2. Reduction to the heavy/light moduli spaces $\overline{\mathrm{M}}_{p,q}$

The goal of this section is to reduce the study of derived category of $\overline{\mathcal{M}}_{0,n}$ to the study of full equivariant exceptional collections on heavy/light moduli spaces $\overline{\mathrm{M}}_{p,q}$ (see Notation 2.10.) In particular, we will prove our main Theorem 1.1 conditionally on the validity of Theorems 1.2, 3.20, 1.6, 1.11, and 1.12, which will be proved in subsequent sections. In fact, the same analysis proves the existence of full equivariant exceptional collections for a large range of Hassett spaces (see Theorem 2.11), which includes $\overline{\mathcal{M}}_{0,n}$ .

We first recall some subtle differences between various notions of exceptional collections ‘permuted by a group’, as investigated by Elagin [Reference ElaginEla09].

1. (a) $R\mathrm{Hom}(\mathcal{E}_\alpha^\bullet, \mathcal{E}_\beta^\bullet)=0$ if $\mathcal{E}_\alpha^\bullet$ and $\mathcal{E}_\beta^\bullet$ are in the same $\Gamma$ -orbit; and

2. (b) $K_0(X)$ is a permutation $\Gamma$ -lattice.

We construct exceptional collections on $\overline{\mathcal{M}}_{0,n}$ and $\overline{\mathrm{M}}_{p,q}$ that are equivariant in a stronger sense, which implies that the quotient stacks $[\overline{\mathcal{M}}_{0,n}/S_n]$ and $[\overline{\mathrm{M}}_{p,q}/S_p\times S_q]$ admit compatible semi-orthogonal decompositions.

A $\Gamma$ -invariant exceptional collection is called $\Gamma$ -equivariant if every complex $\mathcal{E}_\alpha^\bullet$ is quasi-isomorphic to a complex in $D^b(X/\Gamma_\alpha)\cong D^b_{\Gamma_\alpha}(X)$ , the bounded derived category of $\Gamma_\alpha$ -equivariant coherent sheaves on X, where $\Gamma_\alpha\subset \Gamma$ is the stabilizer of the isomorphism class of $\mathcal{E}_\alpha^\bullet$ .

We refer to [Reference ThomasonTho87, Reference Halpern-LeistnerHal15, Reference ElaginEla09] for background on equivariant sheaves and equivariant derived categories.

We need a stronger version of the equivariant Orlov blow-up lemma [Reference Castravet and TevelevCT20a, Lemma 7.2] to prove that our collections are $\Gamma$ -equivariant.

Let X be a smooth projective variety and let $Y_1,\ldots,Y_n\subset X$ be smooth transversal subvarieties of codimensions $l_1,\ldots,l_n$ . Let $\Gamma$ be a finite group acting on X and permuting $Y_1,\ldots,Y_n$ . For every subset $I\subset\{1,\ldots,n\}$ , we denote by $Y_I$ the intersection $\bigcap_{i\in I}Y_i$ . In particular, $Y_\emptyset=X$ .

Let $q:\,\tilde X\to X$ be the iterated blow-up of the (proper transforms of) $Y_1,\ldots,Y_n$ . Since the intersection is transversal, the blow-ups can be performed in any order. More canonically, the iterated blow-up is isomorphic to the blow-up of the ideal sheaf $\mathcal{I}_{Y_1}\cdot\cdots\cdot\mathcal{I}_{Y_n}$ . Let $E_i$ be the exceptional divisor over $Y_i$ for every $i=1,\ldots,n$ . For any subset $I\subset\{1,\ldots,n\}$ , let $E_I=q^{-1}(Y_I)=\bigcap_{i\in I}E_i$ ; in particular, $E_\emptyset=\tilde X$ . Let $i_I:\,E_I\hookrightarrow\tilde X$ be the inclusion and let $\Gamma_I\subset \Gamma$ be the subgroup of elements $g\in\Gamma$ such that $g.Y_I=Y_I$ . The group $\Gamma$ acts on $\tilde X$ and the morphism q is $\Gamma$ -equivariant.

The blocks are ordered in any linear order that respects the following partial order: $B_{I^1,J^1}$ precedes $B_{I^2,J^2}$ if $\sum_{i=1}^n J^1_iE_i\ge \sum_{i=1}^n J^2_iE_i$ (as effective divisors).

Finally, we need to check that the collection can be reordered in blocks permuted by $\Gamma$ . We first order the objects by decreasing value of $\sum J_i$ and observe that objects $B^{\beta_1}_{I^1,J^1}$ and $B^{\beta_2}_{I^2,J^2}$ with the same value of this sum are orthogonal unless $J^1=J^2$ , which in turn forces $I^1=I^2$ . Therefore, we simply arrange the objects with the same sum $\sum J_i$ into the same groups and in the same order as the objects in the collections $\{F_I^\beta\}$ , grouping together the objects permuted by $\Gamma$ .

Next, we need to review Hassett moduli spaces in genus 0.

Notation 2.7. Fix positive rational weights $\mathbf{a}=(a_1,\ldots,a_n)$ with each $a_i\le1$ and $\sum a_i\gt 2$ . Let $\overline{\mathrm{M}}_{\mathbf{a}}$ be the Hassett space of weighted pointed stable rational curves [Reference HassettHas03], i.e., pairs $(C,\sum a_ip_i)$ such that C is a nodal genus-0 curve, the points $p_i$ are smooth, the $\mathbb{Q}$ -line bundle $\omega_C(\sum a_ip_i)$ is ample, and the points $\{p_i\,:\,i\in I\}$ can become equal on C if and only if $\sum_{i\in I}a_i\le 1$ . For example, $\overline{\mathcal{M}}_{0,n}=\overline{\mathrm{M}}_{(1,\ldots,1)}$ . Every $\overline{\mathrm{M}}_{\mathbf{a}}$ is a smooth projective variety. The polytope of weights has a wall-and-chamber structure with walls

(2.1) \begin{equation}\sum\limits_{i\in I}a_i=1\quad \hbox{ for every subset $I\subset\{1,\ldots,n\}$}.\end{equation}

Moduli spaces within the interior of each chamber are isomorphic and carry the same universal family. Unlike in the variation of GIT quotients, there are no moduli spaces ‘on the walls’: reducing all weights slightly yields an isomorphic Hassett space (and universal family) in the interior of an adjacent chamber. There exist birational reduction morphisms $\overline{\mathrm{M}}_{\mathbf{a}}\to\overline{\mathrm{M}}_{{\mathbf{a}}'}$ every time the weight vectors are such that $a_i\ge a_i'$ for every i.

Notation 2.8. Now fix positive rational weights $\mathbf{a}=(a_1,\ldots,a_n)$ with each $a_i\le1$ and $\sum a_i=2$ . Let

\[\mathrm{X}_{\mathbf{a}}=(\mathbb{P}^1)^n\mathbin{/\mkern-6mu/}_{\mathbf{a}}\mathrm{PGL}_2 \]

be the GIT quotient of $(\mathbb{P}^1)^n$ by the diagonal action of $\mathrm{PGL}_2$ with respect to the fractional polarization $\mathcal{O}(a_1,\ldots,a_n)$ . The polytope of GIT weights can be identified with the boundary face $\sum a_i=2$ of the polytope of Hassett weights and inherits its chamber structure with walls (2.1), which encodes the variation of GIT (see [Reference Dolgachev and HuDH98]). Polarizations within the interior of each chamber have no strictly semistable points and carry a universal $\mathbb{P}^1$ -bundle with n sections. More generally, there always exists a morphism

\[\overline{\mathrm{M}}_{\mathbf{a}}:=\overline{\mathrm{M}}_{(a_1+\epsilon,\ldots,a_n+\epsilon)}\to \mathrm{X}_{\mathbf{a}}, \]

for some $0\lt \epsilon\ll1$ depending on ${\mathbf{a}}$ , which is an isomorphism if there are no strictly semistable points (i.e., away from the walls) [Reference HassettHas03, Theorem 8.2]. In the presence of strictly semistable points, $\mathrm{X}_{\mathbf{a}}$ can acquire isolated singularities (cones over $\mathbb{P}^r\times\mathbb{P}^s$ ), and $\overline{\mathrm{M}}_{\mathbf{a}}$ is the blow-up of these singularities (the ‘Kirwan resolution’). In any case, the universal family of a Hassett space $\overline{\mathrm{M}}_{\mathbf{a}}$ ‘near the GIT face’ has fibers with at most two irreducible components.

Proof. Suppose that $\mathcal{U}\to \overline{\mathrm{M}}_{\mathbf{a}}$ is a $\mathbb{P}^1$ -bundle. We claim that reducing all weights $a_i$ to $b_i$ such that $a_i\gt b_i$ for all i and $\sum b_i=2$ does not change stability. Indeed, suppose that $\sum_{i\in I}a_i\gt 1\geq\sum_{i\in I}b_i$ for some subset I. Then

\[\sum\limits_{i\in I^c}a_i\gt \sum\limits_{i\in I^c}b_i=2-\sum\limits_{i\in I}b_i\geq1. \]

So curves with two components, with points labeled by I (respectively, $I^c$ ) on the corresponding components, are stable, which contradicts the assumption.

We can reduce the weights $a_i$ slightly so that no partial sum is equal to 1. By [Reference HassettHas03, Proposition 5.4], the universal family $\mathcal{U}$ is the Hassett space $\overline{\mathrm{M}}_{(a_1, \ldots, a_n, \epsilon)}$ for $\epsilon\ll 1$ . Thus the universal family over $\mathcal{U}$ remains a $\mathbb{P}^1$ -bundle, and hence $\mathcal{U}$ is also a GIT quotient.

We denote $\overline{\mathrm{M}}_n:=\overline{\mathrm{M}}_{n,0}$ . Note that $\mathrm{X}_{p,q}\cong\overline{\mathrm{M}}_{p,q}$ if and only if p or q is odd. If $p=2r$ and $q=2s$ , then $\overline{\mathrm{M}}_{p,q}$ is a divisorial ‘Kirwan resolution’ of singularities of $\mathrm{X}_{p,q}$ . It has exceptional divisors $\mathbb P^{r+s-2}\times\mathbb P^{r+s-2}$ with normal bundle $\mathcal{O}(-1,-1)$ over each of the ${1\over 2}\binom{p}{r}{\binom{q}{s}}$ singular points of $\mathrm{X}_{p,q}$ .

We reduce the proof of Theorem 1.1 to the analysis of $\overline{\mathrm{M}}_{p,q}$ . We emphasize that all spaces $\overline{\mathrm{M}}_{p,q}$ are needed to prove Theorem 1.1 for $n\gg0$ . The same argument also proves the following more general result.

We note that full, exceptional collections on some of the Hassett spaces $\overline{\mathrm{M}}_{{\mathbf{a}}}$ have been constructed in [Reference Ballard, Favero and KatzarkovBFK19]. More generally, full, exceptional collections on many GIT quotients have been constructed in [Reference Halpern-LeistnerHal15, Reference Ballard, Favero and KatzarkovBFK19]. These collections do not satisfy any invariance requirements.

In the remainder of this section, we discuss the proof of Theorem 2.11. All along, we assume that the spaces $\overline{\mathrm{M}}_{p,q}$ have a full $S_p\times S_q$ -equivariant exceptional collection, which will be proved in the subsequent sections.

Proof of Theorem 2.11. Let $\Gamma_{\mathbf{a}}$ be the stabilizer of the vector $\mathbf{a}$ in the group $S_n$ . Choose $p\le n$ such that $a_1=\cdots=a_p\gt a_{p+1}$ . We consider four cases

\[\textrm{(i)}\ a_1=1, p\geq2;\quad \textrm{(ii)}\ a_1\lt 1, pa_1\geq2;\quad \textrm{(iii)}\ a_1=1, p=1;\quad \textrm{(iv)}\ a_j\geq{2\over j}\ \hbox{ for some }\textit{j};\]

and the following claim.

We prove cases (i)–(iv) and Claim 2.12 simultaneously by induction on the dimension of $\overline{\mathrm{M}}_{\mathbf{a}}$ using the existence of full $S_p\times S_q$ -equivariant exceptional collections on $\overline{\mathrm{M}}_{p,q}$ . Note that all statements are clear in dimension 1, as in this case $\overline{\mathrm{M}}_{{\mathbf{a}}}\cong \mathbb{P}^1$ .

2.1 Proof of Claim 2.12

We connect the weights by a $\Gamma_{\mathbf{a}}$ -invariant homotopy ${\mathbf{a}}(t)=t{\mathbf{a}}'+(1-t){\mathbf{a}}$ . The Hassett chamber structure is semi-continuous in the following sense: the reduction map $\overline{\mathrm{M}}_{b_1,\ldots,b_n}\to \overline{\mathrm{M}}_{b_1-\epsilon,\ldots,b_n-\epsilon}$ is an isomorphism for $0\lt \epsilon\ll1$ . It follows that our reduction map $\overline{\mathrm{M}}_{{\mathbf{a}}(0)}=\overline{\mathrm{M}}_{\mathbf{a}}\to \overline{\mathrm{M}}_{{\mathbf{a}}'}=\overline{\mathrm{M}}_{{\mathbf{a}}(1)}$ factors into the sequence of $\Gamma_{\mathbf{a}}$ -equivariant reduction maps of the following form: $f_t:\,\overline{\mathrm{M}}_{{\mathbf{a}}(t-\epsilon)}\to \overline{\mathrm{M}}_{{\mathbf{a}}(t)}$ for $0\lt \epsilon\ll1$ whenever there is a subset of indices $J\subset\{1,\ldots,n\}$ such that

(2.2) \begin{equation}\sum_{j\in J}a_j(0)\gt 1=\sum_{j\in J}a_j(t).\end{equation}

Arguing by induction on the number of wall crossings, it suffices to analyze a single map $f_t$ , described in [Reference HassettHas03] (see also [Reference Bergström and MinabeBM13]). Let $J_1,\ldots,J_s$ be a full list of subsets satisfying (2.2). The group $\Gamma_{\mathbf{a}}$ permutes the $J_i$ . The reduction map $f_t$ blows up the loci $\overline{\mathrm{M}}_{{\mathbf{a}}(t)}(J_i)$ in $\overline{\mathrm{M}}_{{\mathbf{a}}(t)}$ where the points in $J_i$ come together. Since $\sum_{j\in J_i}a_j(t)=1$ , the loci $\overline{\mathrm{M}}_{{\mathbf{a}}(t)}(J_i)$ intersect transversely: $\overline{\mathrm{M}}_{{\mathbf{a}}(t)}(J_i)\cap \overline{\mathrm{M}}_{{\mathbf{a}}(t)}(J_j)\neq\emptyset$ if and only if $J_i\cap J_j=\emptyset$ , in which case, the intersection is the locus at which points in $J_i$ (respectively, $J_j$ ), coincide. Clearly, the loci $\overline{\mathrm{M}}_{{\mathbf{a}}(t)}(J_i)$ and their intersections are themselves Hassett spaces of the form $\overline{\mathrm{M}}_{{\mathbf{a}}''}$ , with the vector ${\mathbf{a}}''$ having at least one weight 1 (for marked points that correspond to combined points indexed by subsets $J_1,\ldots,J_s$ ). These spaces have equivariant exceptional collections by cases (i) and (iii) in smaller dimension. The theorem follows by Lemma 2.6.

Cases (i) and (ii). If $p=2$ (case (i)), we apply Claim 2.12 to the weight vector ${\mathbf{a}}'=(1,1,\epsilon,\ldots,\epsilon)$ of the Losev–Manin space and use the main result of [Reference Castravet and TevelevCT20a]. Let $p\gt 2$ . In both (i) and (ii), we apply Claim 2.12 to the weight vector ${\mathbf{a}}'=(a+\epsilon_2, \ldots, a+\epsilon_2, b+\epsilon_2, \ldots, b+\epsilon_2)$ corresponding to $\overline{\mathrm{M}}_{p,q}$ with $q=n-p$ , $0\lt \epsilon_2\ll 2$ (see Notation 2.10). Concretely, if $q=0$ , then $a={2}/{p}$ and we choose $\epsilon_2$ such that $a+\epsilon_2=({2}/{p})+\epsilon_2\lt a_1$ , while if $q\gt 0$ , then $a=({2}/{p})-\epsilon_1$ , $b={p\epsilon_1}/{q}$ ( $0\lt \epsilon_1\ll 1$ ) and we choose $\epsilon_2$ such that $a+\epsilon_2\lt a_1$ and $b+\epsilon_2\lt a_n$ .

Case (iii). Let $A=a_2+\cdots+a_n\gt 1$ . Consider $a_1'=1$ , $a_i'=a_i/(A-\epsilon)$ for $i\ge2$ for some fixed $0\lt \epsilon\lt \min(a_n,A-1)$ . By Claim 2.12, it suffices to show that $\overline{\mathrm{M}}_{{\mathbf{a}}'}$ admits a $\Gamma_{\mathbf{a}}$ -equivariant exceptional collection. Note that $\overline{\mathrm{M}}_{{\mathbf{a}}'}\cong\mathbb{P}^{n-3}$ by [Reference HassettHas03, Section 6.2] since $\sum_{i\geq 2, i\neq j} a'_i\leq\sum_{i=2}^{n-1}a'_i=(A-a_n/A-\epsilon)\lt 1$ for all $j\geq2$ . Note that the standard collection $\mathcal{O},\mathcal{O}(1),\ldots,\mathcal{O}(n-3)$ on $\mathbb{P}^{n-3}$ is invariant under any group. For equivariance, we need a bit more. All groups appearing are contained in the symmetric group $S_{n-1}$ that acts on $\mathbb P^{n-3}$ by permuting $n-1$ fixed points in general linear position. The homomorphism $S_{n-1}\to\mathrm{PGL}_{n-2}$ can be factored through $\mathrm{GL}_{n-2}$ , and therefore $\mathcal{O}(1)$ is an $S_{n-1}$ -linearized line bundle. The corresponding action on $k^{n-2}$ is the irreducible $(n-2)$ -dimensional representation of $S_{n-1}$ , [Reference Keel and TevelevKT09, Lemma 2.3].

Case (iv). We can assume without loss of generality that j is the largest index with the property $a_j\ge2/j$ . The statement follows from case (i) if $a_j=1$ . If $a_j\lt 1$ , we reduce to case (ii) using Claim 2.12 by taking the second weight vector $a_1'=\cdots =a_j'=a_j$ , $a_i'=a_i$ for $i\gt j$ .

3. Derived category of the moduli stack $\mathcal{M}_n$ of n points on $\mathbb P^1$

The goal of this section is to prove Theorem 1.2, which gives an equivariant full exceptional collection in $D^b(\overline{\mathrm{M}}_n)$ for n odd. As a GIT quotient without strictly semistable points, $\overline{\mathrm{M}}_n$ is isomorphic to an open substack of the quotient stack $\mathcal{P}_n=[(\mathbb P^1)^n/\mathrm{PGL}_2]$ of $\mathbb P^1$ -bundles with n sections.

The proof relies on the windows theorem of Halpern and Leistner.

For a choice of a Kempf–Ness (KN) stratification of the unstable locus $X^{us}$ with data $Z_i$ , $S_i$ , $\lambda_i$ , and $\sigma_i:Z_i\hookrightarrow S_i$ , define the integers

\[\eta_i=\text{weight}_{\lambda_i}\det(N^\vee_{S_i|X})_{|Z_i}\gt 0\]

and choose integers $w_i$ . Define the full subcategory $\mathcal{G}_w$ of all objects $F\in D^b[X/G]$ such that the cohomology sheaves of $\sigma_i^*F$ have weights in $[w_i, w_i+\eta_i)$ for all i. Then the restriction functor $i^*:\mathcal{G}_w\rightarrow D^b[X^{ss}/G]$ is an equivalence of categories.

In order to apply the windows theorem, we have to:

1. (1) compute a full exceptional collection in the derived category of the quotient stack $[(\mathbb P^1)^n/\mathrm{PGL}_2]$ (Theorem 3.6);

2. (2) compute the Kempf–Ness stratification and all relevant data in our setup for a wide range of GIT quotients (Theorem 3.15); and

3. (3) choose integers $w_K$ wisely to squeeze as many objects as possible from the exceptional collection on the quotient stack $[(\mathbb P^1)^n/\mathrm{PGL}_2]$ into the window $\mathcal{G}_w$ and show that the remaining objects of the exceptional collection, when restricted to the semistable locus, are generated by objects in the exceptional collection of the window; this is done by induction on the score (3.4), a useful concept that will be refined and generalized in subsequent sections.

Relatively simple calculations in this section serve as a model for more convoluted applications of the windows theory in the subsequent sections.

In what follows, n is arbitrary and $\mathcal{P}_n=[(\mathbb P^1)^n/\mathrm{PGL}_2]$ . We first analyze the derived category of $\mathcal{P}_n$ . An associated $\mathbb P^1$ -bundle of a $\mathrm{PGL}_2$ -torsor is the ‘universal $\mathbb P^1$ -bundle’, i.e., a representable morphism $\pi=\pi_{n+1}:\,\mathcal{P}_{n+1}\to\mathcal{P}_n$ induced by the first projection $(\mathbb P^1)^{n+1}=(\mathbb P^1)^n\times \mathbb P^1\to(\mathbb P^1)^n$ .

We denote $\sigma_1,\ldots,\sigma_n:\,\mathcal{P}_n\to\mathcal{P}_{n+1}$ the universal sections. They are representable morphisms induced by the big diagonals $(\mathbb P^1)^{n}\cong\Delta_{i,n+1}\hookrightarrow(\mathbb P^1)^{n+1}$ , which are defined by $x_i=x_{n+1}$ , for $i=1,\ldots,n$ . We identify the category of coherent sheaves on $\mathcal{P}_n$ with the category of $\mathrm{PGL}_2$ -equivariant coherent sheaves on $(\mathbb P^1)^n$ and likewise for their bounded derived categories. For $(i_1,\ldots,i_n)\in\mathbb{Z}^n$ with $\sum i_k$ even, the line bundle $\mathcal{O}_{(\mathbb P^1)^n}(i_1,\ldots,i_n)$ has a unique $\mathrm{PGL}_2$ -linearization and thus descends to $\mathcal{P}_n$ .

Definition 3.3. Let N be the set $\{1,\ldots,n\}$ . Fix a subset $E\subset N$ with $e=|E|$ and an integer $l\ge0$ such that $e+l$ is even. For $j=1,\ldots,n$ , let $i_j$ be 1 if $j\in E$ and 0 otherwise. Let

\[F_{l,E}=\pi_*N_{l,E},\]

where $N_{l,E}=\mathcal{O}(i_1,\ldots,i_n,l)$ is a line bundle on $\mathcal{P}_{n+1}$ .

Corollary 3.5. There are isomorphisms of $\mathrm{SL}_2$ -equivariant vector bundles

\[F_{l,E}^\vee\cong \pi_*\mathcal{O}(-i_1,\ldots,-i_n,l)\cong \mathcal{O}(-i_1,\ldots,-i_n)\otimes V_l.\]

If n is even, then $F_{l,E}\cong F_{l,E^c}^\vee\otimes F_{0,N}$ , where $E^c=N\setminus E$ .

Here (and everywhere), $F^\vee$ denotes the dual complex of $F\in D^b(X)$ .

Theorem 3.6. $D^b(\mathcal{P}_n)$ has a full strong $S_n$ -equivariant exceptional collection

\[\{ F_{l,E} : \hbox{$l+e$ even}\}\]

of infinitely many vector bundles. The collection is arranged into blocks $F_{l,e}$ given by the $S_n$ -orbits, which are ordered by increasing e, and for fixed e, arbitrarily.

The forgetful map $\pi_i:\,\mathcal{P}_{n+1}\to\mathcal{P}_{n}$ has the following properties:

\[(L\pi_i)^*F_{l,E}\cong F_{l,E},\quad (R\pi_i)_*F_{l,E}^\vee\cong\begin{cases}0 & i\in E,\cr F_{l,E}^\vee &i\not\in E.\cr\end{cases}\]

It remains to prove fullness, i.e., that any complex $F\in D^b(\mathcal{P}_n)$ can be obtained by finitely many extensions starting with objects in the exceptional collection $\{F_{l,E}\}$ . This is a special case of [Reference ElaginEla09, Theorem 2.10]. Alternatively, we give a simple ad hoc argument.

Viewing $F\in D^b(\mathcal{P}_n)$ as a bounded complex of coherent sheaves on $(\mathbb P^1)^n$ , let k be the maximum index such that (in $D^b((\mathbb P^1)^n)$ ) we have

\[R\mathrm{Hom}(L_k,F)\ne0\]

or we let $k=0$ if $R\mathrm{Hom}(L_j,F)=0$ for every j. We argue by induction on k. If $k=0$ , then $F\cong 0$ because $\mathcal{O}(i_1,\ldots,i_n)$ form a full exceptional collection in $D^b((\mathbb P^1)^n)$ . Otherwise, consider the left mutation triangle $F'\to R\mathrm{Hom}(L_k,F)\otimes L_k\to F\to$ in $D^b((\mathbb P^1)^n)$ . We first prove that this is an exact triangle in $D^b(\mathcal{P}_n)$ . Since F is an $\mathrm{SL}_2$ -equivariant complex and $L_k$ is an $\mathrm{SL}_2$ -equivariant line bundle, it suffices to prove that $-\text{Id}\in\mathrm{SL}_2$ acts trivially on $R\mathrm{Hom}(L_k,F)\otimes L_k$ . This is clear since $-\text{Id}\in\mathrm{SL}_2$ acts either trivially or by multiplication by $-1$ on both terms of the tensor product. It follows that this triangle is $\mathrm{PGL}_2$ -equivariant. To finish the proof, we need to prove that F’ and $R\mathrm{Hom}(L_k,F)\otimes L_k$ are generated by $F_{l,E}$ . For F’, this follows by induction, since $R\mathrm{Hom}(L_k,F')=0$ . Here $R\mathrm{Hom}(L_k,F)\otimes L_k$ is isomorphic to a direct sum of the vector bundles $F_{l,E}$ : the $\mathrm{SL}_2$ -representation $R\mathrm{Hom}(L_k, F)$ is a direct sum of irreducible representations $V_{l_i}$ , for integers $l_i\geq0$ . Since $R\mathrm{Hom}(L_k, F)\otimes L_k$ is a $\text{PGL}_2$ -equivariant vector bundle, we must have $\sum_{i=1}^r l_i+e$ is even and we are done by Lemma 3.4.

Proposition 3.7. On $\mathcal{P}_n$ , we have exact sequences

\begin{align*}0&\to F_{l-1,E\setminus k}\to F_{l,E}\to Q^k_{l,E}\to 0\quad \hbox{for$k\in E$},\\0&\to F^\vee_{l-1,E\cup\{k\}}\to F^\vee_{l,E}\to S^k_{l,E}\to 0\quad \hbox{for $k\not\in E$},\end{align*}

where $Q^k_{l,E}=\mathcal{O}(i_1,\ldots,i_k+l,\ldots,i_n)$ and $S^k_{l,E}=\mathcal{O}(-i_1,\ldots,-i_k+l,\ldots,-i_n)$ .

On $(\mathbb{P}^1)^n\times\mathbb{P}^1$ , we have $\omega_\pi=\mathcal{O}(0,\ldots, 0, -2)$ and

\[\mathcal{O}(\sigma_j)=\mathcal{O}(0,\ldots, 0, 1, 0,\ldots,0,1)\]

with 1 in position j and $n+1$ , since the section $\sigma_j$ is the $\Delta_{j, n+1}$ diagonal. For $i\neq j$ , we have on $(\mathbb{P}^1)^n$

\[\sigma_i^*\mathcal{O}(\sigma_j)=\mathcal{O}(0,\ldots, 0, 1, 0,\ldots, 0, 1, 0,\ldots,0),\]

with 1 in position j and j, while when $i=j$ we have

\[\sigma_j^*\mathcal{O}(\sigma_j)=\mathcal{O}(0,\ldots, 0, 2, 0,\ldots, 0)\]

with 2 in position j. Hence, the tautological line bundle $\psi_i=\sigma_i^*\omega_\pi$ on $\mathcal{P}_n$ is identified with the $\mathrm{PGL}_2$ -equivariant line bundle $\mathcal{O}(0,\ldots,0,-2,0,\ldots,0)$ , where $-2$ is in position i. The line bundle $\delta_{ij}=\sigma_i^*\mathcal{O}(\sigma_j)$ is identified with the $\mathrm{PGL}_2$ -equivariant line bundle $\mathcal{O}(0,\ldots,0,1,0,\ldots,0,1,0,\ldots,0)$ , where 1 is in position i and j. Furthermore, on $\mathcal{P}_{n+1}$ ,

\[N_{l,E}\cong \omega^{\otimes {({(e-l)}/{2}})}_{\pi}(E),\]

where, by abuse of notation, we let $\mathcal{O}(E)=\mathcal{O}(\sum_{j\in E}\sigma_j)$ .

Proof. By the Grothendieck–Verdier duality (see Remark 3.12), we have

(3.1) \begin{equation}R\pi_*\mathcal{O}(-i_1,\ldots,-i_n,-l-2)\cong F_{l,E}^\vee[-1],\end{equation}

where $i_j=1$ if $j\in E$ and $i_j=0$ otherwise. But by Corollary 3.5 when $-l-2\geq 0$ , $F_{-l-2,E}^\vee\cong R\pi_*\mathcal{O}(-i_1,\ldots,-i_n,-l-2)$ .

In the remainder of this section, we study derived categories of open substacks of $\mathcal{P}_n$ , with an eye toward proving Theorem 1.2.

Proof of Proposition 3.13. We have $\mathcal{U}=[U/\mathrm{PGL}_2]$ for an open equivariant subset $U\subset(\mathbb P^1)^n$ . It is enough to show that any equivariant coherent sheaf F on U can be obtained by a finite number of equivariant extensions starting with restrictions of the vector bundles $F_{l,E}$ . This follows from Theorem 3.6 since F is a restriction of an equivariant coherent sheaf on $(\mathbb P^1)^n$ (see, e.g., [Reference ThomasonTho87]). For the rest of the proposition, we restrict isomorphisms on $\mathcal{P}_n$ to $\mathcal{U}$ .

Open substacks $\mathcal{U}\subseteq\mathcal{P}_n$ include all Hassett spaces when the universal family is a $\mathbb P^1$ -bundle. More precisely, fix weights ${\mathbf{a}}=(a_1,\ldots,a_n)$ with $\sum a_i=2$ . Recall from the introduction that the GIT quotient $\mathrm{X}_{\mathbf{a}}$ is a quotient of the semistable locus $(\mathbb{P}^1)^n_{ss}$ for the fractional $\mathrm{PGL}_2$ -polarization given by ${\mathbf{a}}$ . We denote by $\mathcal{M}_{\mathbf{a}}$ the stack quotient $[(\mathbb{P}^1)^n_{ss}/\mathrm{PGL}_2]$ , which is an open substack of $\mathcal{P}_n$ . If there are no strictly semistable points then $\mathrm{PGL}_2$ acts on $(\mathbb{P}^1)^n_{ss}$ freely and $\mathcal{M}_{\mathbf{a}}\cong\overline{\mathrm{M}}_{\mathbf{a}}\cong\mathrm{X}_{\mathbf{a}}$ . In particular, for every partition $P\coprod Q=\{1,\ldots,n\}$ with $p=|P|\ge 2$ and $q=|Q|$ , we define an open substack $\mathcal{M}_{p,q}\subset \mathcal{P}_n$ of $\mathbb P^1$ bundles such that at most $p/2$ sections indexed by P (heavy points) are allowed to coincide and if $p=2r$ , then r heavy points are allowed to coincide with at most $q/2$ points indexed by Q (light points). The corresponding Hassett space is $\overline{\mathrm{M}}_{p,q}$ and the GIT quotient is $\mathrm{X}_{p,q}$ (see Notation 2.10). We have $\mathcal{M}_{p,q}=\overline{\mathrm{M}}_{p,q}=\mathrm{X}_{p,q}$ unless both p and q are even. We use the notation $\mathcal{M}_n$ , $\overline{\mathrm{M}}_n$ , and $\mathrm{X}_n$ if $q=0$ .

Proof. We apply the windows Theorem 3.1 for $X=(\mathbb{P}^1)^n$ and $G=\mathrm{PGL}_2$ . Up to conjugation, a one-parameter subgroup $\lambda$ has the form

\[\lambda(t)=\left[\begin{array}{cccc}t & & & 0\\0 && & t^{-1}\end{array} \right].\]

For every subset $K\subseteq N$ , consider the $\lambda$ -invariant point $z_K=(p_1,\ldots, p_n)$ , with $ p_i=0=[0:1]$ if $i\notin K$ and $p_i=\infty=[1:0]$ otherwise. Then $\text{weight}_{\lambda}\mathcal{O}(1)_{|x}=1$ if $x=0$ and $\text{weight}_{\lambda}\mathcal{O}(1)_{|x}=-1$ if $x=\infty$ . If $\mathcal{L}=\mathcal{O}(a_1,\ldots, a_n)$ , it follows that

(3.2) \begin{equation}\text{weight}_{\lambda}\mathcal{L}_{|z_K}=\sum_{i\in K^c} a_i-\sum_{i\in K} a_i.\end{equation}

Let $\Delta_K$ be a diagonal in $(\mathbb{P}^1)^n$ consisting of points $(p_1,\ldots, p_n)$ such that for all $i\in K$ the points $p_i$ are equal. Let $S_K\subset\Delta_K$ be the complement to the union of smaller diagonals. The KN strata of $X^{us}$ are given by $Z_K=\{z_K\}$ , $S_K$ , and the canonical inclusions $\sigma_K:\,Z_K\hookrightarrow S_K$ , for all $K\subseteq N$ such that $\sum_{i\in K} a_i\gt 1$ . The destabilizing 1-PS is $\lambda$ . Next, we compute $\det(N^\vee_{\Delta_K|(\mathbb{P}^1)^n})$ . We may assume that $n\in K$ . As $\Delta_K$ is a complete intersection in X of big diagonals $\Delta_{jn}$ , for all $j\in K\setminus\{n\}$ , it follows that

\[N^\vee_{\Delta_K|(\mathbb{P}^1)^n}\cong\bigoplus\limits_{j\in K\setminus\{n\}}\mathcal{O}(-\Delta_{jn})_{|\Delta_K},\ \\det(N^\vee_{\Delta_K|(\mathbb{P}^1)^n})\cong\mathcal{O}\bigg({-}\sum\limits_{j\in K\setminus\{n\}}\Delta_{jn}\bigg)_{|\Delta_K}.\]

It follows from (3.2) that

(3.3) \begin{equation}\eta_K:=\text{weight}_{\lambda}{\det(N^\vee_{\Delta_K|(\mathbb{P}^1)^n})_{|z_K}}=2(|K|-1).\end{equation}

Theorem 3.15 then follows from Theorems 3.6 and 3.1 and the following Lemma 3.16.

Proof of Theorem 1.2. We consider the following score function:

(3.4) \begin{equation}S(l,E)=l+\min(e,p-e).\end{equation}

We need to prove that the vector bundles $F_{l,E}$ with $S(l,E)\leq r-1$ form a full exceptional collection. First, we prove exceptionality. We prove that condition $S(l,E)\leq r-1$ implies that the weights of the bundles $F_{l,E}$ fit in a window as in Theorem 3.1. Lemma 3.16 implies that the maximum weight of $F_{l,E}$ over all subsets K with fixed $|K|=k$ is $\text{min}\{l+e, l+2p-2k-e\}$ . Similarly, the minimum weight of $F_{l,E}$ is $-\text{min}\{l+e, l+2k-e\}$ . The conditions in Theorem 3.15 for existence of a window are equivalent to requiring that for any pairs (l,E), (l’, E’), if we let $e=|E|$ and $e'=|E'|$

\[\text{min}\{l+e, l+2k-e\}+\text{min}\{l'+e', l'+2(p-k)-e'\}\lt \eta_k=2(k-1),\]

for all $k\ge r+1$ . We now prove that this is the case for the list of pairs (l,E) in Theorem 1.2. We consider three cases.

Case I: $e,e '\le r$ . By assumption, $l+e, l'+e'\le r-1$ . Then, $\text{min}\{l+e, l+2k-e\}+\text{min}\{l'+e', l'+2(p-k)-e'\}\leq(l+e)+(l'+e')\leq 2r-2\lt 2(k-1)$ for all $k\ge r+1$ .

Case II: $e,e '\geq r+1$ . By assumption, $l+p-e, l'+p-e'\leq r-1$ . We have $\text{min}\{l+e, l+2k-e\}+\text{min}\{l'+e', l'+2(p-k)-e'\}\leq(l+2k-e)+(l'+2(p-k)-e')\leq2(r-1)\lt 2(k-1) $ for all $k\ge r+1$ .

Case III: $e \le r$ and $e' \geq r+1$ (or the opposite). By assumption, $l+e\leq r-1$ and $l'+p-e'\leq r-1$ . It follows that $\text{min}\{l+e, l+2k-e\}+\text{min}\{l'+e', l'+2(p-k)-e'\}\leq(l+e)+(l'+2(p-k)-e')\leq 2(r-1)\lt 2(k-1)$ for all $k\ge r+1$ . We finish by applying Theorem 3.6.

Next, we prove fullness. By Proposition 3.13, it suffices to prove the following claim.

For simplicity, let $\mathcal{C}$ be the collection of vector bundles $F_{l,E}$ in Theorem 1.2, i.e., those in the range $S(l,E)\le r-1$ . We prove the equivalent dual statement that every vector bundle $F^\vee_{l,E}$ on $\overline{\mathrm{M}}_p$ is generated by the dual collection $\mathcal{C}^\vee$ . We argue by induction on the score and for a fixed score, by induction on l. Clearly, the statement holds when $S(l,E)\leq r-1$ . Let $a\geq r$ and assume that all the bundles $F^\vee_{l,E}$ with $S(l,E)\lt a$ are generated by $\mathcal{C}^\vee$ . Let $F^\vee_{l,E}$ be a bundle such that $S(l,E)=a$ . We consider two cases: $e=|E|\leq r$ and $e=|E|\geq r+1$ .

Assume $e\leq r$ . Let $I\subseteq\{1,\ldots, p\}$ with $I\cap E=\emptyset$ and $|I|=r+1$ . We consider the Koszul complex for the diagonal $\Delta=\Delta_{I\cup\{x\}}\subseteq (\mathbb P^1)^p\times \mathbb{P}^1_x$ ,

\begin{align*}0\leftarrow\mathcal{O}_{\Delta}\leftarrow\mathcal{O}&\leftarrow\bigoplus_{i\in I}\mathcal{O}(-{\mathbf{e_i}}-{\mathbf{e_x}})\leftarrow\bigoplus_{i,k\in I}\mathcal{O}(-{\mathbf{e_i}}-{\mathbf{e_k}}-2{\mathbf{e_x}})\leftarrow\cdots\\&\leftarrow\mathcal{O}\bigg({-}\sum_{i\in I}{\mathbf{e_i}}-(r+1){\mathbf{e_x}}\bigg)\leftarrow 0.\end{align*}

Here, $\{{\mathbf{e_i}}\}$ for $i\in P$ is the standard basis of $\mathrm{Pic}(\mathbb{P}^1)^{p}\cong\mathbb{Z}^{p}$ .

We tensor this resolution with $\mathcal{O}(-\sum_{j\in E}{\mathbf{e_j}}+l {\mathbf{e_x}})$ and take derived pushforwards of its terms via the projection map $\pi:\,(\mathbb P^1)^{p+1}\to(\mathbb P^1)^p$ , which we then restrict to the semistable locus in $(\mathbb P^1)^p$ . Since $\pi(\Delta)$ is in the unstable locus, we obtain the following objects:

\[0\quad F^\vee_{l, E}\quad \bigoplus_{i\in I} F^\vee_{l-1, E\cup\{i\}}\quad \cdots\quad \bigoplus_{J\subseteq I, |J|=j} F^\vee_{l-j, E\cup J}\quad \cdots\quad \bigoplus_{J\subseteq I, |J|=l} F^\vee_{0, E\cup J}\quad 0\]

\[\bigoplus_{J\subseteq I, |J|=l+2} F^\vee_{0, E\cup J}[-1]\quad \cdots\quad \bigoplus F^\vee_{j-l-2, E\cup J}[-1]\quad \cdots\quad F^\vee_{r-l-1, E\cup I}[-1],\]

where the terms $F^\vee_{l-j, E\cup J}$ appear as $R^0\pi_*$ as long as $0\leq j\leq l$ , the 0 term appears when $j=l+1$ (which happens if $l\leq r$ ; if $l\gt r$ , then the last term that appears is $F^\vee_{l-r-1, E\cup I}$ ), while the terms $F^\vee_{j-l-2, E\cup J}$ appear as $R^1\pi_*$ in the range $l+2\leq j\leq r+1$ by the Grothendieck–Verdier duality (see 3.1).

We claim that the first object $F^\vee_{l, E}$ is generated by the remaining objects, which are all generated by $\mathcal{C}^\vee$ by induction. It will follow that $F^\vee_{l, E}$ is generated by $\mathcal{C}^\vee$ . The first claim is a special case of the following lemma, which is used repeatedly in the remainder of the paper.

We now prove that the remaining terms are generated by $\mathcal{C}^\vee$ by induction. Consider the two types of terms.

1. (a) The terms $F^\vee_{\tilde l,\tilde E}=F^\vee_{l-j, E\cup J}$ have scores $S(\tilde l,\tilde E)=(l-j)+\min\{e+j,p-e-j\}\leq (l-j)+(e+j)=l+e=S(l,E)=a$ , and we are done by induction, since $\tilde l\leq l$ , with equality if and only if $(\tilde l,\tilde E)=(l,E)$ .

2. (b) The terms $F^\vee_{\tilde l,\tilde E}=F^\vee_{j-l-2, E\cup J}$ ( $l+2\leq j\leq r+1$ ) have scores $S(\tilde l,\tilde E)=(j-l-2)+\min\{e+j,p-e-j\}\leq (j-l-2)+(p-e-j)=p-e-l-2=p-a-2$ since $S(l,E)=l+e=a$ . But $p-a-2\lt a$ since $a\geq r$ . It follows that these terms are generated by $\mathcal{C}^\vee$ by induction.

Assume that $e\geq r+1$ . Let $I\subseteq E$ , $|I|=r+1$ and let $E'=E\setminus I$ . As before, we tensor the above Koszul resolution of $\Delta_{I\cup\{x\}}\subseteq (\mathbb{P}^1)^p\times\mathbb{P}^1$ with $\mathcal{O}(-\sum_{k\in E'}{\mathbf{e_k}}+(r-1-l) {\mathbf{e_x}})$ , take derived pushforwards via the projection map $\pi$ and apply Lemma 3.18:

\[0\quad F^\vee_{r-1-l, E'}\quad \cdots\quad \bigoplus_{J\subseteq I, |J|=j} F^\vee_{r-1-l-j, E'\cup J}\quad \cdots \bigoplus_{J\subseteq I, |J|=r-1-l} F^\vee_{0, E'\cup J}\quad 0\]

\[\bigoplus_{J\subseteq I, |J|=r+1-l} F^\vee_{0, E'\cup J}\quad \cdots\bigoplus_{J\subseteq I, |J|=j} F^\vee_{l+j-r-1, E'\cup J}\quad \cdots\quad F^\vee_{l, E},\]

where the terms $F^\vee_{r-1-l-j, E'\cup J}$ appear as $R^0\pi_*$ for $0\leq j\lt r-l$ , the 0 term appears when $j=r-l$ , and the terms $F^\vee_{l+j-r-1, E'\cup J}$ appear as $R^1\pi_*$ for $r+1-l\leq j\lt r+1$ . We consider the two types of terms and compare their scores with $a=S(l,E)=l+p-e$ .

1. (a) The terms $F^\vee_{\tilde l,\tilde E}=F^\vee_{r-1-l-j, E'\cup J}$ have scores $S(\tilde l,\tilde E)=(r-1-l-j)+\min\{e'+j,p-e'-j\}\leq (r-1-l-j)+(e'+j)=e-l-2$ (where $e'=|E'|=e-r-1$ ). But since we assume that $a=l+p-e\geq r$ , we have $e-l\leq r+1$ ; hence, $S(\tilde l,\tilde E)\leq r-1$ , i.e., $F^\vee_{\tilde l,\tilde E}$ is in $\mathcal{C}^\vee$ .

2. (b) The terms $F^\vee_{\tilde l,\tilde E}=F^\vee_{l+j-r-1, E'\cup J}$ have scores $S(\tilde l,\tilde E)=(l+j-r-1)+\min\{e'+j,p-e'-j\}\leq (l+j-r-1)+p-e'-j=l+p-e=S(l,E),$ and we are done by induction, since $\tilde l=l+j-r-1\leq l$ (as $j\leq r+1$ ) with equality if and only if $j=r+1$ , i.e., $J=I$ (i.e., $(\tilde l,\tilde E)=(l,E)$ ).

The order of the blocks given by the $S_p\times S_q$ -orbits $F_{l,e_p,e_q}$ is first by increasing $e_q$ , then by increasing $e_p$ , and finally arbitrary in l.

4. Functions of subsets, score, and inequalities for pairs (l, E)

In this section, we collect numerical functions of subsets and inequalities used throughout the paper. The reader may wish to read Notation 4.1 and 4.3 and skip the rest of this section, referring back to it as needed.

Let $l\in\mathbb{Z}$ and let E, T be subsets of the set $N=\{1,\ldots, n\}$ . Unless otherwise noted, in this section we do not impose any condition on the parity of $l+e$ . In all the sections that follow, we will always assume that $e+l$ is even, as all the considered pairs (l,E) belong to one of the groups 1A, 1B, and 2 or 2A and 2B. The reason we do not impose parity conditions on $l+e$ in this section is for induction purposes (proof of Corollary 4.11).

Notation 4.1. We denote

(4.1) \begin{align}f_{T,E,l}&=|E\cap T|-{e-l\over 2}={|E\cap T|-|E\cap T^c|+l\over2},\quad T^c=N\setminus T,\end{align}

(4.2) \begin{align}\alpha_{T,E,l}&=\max\{0,-f_{T,E,l}\}.\end{align}

(4.3) \begin{align}m_{T,E,l}&=\max\{0,f_{T,E,l}\}.\end{align}

We use the notation $f_T$ , $\alpha_T$ , and $m_T$ when E and l are clear from the context.

In the first case, we have $|Q|=q=2s+1\ge 1$ , and in the second case we have $|\tilde{Q}|=q+1=2s+2\ge 0$ . In particular, q is always odd and can be equal to $-1$ in the second case.

For $l\in\mathbb{Z}$ , $E\subseteq P\cup Q$ (respectively, $E\subseteq P\cup \tilde Q$ ), we define the score

\begin{align*}S(l,E)&=l+\min\{e_p,p-e_p\}+\min\{e_q,q-e_q\}\\(\hbox{ respectively,}\ S'(l,E)&=l+\min\{e_p,p-e_p\}+\min\{e_q,q+1-e_q\}).\end{align*}

We use different notation for S(l,E) and S’(l,E) because sometimes E can be regarded as a subset in both Q or $\tilde{Q}$ . We emphasize that whenever we consider $E\subseteq P\cup Q$ , we assume that $q=2s+1\geq1$ , while whenever we consider $E\subseteq P\cup \tilde Q$ , we assume that $q=2s+2\geq0$ .

For the purposes of this section, we introduce the following.

1. (i) For $l\geq0$ , $E\subseteq P\cup Q$ , and (l,E) satisfying conditions 1A, or 1B, or 2 of Theorem 1.6 and Remark 1.14, we have $S(l,E)\leq r+s-1$ .

2. (ii) For $l\geq0$ , $E\subseteq P\cup \tilde Q$ , and (l,E) satisfying conditions 1A, or 1B, or 2A, or 2B of Theorem 1.11 and Remark 1.14, we have $S'(l,E)\leq r+s$ .

Moreover, equality in part (i) holds if and only if either $l+e_p=r-1$ , $e_q=s$ (group 1A), or $l+e_p=r-1$ , $e_q=s+1$ (group 1A), or $l+(p-e_p)=r-1$ , $e_q=s$ (group 1B), or $l+(p-e_p)=r-1$ , $e_q=s+1$ (group 1B), or $e_p=r$ , $l+e_q=s-1$ (group 2), or $e_p=r$ , $l+(q-e_q)=s-1$ (group 2).

Similarly, equality in part (ii) holds if and only if either $l+e_p=r-1$ , $e_q=s+1$ (group 1A), or $l+(p-e_p)=r-1$ , $e_q=s+1$ (group 1B), or $e_p=r$ , $l+q+1-e_q=s$ (group 2A), or $e_p=r$ , $l+e_q=s$ (group 2B).

Lemma 4.6. Let $l\in\mathbb{Z}$ (positive or negative), $E\subseteq P\cup \tilde Q$ . Then

(4.4) \begin{align}\max_{P\cup \tilde Q=T\sqcup T^c} f_{T,E,l}&=S'(l,E)/2,\end{align}

(4.5) \begin{align}\max_{P\cup \tilde Q=T\sqcup T^c}- f_{T,E,l}&=S'(l,E)/2-l.\end{align}

Here, we assume that $T=T_p\coprod T_q$ (so $T^c=T^c_p\coprod T^c_q$ , where $T^c_p=P\setminus T_p$ and $T^c_q=Q\setminus T_q$ ) is a subset of markings split into heavy and light markings such that $|T_p|=r$ and $|T_q|=s+1$ . Equality in (4.4) is attained if and only if $(T_p\subseteq E_p$ or $E_p\subseteq T_p)$ and $(T_q\subseteq E_q$ or $E_q\subseteq T_q)$ , while equality in (4.5) is attained if and only if $(T^c_p\subseteq E_p$ or $E_p\subseteq T^c_p)$ and $(T^c_q\subseteq E_q$ or $E_q\subseteq T^c_q)$ .

Lemma 4.7. Let $E\subseteq P\cup \tilde Q$ . Suppose that (l,E) satisfies conditions 1A, or 1B, or 2A, or 2B of Theorem 1.11 and Remark 1.14. Then

(4.6) \begin{equation}m_{T,E,l}=\max\{0,f_{T,E,l}\}\leq (r+s)/2,\end{equation}

for every subset of markings $T=T_p\coprod T_q$ split into heavy and light markings such that $|T_p|=r$ , $|T_q|=s+1$ . We call T critical for (l,E) if equality holds in (4.6).

Then T is critical if and only if we have one of the cases listed in the following table, where we also list critical subsets:

In addition, we have

(4.7) \begin{equation}f_{T^c,E,l}\ge -(r+s)/2.\end{equation}

The inequality (4.7) is strict unless $l=0$ and T is critical for (l,E); in particular, it appears in the above table.

When, in addition, $l+e$ is even, a set T is critical for (l,E) if and only if $r+s$ is even and we are in one of the cases listed in the above table.

We use analogues of Lemmas 4.6 and 4.7 for $P\cup Q$ as follows.

Lemma 4.9. Let $l\in\mathbb{Z}$ (positive or negative), $E\subseteq P\cup Q$ , $y\in Q$ . Then

(4.8) \begin{equation}\max_{P\cup (Q\setminus\{y\})=T\sqcup T^c} f_{T,E,l}\leq S'(l,E\setminus\{y\})/2.\end{equation}

Here, we assume that $T=T_p\coprod T_q$ (so $T^c=(P\setminus T_p)\coprod (Q\setminus T_q)$ ) is a subset of markings split into heavy and light markings such that $|T_p|=r$ and $|T_q|=s$ and the score function $S'(l,E\setminus\{y\})$ is considered on $P\cup (Q\setminus\{y\})$ (i.e., $\tilde Q=(Q\setminus\{y\})$ in Notation 4.3).

Equality in (4.8) is attained if and only if $y\notin E$ . In this case, equality is attained for some $f_{T,E,l}$ if and only if $(T_p\subseteq E_p$ or $E_p\subseteq T_p)$ and $(T_q\subseteq E_q$ or $E_q\subseteq T_q)$ .

Lemma 4.10. Let $E\subseteq P\cup Q$ . Suppose that (l,E) satisfies conditions 1A, or 1B, or 2 of Theorem 1.6 and Remark 1.14. Let $y\in Q$ and $T=T_p\coprod T_q\subset P\cup (Q\setminus\{y\})$ be split into heavy and light markings such that $|T_p|=r$ and $|T_q|=s$ . Then

(4.9) \begin{equation}m_{T,E,l}=\max\{0,f_{T,E,l}\}\le (r+s-1)/2.\end{equation}

We call T critical if equality holds in (4.9), which happens if and only if $y\notin E$ and we are in one of the following cases:

In particular, $m_{T,E,l}\le (r+s)/2-|E\cap\{y\}|$ .

If, in addition, $l+e$ is even and T is critical, we must have that $r+s$ is odd.