# Thursday, November 17 ## Constructing moduli of semistable sheaves :::{.remark} Today: a sketch of a proof of existence of a moduli space of semistable sheaves. Setup: let $X\in\Proj\Var$ or $X\in \Sch$, fix a Hilbert polynomial $P(n)$, and fix $\mce\in \Coh(X)$ with $p(\mce, n) = P(n)$; we want to construct the moduli space $M(X, P(n))$. Using that $\mce(n)$ is globally generated for $n > n_0 \gg 0$, there is a surjection $H^0(X; \mce(n)) \tensor \OO_X \surjects \mce(n)$ and thus a surjection $H^0(X; \mce(n) )\tensor \OO_X(-n) \surjects \mce$. Note $V \da H^0(X; \mce(n))$ is a vector space of dimension $\deg P(n)$ there is a surjection of sheaves $V\tensor\mcw \surjects \mce$ making $\mce\in \Hilb(V\tensor \mcw, P(n))$. Grothendieck embeds this into $\Gr(V\tensor W, P(n))$, and more generally $\Quot(\mcy, P(n))\embeds \Gr(G, a)$. At this point, $\Quot$ includes the data of a choice of basis of $V$, so we'll quotient by an action $\SL(V)\actson V \leadsto \SL(V)\actson \Hilb(V\tensor \mcw, P(n))$. ::: :::{.lemma title="A key computational lemma"} $\SL(V)\actson \Gr(V\tensor W \to U_a)\embeds \PP^N$, so we need a lift $\SL(V)\actson \PP^N$ with a linearization $\SL(V)\actson \AA^{N+1}$. In this situation, we'll have the Hilbert-Mumford numerical criterion to check if $[V\tensor W\surjects U]$ is (semi)stable. The condition will turn out to be \[ \forall H \subseteq V,\qquad {\dim H \over \dim \im(H \tensor W \injects V\tensor W)} \leq {\dim V\over \dim U} .\] ::: :::{.remark} Let $V = H^0(\mce(n))$, then $V\cdot \mce = \mce$, i.e. $V$ spans the stalks, and any subspace $H \subseteq V$ defines a subsheaf $\mcf \da H\cdot \mce \leq \mce$. The criterion yields $\bar{p}_\mcf(n) \leq \bar{p}_\mce(n)$. For a single sheaf $\mce$, $n$ depends on $\mce$ and this is easy, but boundedness in families is difficult in general. To define the $\PP^N$ appearing in the lemma, we'll need to discuss Grassmannians. ::: ## Grassmannians :::{.remark} For a fixed $B$, a SES $A\injects B\surjects C\in \kmod$ of dimensions $a,b,c$ respectively, note $\Gr_{a, b} = \Gr_{b, c}$ where the former parameterizes subspaces and the latter quotients. There are several levels of generality in which Grassmannians can be defined: - Over $k\in \Field$, points of $\Gr(B)$ correspond to $\ts{A\injects B}$ or equivalently $\ts{B\surjects C}$ - In families, in which cases quotients are preferred. ::: :::{.remark} Let $B = k^n$, how does one parameterize subspaces? Any subspace $A$ has a basis $A = \gens{v_1,\cdots, v_a}$. Fixing a basis $k^n = \gens{e_1,\cdots, e_b}$, one can form a matrix $M_B\in \Mat_{a\times b}(k)$ whose rows are the $v_i$. There is an action $\GL_a\actson M_B$ by conjugation. Recall that **Plucker coordinates** are the components of $(P_I)$, the determinants of all $a\times a$ minors where $\abs{I} = a$ is an index set. For $I = \ts{1,\cdots, b}$, there are $b\choose a$ such minors. We can regard $(P_I) \in P^{{b\choose a} - 1} = \PP(\Extalg^a B)$ where $B = \gens{e_1,\cdots, e_b}$ and $\Extalg^a = \gens{e_{i_1} \wedgeprod \cdots e_{i_a}}$. Writing $v_i$ in the $e_i$ basis, their Plucker coordinate is $v_1 \wedgeprod \cdots \wedgeprod v_a = \sum P_I e_I \in \Extalg^a A \cong k \subseteq \Extalg^a B$. ::: :::{.claim} Each point $(P_I)\in \PP^N$ defines $A \subseteq B$ uniquely, so $\Gr_{a, b} \embeds \PP^N$. ::: :::{.proof title="?"} Let $M$ be a matrix of rank $a$, and change basis so that $M$ is of the form $M = [I | M']$, where entries of $M'$ encode some of the Plucker coordinates. For example, $M'_{0, 0}$ is the determinant of a certain submatrix: ![](figures/2022-11-17_13-32-20.png) We can also see that $\Gr_{a, b} = \union \AA^{a(b-a)}$ where each $\AA^{a(b-a)} \embeds\AA^N = \ts{P_I\neq 0}$, yielding $\Gr_{a, b} \embeds \PP^N \contains \Union_N \AA^N$. There is in fact a closed embedding $\Union \AA^{a(b-a)} \embeds \Union \AA^N$ given by algebraic equations. ::: :::{.corollary title="?"} $\dim \Gr_{a, b} = ac = a(b-a)$ when parameterizing quotients. ::: :::{.remark} What does the Hilbert-Mumford criterion say in this situation? Let $K\injects V\tensor W\surjects U$, and pick a basis to get $P_I(K) = P_I(U)$ and $v_{i_1}\wedgeprod \cdots \wedgeprod v_{i_n} = \sum p_{i_1, \cdots, i_a} e_{i_1} \wedgeprod \cdots \wedgeprod e_{i_a}$. Letting $\ts{f_i\tensor g_j}$ be a basis for $V\tensor W$, consider how to linearize the action $\SL(V)\actson V\tensor W$: pick a $\GG_m \subseteq \SL(V) \subseteq \SL(V\tensor W$ so $t.f_i = t^{r_i} f_i$ with $\weight(f_i) = r_i$ and $\sum r_i = 0$. Then $\weight(f_i \tensor g_j) = r_i$ since there is no action on $W$. Check that \[ \weight\qty{ (f_{i_1} \tensor g_{j_1} ) \wedgeprod (f_{i_2}\tensor g_{j_2} ) \wedgeprod \cdots \wedgeprod ( f_{i_a} \tensor g_{j_a} ) } = \sum_{s=1}^a r_{i_s} .\] Now the subspace $K\to V\tensor W$ or quotient $V\tensor W\to U$ is GIT is stable (resp. semistable) iff for all $\lambda: \GG_m \to \SL(V)$, there exists some $P_I$ such that - $\weight \leq 0$ for semistability, - $\weight < 0$ for stability, with $P_I(K) \neq 0$, by the numerical criterion. This translates to having a nonzero $a\times a$ minor for any choice of basis in $V$. ::: :::{.remark} Let $V = \gens{f_1,\cdots, f_n}$, then there are subspaces $H_{n-1} = \gens{f_2,\cdots, f_n}, H_3 = \gens{f_3,\cdots, f_n}, \cdots, H_1 = \gens{f_n}$. This corresponds to an ordered list of weights $r_1\leq r_2\leq \cdots \leq r_n$. ::: :::{.exercise title="?"} Try this in dimension 2, where $V = \gens{f_1, f_2}$ with weights $-r, r$ resp. and write $V\tensor W = f_1 W \oplus f_2 W$. Check $\weight\qty{\Extalg(f_i\tensor g_j)} = \sum r_i = r$. :::