# Tuesday, November 15 ## Moduli of semistable sheaves :::{.remark} See this very interesting paper posted today! ::: :::{.remark} Next goal: constructing moduli spaces of stable sheaves and how to reduce it to GIT, after Seshadri, Narasimhan, Mumford (on curves), Gieseker (surfaces), Maruyama (higher dimensional varieties), Simpson (completed for higher-dimensional varieties). We'll follow the treatment in [Simpson's 1994 paper, "Moduli of representations of fundamental groups..."](http://www.numdam.org/item/PMIHES_1994__79__47_0.pdf). Setup: let $X \subseteq \PP^N$ be a projective variety with $\OO_X(1)$ and $\mce \in \Coh(X)$ and Hilbert polynomial $p(\mce, n) = \chi(\mce(n))$. One can easily prove by induction that $p$ is in fact a polynomial, and it turns out to have terms of the form $p(\mce, n) = r {n^d\over d!} + a {n^{d-1}\over (d-1)!} + \cdots$. We define - $d(\mce) \da \dim \supp(\mce)$ -- for $\mce \in \Pic(X)$, this is $\dim X$, but in general could be smaller. It turns out $d(\mce) = d \da \deg p(\mce, n)$. - A "generalized rank" of $\mce$ by $r(\mce) = r$, the leading coefficient in $p(\mce, n)$ above. - $\mu(\mce) \da a/r$ the slope. - $\bar p(\mce, n) \da {1\over r}p(\mce, n) = {n^d\over d!} + {a\over r} {n^{d-1}\over (d-1)!} + \cdots$, noting that the first nontrivial coefficient is the slope $a/r$. ::: :::{.definition title="Pure dimensional"} We say $\mce$ is **pure dimensional** iff it has no subsheaves of strictly smaller support, i.e. for all nonzero $\mcf \leq \mce$, one has $d(\mcf) = d(\mce)$. On affine schemes, this is Serre condition 1, and this says there are no embedded components (corresponding to primes; take the primary decomposition). ::: :::{.example title="Pure dimension"} For a curve with many irreducible components, there are *no* sheaves supported only a single point. If $X = X_1\union X_2$, $\OO_X$ has the subsheaf $\OO_{X_1}\cdot I_{X \sm X_1}$. For a nodal curve, this yields $\OO_{X_1}(-p)$, regular functions on $X_1$ that vanish at $p$: \begin{tikzpicture} \fontsize{45pt}{1em} \node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/Moduli/sections/figures}{2022-11-15_13-07.pdf_tex} }; \end{tikzpicture} ::: :::{.definition title="$p\dash$(semi)stable or Hilbert (semi)stable"} We say $\mce$ is **$p\dash$(semi)stable** or **Hilbert (semi)stable** iff 1. $\mce$ is pure dimensional, so any subsheaf has the same dimension, 2. For any nonzero subsheaf $\mcf \leq \mce$, there is an inequality of reduced Hilbert polynomials $\bar p(\mcf, n) \leq \bar p(\mce, n)$, resp. $\bar p(\mcf, n) < \bar p(\mce, n)$, where $f \leq g \iff f(N) \leq g(N)$ for all $N\geq N_0 \gg 0$, or equivalently $f\leq g$ in the lexicographic order. ::: :::{.theorem title="?"} For any Hilbert polynomial $P(n)$, there exists a moduli space $M(\OO_X, p)$ of semistable sheaves on $X$ with $p(\mce, n) = P(n)$, which has a (semi)stable locus. This gives a bijection on points: \[ M(\OO_X, P) &\mapstofrom \ts{\text{semistable } \mce \in \Sh(X),\, p(\mce, n) = P(n) }\modiso \qquad \text{under $\gr\dash$equivalence} \\ M(\OO_X, P)^\stable &\mapstofrom \ts{\text{stable } \mce\in \Sh(X),\, p(\mce, n) = P(n) } .\] ::: ## Construction of $M(\OO_X, P)$ :::{.remark} This will essentially be a quotient by $\SL(V)$ for some $V$. Let $\mce \in \Coh(X)$, then Serre yields that for all $n\gg n_0$, 1. $H^{>0}(\mce(n)) = 0$, $\dim H^0(\mce(n)) < \infty$, and 2. The twist $\mce(n)$ is generated by global sections. Thus there is a surjection \[ 0 \to K \da \ker f \to H^0(\mce(n)) \tensor \OO_X \surjectsvia{f} \mce(n) \to 0 .\] Note that a global section $s\in \globsec{\mcf}$ is equivalent to a morphism \[ \OO_X &\to \mcf \\ 1 &\mapsto s .\] Untwisting this surjection yields \[ 0 \to K(-n) \to H^0(\mce(n))\tensor \OO_X(-n) \surjectsvia{\tilde f} \mce \to 0 .\] ::: :::{.definition title="Hilbert/Quot scheme (due to Grothendieck)"} Let $V\in \kmod^\fd$ and $\mcw\in \Sh(X)$ (e.g. $\mcw =\OO_X$), and define $\Hilb(V\tensor \mcw, P)$ to be the moduli space of quotients $V\tensor \mcw\to \mce \to 0$ with $p(\mce, n) = P(n)$, i.e. the scheme of quotient sheaves of $V\tensor \mcw$. More generally, one can define $\Hilb(\mcg, P) = \Quot(\mcg, P)$ to be the scheme of quotients $\mcg\to \mce \to 0$ with $p(\mce, n) = P(n)$. ::: :::{.theorem title="?"} $\Quot(\mcg, P)$ exists as a scheme and admits a universal family, yielding a fine moduli space. Moreover, one can embed it into some Grassmannian, yielding $\Quot(\mcg, P) \embeds \Gr_{r, n}$. ::: :::{.remark} Note that if $V$ is a vector space, every dimension $r$ subspace yields a codimension $r$ quotient, so $\Gr_{r, N}$ also parameterizes quotients, and we choose quotients as they are better behaved from a commutative algebraic POV. ::: :::{.proof title="?"} Let $\mcg$ be fixed and consider quotients $\mcg\to \mce \to 0$ as $\mce$ varies. Take the sheaf kernel to obtain \[ 0\to K\to \mcg\to\mce \to 0 ,\] and twist by $\OO_X(n)$ for $n\gg 0$ to obtain \[ 0\to H^0(K(n)) \to H^0(\mcg(n)) \to H^0( \mce(n) ) \to 0 \] using Serre vanishing. This is a SES of vector spaces $0\to V\to k^N \to U\to 0$ for some $N$, and thus we get a point in the Grassmannian. We know $\dim U = p(\mce, n)$ (maybe the degree..?), and as we vary the quotients, $p(\mce, n)$ does not vary. Note that one needs to show that the number of such quotients to be bounded so that $n$ can be chosen uniformly for all $\mce$, which we'll not prove here. Conversely, suppose $H^0(\mcg(n))\surjects U\to 0$, we can produce a sheaf? Take the kernel of vector spaces to get a SES \[ 0\to K\to H^0(\mcg(n)) \to U\to 0 ,\] where sections of $K$ generate a subsheaf of $\mcg(n)$, say $\mck(n)\da K\cdot \mcg(n) \leq \mcg(n)$. Untwisting yields $0\to \mck \injectsvia{f} \mcg\to \coker f \to 0$. Again, once $P$ is fixed, $n$ can be chosen uniformly. This yields the embedding $\Quot(\mcg, P) \embeds \Gr_{r, N}$ as a closed subscheme, since it turns out that each quotient $U$ is defined by polynomial equations and thus algebraic conditions. ::: :::{.remark} Note that this is a closed subscheme, which is easier to handle than a closed subvariety: e.g. any equations define an ideal $I$ and $V(I)$ is a closed subscheme, say of $\AA^n$, whereas it is only subvariety iff $I = \sqrt I$. This is equivalent to asking if $V(I)$ is *reduced*. ::: :::{.remark} Note that $V\da H^0(\mce(n))$ is fixed in the proof, and fixing this is equivalent to choosing a basis of $H^0(\mce(n))$, so $\Quot(\mcg, P)$ encodes a choice of basis. To forget this choice, we need to quotient by change of basis, and we'll have $M(\OO_X, P) \da \Quot(V\tensor \OO_X(-n), P) \modmod \SL(V)$. Note that the Grassmannian has a Plucker embedding into $\PP^N$ for some large $N$. We have $\SL(V)\actson \Quot(\mcg, P)$, so we can apply the Hilbert-Mumford numerical criterion to the induced action $\SL(V)\actson \PP^N$ -- doing this precisely yields the (semi)stability criterion $\bar p(\mcf, n) \leq \bar p(\mce, n)$. The hard part will be boundedness -- e.g. consider $X$ a curve and $\mcf = \OO_X(n) \oplus \OO_X(-n)$, which all have the same Hilbert polynomial and thus yields an unbounded family. Starting with semistable sheaves yields a bounded family. Maruyama handled boundedness for low dimensions and Simpson proved it for the remaining dimensions, so we'll generally skip the boundedness issues when choosing $n$. ::: :::{.remark} Next time: the Plucker embedding, seeing what points look like under the embedding, and seeing the polynomial criterion drop out of the calculation. Later: moduli of varieties. :::