations must be induced by the deformation of $[v]$ inside ${\mathbb {P}}(\wedge ^4V_8)$ , which should provide $69$ parameters. But recall that the family ${\mathcal {F}}_H$ of Hecke lines inside $SU_C(2)$ is a subvariety of $G(2,V_8)$ , birationally fibered over the curve C, with one fiber isomorphic to $D\simeq \operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}_C(c))$ for some point $c\in C$ . So we expect one extra deformation of D to be obtained by deforming c in the curve C. That these deformations are independent is essentially the content of

Lemma 5.6. $h^0(D,{\mathcal {N}}_{D/G})\ge 70.$

Proof. The locus in $\wedge ^4 V_6\simeq \wedge ^2V_6^\vee $ corresponding to skew-symmetric forms of rank at most $2$ is desingularized by the total space of $\wedge ^4 {\mathcal {U}}_4 $ over the Grassmannian $G(4,V_6)$ [Reference WeymanWey03, (6.4.2)]. As a consequence of this and of [Reference Benedetti, Filippini, Manivel and TanturriBFMT20a, Proposition 2.3], the Pfaffian locus D is desingularized by the zero locus $Z:=Z_6(v)$ inside $Fl(2,6,V_8)$ of a (general) section of the bundle ${\mathcal {V}}=\wedge ^4 (V_8/{\mathcal {U}}_2)/\wedge ^4 ({\mathcal {U}}_6/{\mathcal {U}}_2)$ . This bundle is an extension of irreducible bundles

$$ \begin{align*}0 \to \wedge^3 ({\mathcal{U}}_6/{\mathcal{U}}_2)\otimes (V_8/{\mathcal{U}}_6) \to {\mathcal{V}} \to \wedge^2 ({\mathcal{U}}_6/{\mathcal{U}}_2)\otimes \det(V_8/{\mathcal{U}}_6) \to 0.\end{align*} $$

By dimension count, Z is in fact isomorphic to D via the natural projection. Under this isomorphism and by Lemma 4.7, ${\mathcal {N}}_{D/G}$ can be identified with the restriction of ${\mathcal {N}}:=\wedge ^2 ({\mathcal {U}}_6/{\mathcal {U}}_2)\otimes \det (V_8/{\mathcal {U}}_6)$ to Z. In order to compute the cohomology of this restriction, we can tensorize with ${\mathcal {N}}$ the Koszul complex $\wedge ^\bullet {\mathcal {V}}^\vee $ of the global section of ${\mathcal {V}}$ , whose zero locus is $Z\subset Fl(2,6,V_8)$ . This gives the following resolution of ${\mathcal {N}}_{D/G}$ by locally free sheaves on $Fl(2,6,V_8)$

$$ \begin{align*}0\to \wedge^\bullet {\mathcal{V}}^\vee \otimes {\mathcal{N}} \to {\mathcal{N}}_{D/G} \to 0. \end{align*} $$

By applying the Bott–Borel–Weil theorem we can compute the cohomology groups of the bundles $\wedge ^k {\mathcal {V}}^\vee \otimes {\mathcal {N}}$ , for all $k \ge 0$ . Those that do not vanish are the following:

$$ \begin{gather*} H^0(\wedge^0 {\mathcal{V}}^\vee \otimes {\mathcal{N}})=\wedge^4 V_8, \\H^0(\wedge^1 {\mathcal{V}}^\vee \otimes {\mathcal{N}})={\mathbb{C}},\\H^2(\wedge^3 {\mathcal{V}}^\vee \otimes {\mathcal{N}})={\mathbb{C}} , \;\; H^3(\wedge^3 {\mathcal{V}}^\vee \otimes {\mathcal{N}})={\mathbb{C}}^2, \\H^4(\wedge^4 {\mathcal{V}}^\vee \otimes {\mathcal{N}})= H^5(\wedge^4 {\mathcal{V}}^\vee \otimes {\mathcal{N}})=\wedge^4 V_8,\\H^4(\wedge^5 {\mathcal{V}}^\vee \otimes {\mathcal{N}})=\mathfrak{sl}(V_8)\oplus {\mathbb{C}}^3, \quad H^5(\wedge^5 {\mathcal{V}}^\vee \otimes {\mathcal{N}})=\mathfrak{sl}(V_8)\oplus {\mathbb{C}}^4 , \quad H^6(\wedge^5 {\mathcal{V}}^\vee \otimes {\mathcal{N}})={\mathbb{C}},\\H^6(\wedge^7 {\mathcal{V}}^\vee \otimes {\mathcal{N}})= H^7(\wedge^7 {\mathcal{V}}^\vee \otimes {\mathcal{N}})={\mathbb{C}},\\H^8(\wedge^9 {\mathcal{V}}^\vee \otimes {\mathcal{N}})= H^9(\wedge^9 {\mathcal{V}}^\vee \otimes {\mathcal{N}})={\mathbb{C}},\\H^{12}(\wedge^{13} {\mathcal{V}}^\vee \otimes {\mathcal{N}})= H^{13}(\wedge^{13} {\mathcal{V}}^\vee \otimes {\mathcal{N}})= {\mathbb{C}}. \end{gather*} $$

A direct consequence is that $\chi ({\mathcal {N}}_{D/G})=70$ . Moreover, observe that

$$ \begin{align*}H^q(\wedge^k {\mathcal{V}}^\vee \otimes {\mathcal{N}})=0\quad \mathrm{for} \; q-k>1.\end{align*} $$

Since these groups give the first page of the spectral sequence in cohomology induced by the Koszul complex of ${\mathcal {O}}_{Z}$ twisted by ${\mathcal {N}}$ , this implies that $H^i({\mathcal {N}}_{D/G})=0$ for $i>1$ . Therefore, $h^0({\mathcal {N}}_{D/G})=\chi ({\mathcal {N}}_{D/G})+h^1({\mathcal {N}}_{D/G})\ge 70$ .

5.1.4 An affine module M

As usual $V_6$ denotes a six-dimensional vector space. Let S be the coordinate ring of $\wedge ^2 V_6$ . The ideal I of the cone over $G(2,V_6)$ is generated by the submaximal Pfaffians of the generic skew-symmetric matrix of size $6$ ; the $\operatorname {\mathrm {GL}}(6)$ -module generated by these submaximal Pfaffians is $\wedge ^4V_6^\vee \subset S^2(\wedge ^2V_6^\vee )$ . The square of I is then generated by the symmetric square of this module. The module decomposes [Reference van Leeuwen, Cohen and LisservLCL92] as

$$ \begin{align*}S^2(\wedge^4V_6^\vee) = S_{221111}V_6^\vee\oplus S_{2222}V_6^\vee.\end{align*} $$

The first component is $\wedge ^2V_6^\vee \otimes \det V_6^\vee $ and must be interpreted as parametrizing quartics that are multiples of linear forms by the Pfaffian cubic. The ideal they generate is $S_+I_{P}$ , where $S_+\subset S$ is the irrelevant ideal, and $I_P$ denotes the ideal of the Pfaffian hypersurface.

Consider the exact sequence

$$ \begin{align*}0\to S_+I_P \to I^2 \to M:=I^2/ S_+I_P \to 0.\end{align*} $$

The quotient module M is generated by $S_{2222}V_6^\vee $ . According to [Reference Grayson and StillmanGS], the minimal resolution $R_\bullet $ of M has the Betti numbers of Table 1.

Table 1

Betti table of M.

The minimal resolution is $\operatorname {\mathrm {GL}}_6$ -equivariant, and it is not difficult to write it in terms of Schur functors. Indeed, we know that the quartic generators are parametrized by $S_{2222}V_6^\vee $ , so the first syzygy module must be contained in $S_{2222}V_6^\vee \otimes \wedge ^2V_6^\vee $ , and it turns out that there is a unique $\operatorname {\mathrm {GL}}_6$ -module of the correct dimension inside this tensor product. Proceeding inductively, we arrive at the following conclusion: The minimal $\operatorname {\mathrm {GL}}_6$ -equivariant resolution of the S-module M has the following shape:

Here, vertical arrows have degree one and diagonal arrows have degree two. Notice that the complex in bold reproduces the resolution of the Pfaffian ideal I itself.

Remark 5.7. As J. Weyman observed, one could also obtain this resolution by considering the natural resolution of the Pfaffian hypersurface given by the total space of the vector bundle $\wedge ^2{\mathcal {U}}$ over the Grassmannian $G(4,V_6)$ . The morphism $\pi $ from $\operatorname {\mathrm {Tot}}(\wedge ^2{\mathcal {U}})$ to $\wedge ^2V_6$ is a resolution of singularities, and one can check that M is the push-forward by $\pi $ of the module given by the pull-back of the line bundle ${\mathcal {O}}(2)$ from the Grassmannian. Applying the geometric technique from [Reference WeymanWey03], one can extract the minimal resolution of M from the collection of $\operatorname {\mathrm {GL}}(V_6)$ -modules given by

$$ \begin{align*}F_i = \bigoplus_{j\ge 0} H^j(G(4,V_6),{\mathcal{O}}(2)\otimes\wedge^{i+j}(\wedge^2{\mathcal{U}})^\perp).\end{align*} $$

Here, $(\wedge ^2{\mathcal {U}})^\perp $ is the kernel of the natural projection $\wedge ^2V_6^\vee \to \wedge ^2{\mathcal {U}} ^\vee $ . The bundle $(\wedge ^2{\mathcal {U}})^\perp $ is not semisimple but is an extension of ${\mathcal {O}}(-1)$ by ${\mathcal {U}}^\vee \otimes Q^\vee $ . Remarkably, it is the contribution of ${\mathcal {O}}(-1)$ that reproduces the minimal resolution of I (twisted) inside that of M.

5.1.5 Relativizing M

Now, we want to use these results in the relative setting. Since $\wedge ^4 {\mathcal {Q}}$ is a vector bundle on $G(2,V_8)$ which is locally isomorphic to $\wedge ^2 V_6$ , we can relativize the construction of I and $I_P$ and M. For convenience, let us restrict to the complement ${\mathcal {X}}$ of the zero section inside the total space of this vector bundle. We get sheaves of ${\mathcal {O}}_{{\mathcal {X}}}$ -modules and ideals that we denote, respectively, by ${\mathcal {I}}', {\mathcal {I}}^{\prime }_P, {\mathcal {M}}'$ . Note that, since we avoid the zero section, we get an exact sequence

$$ \begin{align*}0\to {\mathcal{I}}^{\prime}_P \to {\mathcal{I}}^{\prime2} \to {\mathcal{M}}' \to 0.\end{align*} $$

Then we consider $v\in \wedge ^4V_8$ as a general section of $\wedge ^4 {\mathcal {Q}}$ , that we interpret as a morphism from $G=G(2,V_8)$ to the total space of $\wedge ^4 {\mathcal {Q}}$ . By the definition of orbital degeneracy loci [Reference Benedetti, Filippini, Manivel and TanturriBFMT20b, Definition 2.1], the ideal of $D_{Z_1}(v)$ is ${\mathcal {I}}_P:={\mathcal {I}}^{\prime }_P\otimes {\mathcal {O}}_G$ and the ideal of $D=D_{Z_6}(v)$ is ${\mathcal {I}}_D:={\mathcal {I}}'\otimes {\mathcal {O}}_G$ . Let us also denote ${\mathcal {M}}={\mathcal {M}}'\otimes {\mathcal {O}}_G$ . Of course, these tensor products are taken over ${\mathcal {O}}_{{\mathcal {X}}}$ .

Lemma 5.8. There is an exact sequence

$$ \begin{align*}0\to {\mathcal{I}}_P \to {\mathcal{I}}_D^2 \to {\mathcal{M}} \to 0.\end{align*} $$

Proof. By the right exactness of tensor product, here by ${\mathcal {O}}_G$ , we get an exact sequence

$$ \begin{align*}{\mathcal{I}}_P \to {\mathcal{I}}_D^2 \to {\mathcal{M}} \to 0.\end{align*} $$

But the map ${\mathcal {I}}_P \subset {\mathcal {I}}_D^2$ (which expresses the fact that D is contained in the singular locus of the Pfaffian hypersurface) clearly remains an injection, and we are done.

In order to control ${\mathcal {M}}$ , we will now consider the complex of vector bundles induced by the resolution we constructed for M. We can deduce a resolution of ${\mathcal {M}}'$ and then tensor out again by ${\mathcal {O}}_G$ . In order to prove that we get a resolution of ${\mathcal {M}}$ (the resolution given just below), we need to check that the Tor-sheaves of ${\mathcal {O}}_{\mathcal {X}}$ -modules $\mathcal {T}or_i({\mathcal {M}}',{\mathcal {O}}_G)$ vanish for $i>0$ . All the Tor-sheaves we compute in the sequel will also be for ${\mathcal {O}}_{\mathcal {X}}$ -modules.

Lemma 5.9. For any $i>0$ ,

1. 1. $\mathcal {T}or_i({\mathcal {I}}^{\prime }_P,{\mathcal {O}}_G)=0,$

2. 2. $\mathcal {T}or_i({\mathcal {O}}_{\mathcal {X}}/{\mathcal {I}}',{\mathcal {O}}_G)=0,$

3. 3. $\mathcal {T}or_i({\mathcal {I}}'/{\mathcal {I}}^{\prime 2},{\mathcal {O}}_G)=0,$

4. 4. $\mathcal {T}or_i({\mathcal {M}}',{\mathcal {O}}_G)=0.$

Proof. (1) is obvious since ${\mathcal {I}}^{\prime }_P$ is locally free. (2) is a consequence of the generic perfection theorem (see [Reference Eagon and NorthcottEN67]), since I and therefore ${\mathcal {I}}'$ is perfect, and D has the expected dimension. (3) is a consequence of (2) because ${\mathcal {I}}'/{\mathcal {I}}^{\prime 2}$ is a locally free ${\mathcal {O}}_{\mathcal {X}}/{\mathcal {I}}'$ -module (recall that since we have a generality assumption the singular locus is avoided). Finally, to prove (4) observe first that, by the long exact sequence of Tor, $\mathcal {T}or_i({\mathcal {I}}',{\mathcal {O}}_G)=\mathcal {T}or_ {i+1}({\mathcal {O}}_{\mathcal {X}}/{\mathcal {I}}',{\mathcal {O}}_G)=0$ for any $i>0$ . Because of (3) this implies that $\mathcal {T}or_i({\mathcal {I}}^{\prime 2},{\mathcal {O}}_G)=0$ for any $i>0$ . Then we can use the exact sequence of Lemma 5.8 to deduce that $\mathcal {T}or_i({\mathcal {M}}',{\mathcal {O}}_G)=0$ when $i>1$ and that there is an exact sequence

$$ \begin{align*}0{\longrightarrow} \mathcal{T}or_1({\mathcal{M}}',{\mathcal{O}}_G){\longrightarrow} {\mathcal{I}}_P{\longrightarrow} {\mathcal{I}}_D^2{\longrightarrow} {\mathcal{M}}{\longrightarrow} 0.\end{align*} $$

By Lemma 5.8, $\mathcal {T}or_1({\mathcal {M}}',{\mathcal {O}}_G)$ vanishes as well, and we are done.

Lemma 5.10. ${\mathcal {M}}(2)$ is acyclic.

Proof. Twist the previous resolution of ${\mathcal {M}}$ by ${\mathcal {O}}(2)$ , and deduce from the Bott–Borel–Weil theorem that all the bundles in the twisted resolution are acyclic. This implies the claim.

Lemma 5.11. $H^i({\mathcal {I}}_D^2(2))=0$ for any $i>0$ .

Proof. This follows immediately from Lemmas 5.8 and 5.10.

This concludes the proof of Theorem 5.1. Note the following consequence: D has nonobstructed deformations.

Corollary 5.12. $h^0({\mathcal {N}}_{D/G})=70$ and $h^i({\mathcal {N}}_{D/G})=0$ for any $i>0$ .

5.2 Deforming the Pfaffian hypersurface

We already observed that, varying v in $\wedge ^4V_8$ , we only get a codimension one family of deformations of D. The missing dimension is provided by the choice of the point on the curve C, but this is invisible in our constructions. We will nevertheless prove that the special quadric section of the Grassmannian deforms.

Proof. Let us consider the embedding $Q=D_{Z_1}(v)\hookrightarrow G$ from Theorem 5.1. Let $H^{\prime }_{Q/G}$ be the so-called ‘locally trivial Hilbert scheme’ parametrizing locally trivial deformations of $Q\subset G$ , as defined in [Reference Greuel and KarrasGK89, 2.2]. Remark that the construction of [Reference Greuel and KarrasGK89] is done for finite singularities, but their arguments, as the authors underline in the introduction, go through for arbitrary singularities because of [Reference Flenner and KosarewFK87]. Let

$$ \begin{align*}\mathcal{N}^{\prime}_{Q/G}= \operatorname{\mathrm{Ker}} ( \mathcal{N}_{Q/G} \to \mathcal{T}^1_Q),\end{align*} $$

where $\mathcal {T}^1_Q$ denotes the first cotangent sheaf of Q (as defined, for instance, in [Reference SernesiSer06, Section 1.1.3]). In order that the locally trivial Hilbert scheme be smooth at Q, by [Reference Greuel and KarrasGK89, Prop. 2.3], we need that $H^1(Q,\mathcal {N}^{\prime }_{Q/G})=0$ . If this happens, then $h^0(Q,\mathcal {N}^{\prime }_{Q/G})=\dim (H^{\prime }_{Q/G})$ and we will show that this equals 70. By [Reference SernesiSer06, Section 4.7.1], we have an exact sequence

$$ \begin{align*}0 \to T_Q \to T_{G}|_Q \to {\mathcal{N}}_{Q/G} \to \mathcal{T}^1_Q \to 0 .\end{align*} $$

Hence, $\mathcal {N}^{\prime }_{Q/G}$ coincides with the image of $T_{G/Q}$ inside ${\mathcal {N}}_{Q/G}$ , which is exactly the (twisted) Jacobian ideal ${\mathcal {J}}_{Q/G}(2)$ restricted to Q. In turn, the Jacobian ideal of the Pfaffian locus of $6\times 6$ matrices is exactly the ideal of $4\times 4$ skew-symmetric minors. This implies that ${\mathcal {J}}_{Q/G}(2)$ is the twisted ideal ${\mathcal {I}}_D(2)/{\mathcal {I}}_Q(2)$ of D inside ${\mathcal {O}}_Q(2)={\mathcal {O}}_Q(Q)$ . Let us therefore consider the exact sequence

(5.2) $$ \begin{align} 0 \to {\mathcal{I}}_Q(2) \to {\mathcal{I}}_D(2) \to {\mathcal{J}}_{Q/G}(2)\to 0. \end{align} $$

By Lemma 5.3, we have $h^0(G,{\mathcal {I}}_D(2))=71$ and in the proof of the same lemma we showed that $h^i(G,{\mathcal {I}}_D(2))=0$ , for $i>0$ . On the other hand, we have ${\mathcal {I}}_Q(2)={\mathcal {O}}_G$ . Via the long cohomology exact sequence associated to sequence (5.2), we deduce that $h^0(G,{\mathcal {J}}_{Q/G}(2))=70$ and $h^i(G,{\mathcal {J}}_{Q/G}(2))=0$ for $i>0.$ Hence $H^{\prime }_{Q/G}$ is smooth of dimension 70 at $[Q]$ . We have a natural map between Hilbert schemes

$$ \begin{align*}\sigma: H^{\prime}_{Q/G} \to H_D,\end{align*} $$

where $H_D$ is the component of the Hilbert scheme of $G(2,V_8)$ that contains the point $[D]$ defined by D. Both spaces have dimension $70$ and are smooth, respectively, at $[Q]$ and $[D]$ by Corollary 5.12. In order to show that $\sigma $ is dominant, it is enough to check that the induced morphism of tangent spaces is dominant. This is true because $H^0(G,{\mathcal {J}}_{Q/G}(2))$ and $H^0({\mathcal {N}}_{D/G})$ are both dominated by $H^0({\mathcal {I}}_D(2))$ , and the morphism from ${\mathcal {I}}_D(2)$ to ${\mathcal {N}}_{D/G}$ factorizes through ${\mathcal {J}}_{Q/G}(2)$ . This concludes the proof.

5.3 Grassmannian self-duality

Exactly as we constructed the singular quadric hypersurface $D_{Z_1}(v)\subset G(2,V_8)$ , there is another hypersurface $D_{Z_1}(v^\vee )\subset G(2,V_8^\vee )=G(6,V_8)$ . Because of Proposition 3.5, these two hypersurfaces are projectively isomorphic since they are constructed from lines contained in isomorphic Coble quartics. But one should also expect some projective duality statement analogous to Theorem 3.4. Of course, we cannot refer to classical projective duality since we want to consider $D_{Z_1}(v)$ and $D_{Z_1}(v^\vee )$ really as hypersurfaces in Grassmannians, not as subvarieties of the ambient projective spaces. It turns out that a version of projective duality in this setting (and for certain other ambient varieties than Grassmannians) was once proposed in [Reference ChaputCha07] (that remained unpublished). We will refer to it as Grassmannian duality.

The idea is the following. Consider, say, a hypersurface H in $G(2,V_8)$ (or any Grassmannian, but let us restrict to the case we are interested in). At a smooth point $h=[U_2]$ of H, the tangent space to H is a hyperplane in $T_hG(2,V_8)=\operatorname {\mathrm {Hom}}(U_2,V_8/U_2)$ or, equivalently, a line in the dual space $\operatorname {\mathrm {Hom}}(V_8/U_2,U_2)$ . If this line is generated by a surjective morphism, the kernel of this morphism is a four-dimensional subspace of $V_8/U_2$ . Equivalently, this defines a six-dimensional space $U_6$ such that $U_2\subset U_6\subset V_8$ . We get in this way a rational map from H to $G(6,V_8)$ , and we can define the Grassmannian dual $H^\vee $ as the image of this rational map. For more details, see [Reference ChaputCha07, section 1.6]. Chaput has a remarkable biduality theorem generalizing the classical statement, according to which duality for subvarieties of Grassmannians is an involution [Reference ChaputCha07, Theorem 2.1].

So this Grassmannian duality is perfectly natural, and we have:

Proof. Suppose that $U_2$ belongs to $D_{Z_1}(v)$ . By definition, this means that there exists $U_4\supset U_2$ (unique in general) such that

$$ \begin{align*}v \in U_2\wedge (\wedge^3V_8)+(\wedge^2U_4)\wedge (\wedge^2V_8).\end{align*} $$

If we mod out by $ \wedge ^2U_4$ , we get a tensor in $U_2\otimes \wedge ^3(V_8/U_4)\simeq U_2\otimes (V_8/U_4)^\vee $ , that is, a morphism from $V_8/U_4$ to $U_2$ . Generically, this morphism has full rank, and its kernel defines some $U_6\supset U_4$ . So we get a flag $(U_2\subset U_4\subset U_6)$ such that

(5.3) $$ \begin{align} v\in U_2\wedge (\wedge^2U_6)\wedge V_8+(\wedge^2U_4)\wedge (\wedge^2V_8). \end{align} $$

Proof. Using adapted basis, one checks that condition (5.3) implies that

$$ \begin{align*}v^\vee\in U_6^\perp\wedge (\wedge^2U_2^\perp)\wedge V_8^\vee+ (\wedge^2U_4^\perp)\wedge (\wedge^2V_8^\vee).\end{align*} $$

In particular, $v^\vee $ mod $U_6^\perp $ has rank at most four.

Proof. Using a basis of $V_8$ adapted to the flag $(U_2\subset U_4\subset U_6)$ , we can rewrite relation (5.3) in the form

$$ \begin{align*}v=e_1\wedge e_5\wedge e_6\wedge e_7+e_2\wedge e_5\wedge e_6\wedge e_8+v', \quad v'\in (\wedge^2U_4)\wedge (\wedge^2V_8),\end{align*} $$

where $U_2=\langle e_1,e_2\rangle $ and $U_6=\langle e_1,\ldots ,e_6\rangle $ . We can describe infinitesimal deformations of $U_2$ by some infinitesimal deformations of the vectors in the adapted basis, say $e_i\mapsto e_i+\epsilon \delta _i$ , and we must keep a similar relation. Modding out by $U_4$ , we only remain with the relation

$$ \begin{align*}\delta_1\wedge e_5\wedge e_6\wedge e_7+\delta_2\wedge e_5\wedge e_6\wedge e_8=0 \qquad \mathrm{mod} \; U_4,\end{align*} $$

which we can simply rewrite as $\delta _{18}=\delta _{27}$ . This relation describes the tangent hyperplane to $D_{Z_1}(v)$ at $U_2$ , as a hyperplane in $\operatorname {\mathrm {Hom}}(U_2,V_8/U_2)$ , orthogonal to the morphism $e_8^*\otimes e_1-e_7^*\otimes e_2$ . The kernel of this morphism is $U_6/U_2$ , and we are done.

These two lemmas together imply that $D_{Z_1}(v^\vee )$ coincides with the Grassmannian dual to $D_{Z_1}(v)$ . The proof of the theorem is complete.

Note that we can resolve the singularities of $D_{Z_1}(v)$ by considering flags $(U_2\subset U_4)$ as before, which gives a subvariety $\tilde {D}_{Z_1}(v)\subset Fl(2,4,V_8)$ . By considering the flags $(U_2\subset U_4\subset U_6)$ as in the proof of the previous statement, we obtain a subvariety $D_{Z_1}(v,v^\vee )\subset Fl(2,4,6,V_8)$ that resolves simultaneously the singularities of $D_{Z_1}(v)$ and $D_{Z_1}(v^\vee )$ . As for the Coble quartic, we get a diagram

The birational map $\tilde {D}_{Z_1}(v)\dashrightarrow \tilde {D}_{Z_1}(v^\vee )$ must be a flop, resolved by two symmetric contractions.