Some examples of moduli spaces:

Ways of forming moduli spaces:

• GIT,
• Hodge theory over \({\mathbf{C}}\),
• Stacks (e.g. Artin’s method).

These rarely produce compact/complete spaces, so we’ll discuss compactification. Why compactify? Computing things, projectivizing, intersection theory. See Bailey-Borel and toroidal compactifications.

A note on Hodge theory: for an elliptic curve, one can write \(E = {\mathbf{C}}/\left\langle{1, \tau}\right\rangle\) with \(\Im(\tau) > 0\) (so \(\tau\in {\mathbb{H}}\)), one can form \({\mathcal{M}}_1 = \dcosetr{{\mathbb{H}}}{{\operatorname{SL}}_2({\mathbf{Z}})}\). This is Hodge theory: \(\tau\) is a period, and we quotient a bounded symmetric domain by an arithmetic group. Similarly, for PPAVs one can write \({\mathcal{A}}_g = \dcosetr{H_g}{{\mathsf{Sp}}_{2g}({\mathbf{Z}})}\), and for K3 surfaces one has \(F_{2d} = {\mathbb{D}}_{2g}/\Gamma_{2g}\) where \(\omega_X \in {\mathbb{D}}\). One can determine things like Jacobians using Torelli theorems.

Todo: how much do you know, and what are you trying to get out of the course?

A question from me: is every curve a branched cover of \({\mathbf{P}}^1\) over some number of points? Consider maps \(f: X\to {\mathbf{P}}^1\) where \(f(X) \not\subset {\mathbf{P}}^{n-1}\) is not contained in a hyperplane. This biject with basepoint free linear systems – let \({\mathcal{F}}\) be an invertible sheaf, then a linear system is any linear subspace \(V \subseteq H^0(X; {\mathcal{F}})\). Writing \(V = \left\langle{f_i}\right\rangle_{i=0}^n\), the bijection is sending \(p\mapsto {\left[ {f_0(p): \cdots : f_n(p)} \right]} \in {\mathbf{P}}^{n}\). Since \({\mathcal{F}}\) is invertible, locally \({ \left.{{{\mathcal{F}}}} \right|_{{U}} } \cong {\mathcal{O}}_U\) – this map is well-defined precisely when not all the \(f_i(p)\) are zero, which is precisely the basepoint-free condition. A map \(f:X\to {\mathbf{P}}^1\) thus corresponds to two sections which don’t simultaneously vanish. If \(X\) is projective, it admits a very ample line bundle \({\mathcal{L}}\) where the base locus of \(H^0({\mathcal{L}})\) is empty. One can now project away from any point outside of the curve to get a regular map factoring the projective embedding:

Link to Diagram

The projection:

One can continue to project until reaching \({\mathbf{P}}^1\).

Recall \(X\) is hyperelliptic if it admits a 2-to-1 map \(X\to {\mathbf{P}}^1\), so has gonality 2. Gonality 1 curves are isomorphic to \({\mathbf{P}}^1\), and gonality 3 are trigonal.

Plan for the course:

1. GIT (\(1/2\) to \(2/3\) of the course). Goals:

• Construct moduli of vector bundles on a curve, surface, etc.
• Construct \({\mathcal{M}_g}\) (Riemann’s moduli space, not complete, projective, affine, but quasiprojective), \(\overline{{\mathcal{M}_g}}\) (Deligne-Mumford’s moduli of stable curves), \(\overline{{\mathcal{M}}}_{0,n}\), and \(\overline{{ \mathcal{M}_{g, n} }}(V)\) (Kontsevich’s moduli of stable maps, to rigorously define Gromov-Witten invariants).¹
• Sources:
  • Mukai’s book, An Introduction to Invariants and Moduli (this will be our primary source). He covers stable vector bundles on curves.
  • Mumford’s book on GIT and his paper about stability for algebraic curves, although this is perhaps unnecessarily difficult! We’ll need this for \(\overline{{\mathcal{M}_g}}\).

2. Hodge theoretic approaches. Goals:

• Construct \({\mathcal{A}_g}\) the moduli of abelian varieties. See Birkenhake and Lange.
• Construct \(F_{2d}\) the moduli of K3 surfaces. See Huybrechts.

Let \(X\) be a genus \(g\) smooth projective curve. Over \({\mathbf{C}}\), projective implies compact, and non-projective is a Riemann surface with finitely many punctures. More generally, over \(k={ \overline{k} }\) smooth means that \(\dim {\mathbf{T}}_{X, x} = \dim X\) at every point \(x\in X\). Note that \(X^{{\mathrm{sing}}} \subseteq X\) is an algebraic and thus closed subset, so curves have finitely many singularities (nodes, cusps, etc). There is only one topological type of curve, but there are distinct algebraic and conformal structures (which turn out to be equivalent notions for curves).

One can show \({\mathcal{M}}_g({\mathbf{C}})\) is an orbifold of dimension \(3g-3\), i.e. locally a quotient \(M/G\) of a manifold by a finite group. Similarly \({\mathcal{M}}_g\) is a quasiprojective algebraic variety of dimension \(3g-3\) with only quotient singularities. Mumford was the first to ask questions about its geometry, e.g. is it rational?

Lüroth proved this in dimension 1, and as a consequence of the classification of surfaces, the Italian school showed this in dimension 2. See the Castelnuovo criterion, which shows \(X\) is rational iff \(X\) is regular, i.e. \(q \mathrel{\vcenter{:}}= h^1(X; {\mathcal{O}}_X) = 0\) and \(p_2 \mathrel{\vcenter{:}}= h^0(X; 2K_X) = 0\).

One of the classical Italian algebraic geometers (either Severi or Castelnuovo) “proved” the false statement that \({\mathcal{M}_g}\) is unirational for all \(g\). In fact this is only true for \(g\leq 9\). The idea is good though: any curve \(X\hookrightarrow{\mathbf{P}}^n\) can be projected to a curve in \(X \to {\mathbf{P}}^2\) with only finitely many nodes. The coordinates for the nodes can serve as parameters. Having a curve pass through given points is a linear condition, as is saying it is singular at a point (by computing partial derivatives). Being a node is not a linear condition – instead, it is a quadratic algebraic condition coming from the vanishing of a \(2\times 2\) determinant. It’s also not clear that imposing singularity conditions locally are all independent, since singularities at some points can force singularities at others. Mumford proved that \({\mathcal{M}_g}\) is not unirational for \(g\geq 24\), and is in fact general type, which is far from unirational.

Next time: more general introduction, stable curves, a bit about Hodge theory, then starting Mukai’s book.

Goal: showing \({\mathcal{M}_g}\) exists as a quasiprojective complex variety, and can in fact be defined over any field \(k\) or even over \({\mathbf{Z}}\). Here quasiprojective over \({\mathbf{Z}}\) means \(X \subseteq {\mathbf{P}}^n_{/ {{\mathbf{Z}}}}\) is a closed subset given as \(X = V(f_i)\) for homogeneous integral polynomials \(f_i\). Note that \({\mathcal{M}_g}({ \overline{k} }) = \left\{{\text{smooth projective curves of genus } g}\right\} = X({ \overline{k} }) \setminus Z({ \overline{k} }) \subseteq {\mathbf{P}}^n_{/ {{ \overline{k} }}}{_{\scriptstyle / \sim} }\) where \(Z = V(f_i, g_i)\) – this says \({\mathcal{M}_g}\) satisfies exactly the equations \(f_i\) and no more. Anytime objects have isomorphisms, one only gets a coarse moduli space instead of a fine moduli space, which we’ll later describe. Families \({\mathcal{X}}\to S\) yield to maps \(S\to {\mathcal{M}_g}\) over \(\operatorname{Spec}{ \overline{k} }\), and this will be a bijection when \({\mathcal{M}_g}\) is a fine moduli space and \({\mathcal{X}}\) is the pullback of a universal family \({\mathcal{E}}\to{\mathcal{M}_g}\). Since we only have a coarse moduli space, a family yields a map to \({\mathcal{M}_g}\), but these are not in bijection.

We’ll want projective varieties in order to do intersection theory. The most fundamental compactification: the Deligne-Mumford compactification \(\overline{{\mathcal{M}_g}}\) of \({\mathcal{M}_g}\), i.e. the moduli of stable curves of genus \(g\). This is a projective moduli space containing \({\mathcal{M}_g}\) as an open dense subset, and is obtained by adding degenerate curves “at infinity.”

Writing a multi-component curve as \(X = \bigcup X_i\), the numerical condition requires that for every \(X_i\cong {\mathbf{P}}^1\), one has \({\left\lvert {X_i \cap(X\setminus X_i)} \right\rvert} \geq 3\), and for all \(X_i\) of the following form (or \(X_i\cong E\) an elliptic curve), \({\left\lvert {X_i \cap(X\setminus X_i)} \right\rvert} \geq 1\):

This is equivalent to \({\sharp}\mathop{\mathrm{Aut}}X < \infty\). For \(g\geq 2\) and \(C_g\) smooth of genus \(g\), one has \({\sharp}\mathop{\mathrm{Aut}}C_g < \infty\), and for \(g=1\) enforces \(\dim \mathop{\mathrm{Aut}}C_g = 1\). For \(g=0\), note \(\mathop{\mathrm{Aut}}{\mathbf{P}}^1 = \operatorname{PGL}_2\) which has dimension 3, so fixing at least 3 points cuts this down to a finite automorphism group.

The dualizing sheaf \(\omega_X\) is invertible if \(X\) has only nodes. The adjunction formula yields a twist \({ \left.{{\omega_X}} \right|_{{X_i}} } = \omega_{X_i}(X\setminus X_i)\) Then \(\omega_X\) is ample iff \(\deg{ \left.{{\omega_X}} \right|_{{X_i}} } > 0\). One can compute \(\deg \omega_{X_i}(X\setminus X_i) = 2g_i - 2 + {\left\lvert {X_i \cap(X\setminus X_i)} \right\rvert}\), hence the lower bound on the number of intersection points.

If \(\omega_{X_t}\) is ample for all \(t\), then \(\omega_{{\mathcal{X}}}/S\) is relatively ample, which implies \({\mathcal{X}}/S\) is the canonical model. One can contract \((-1)\) curves to get a minimal model, and \((-2)\) curves to get canonical models. See degenerations of elliptic curves to wheels of copies of \({\mathbf{P}}^1\):

See Kodaira’s elliptic fibers – classified by extended Dynkin diagrams \(\tilde A_n, \tilde D_n, \tilde E_n\), and special types \(\tilde A_i^*\) for \(i=0,1,2\):

Defined to formulate Gromov-Witten invariants. Motivated by physics, but originally non-algebraic and used almost complex structures. The second condition yields unique limits since it will yield a relative canonical model, which exist and are unique. This moduli space can be generalized to higher dimensions, see KSBA compactifications.

Constructing \({\mathcal{M}_g}\) and \(\overline{{\mathcal{M}_g}}\):

• Step 1: Parameterize embedded curves \(C_g \hookrightarrow{\mathbf{P}}^N\) by the picking a basis of the linear system \({\left\lvert {2K_X} \right\rvert}\), where \(N = 2(2g-2) - (g-1) - 1 = 3g-4\) and \(\deg C_g = 2(2g-2)\). Use either the Chow variety \(\mathsf{Ch}_{d, N}\), parameterizing cycles/subvarieties of \({\mathbf{P}}^N\) with degree \(d\), or the Hilbert scheme \(\operatorname{Hilb}_h\) parameterizing closed subschemes \(X \hookrightarrow{\mathbf{P}}^N\) with a fixed Hilbert polynomial \(h\). The latter may not yield reduced curves, but closed subschemes are easier than varieties since they are just defined by equations.

• Step 2: Divide by \(\operatorname{PGL}_{N+1}\), using GIT (next week) to produce a space \(X/G\) whose points (ideally) correspond to \(G{\hbox{-}}\)orbits.

Goal: understanding quotients of varieties by general group actions, a basic notion for moduli. The easiest case: finite groups \(G\curvearrowright X\in{\mathsf{Aff}} \mathsf{Alg}{\mathsf{Var}}_{/ {k}}\) for \(k={ \overline{k} }\).

Think of \(X \subseteq {\mathbf{A}}^n_{/ {k}}\) for some \(N\), with coordinates \({\left[ {a_1,\cdots, a_n} \right]}\), so \(X = V(f_1,\cdots, f_n)\). Note \({\mathcal{O}}_{{\mathbf{A}}^n} = k[x_1, \cdots, x_{n}]\), and regular functions on \(X\) are restricted polynomials, so we get a sequence \begin{align*} R = k[X] \leftarrow k[x_1, \cdots, x_{n}]\leftarrow I = \sqrt{\left\langle{f_1,\cdots, f_n}\right\rangle} ,\end{align*} so \(R\in {}_{k} \mathsf{Alg}^{\mathrm{fg}}\) without nilpotents – in fact varieties biject with such algebras. If \(G\curvearrowright R\) any ring, one can take invariants \begin{align*} R^G \mathrel{\vcenter{:}}=\left\{{r\in R {~\mathrel{\Big\vert}~}g^*(r) = r\,\,\forall g\in G}\right\} \end{align*} which is a subring and a \(k{\hbox{-}}\)subalgebra of \(R\). Here \(g^*\) is defined in terms of pullbacks of functions:

Link to Diagram

There is a \(k{\hbox{-}}\)linear averaging map \begin{align*} S: R &\to R^G \\ r &\mapsto {1\over {\sharp}G} \sum_{g\in G} g^*(r) ,\end{align*} noting that \(S\) is not a ring morphism.

Let \(a\in R\) and consider \(p_a(x) \mathrel{\vcenter{:}}=\prod_{g\in G} (x - g^*(a) )\), a polynomial of degree \(n = {\sharp}G\) whose coefficients are in the subring \(R^G\) and are symmetric polynomials in the \(g^*(a)\). Since \(p_a(a) = 0 = a^n + \cdots\), \(a^n\) is a linear combination of \(1,a,\cdots, a^{n-1}]\) with coefficients in \(R^G\) and these symmetric polynomials. So if \(\left\{{a_1,\cdots, a_m}\right\}\) generate \(R\) as a \(k{\hbox{-}}\)algebra, the images of monomials \(S(a_1^{k_1}\cdots a_m^{k_m})\) with \(0 \leq k_i \leq n\) generate \(R^G\). If \(b\in R^G\), one one hand \(b = S(b)\), and on the other hand \(b = \sum c_k a_I^{k_I}\) so \(S(b) = \sum c_k S(a_I^{k_I})\). Thus the \(c_k\) are in the subring generated by elementary symmetric polynomials in the \(g^*(a_i)\).

There is another basis for elementary symmetric polynomials given by Newton sums. Recall

• \(\sigma_1 = \sum x_i\)
• \(\sigma_2 = \sum x_i x_j\)
  
• \(\sigma_n = \prod x_i\)

The Newton sums are

• \(N_1 = \sum x_i\)
• \(N_2 = \sum x_i^2\)
• \(N_n = \sum x_i^n\),

and one can inductively show that one can be written in terms of the other.

The advantage is that the averaging operator commutes with sums, so the \(c_k\) like in the subring generated by Newton sums of the \(S(a_i^{k_i})\)

The right classes of groups to take: geometrically reductive and linearly reductive.⁶

Over \({\mathbf{C}}\) these coincide, and are e.g. \(\operatorname{GL}_n({\mathbf{C}})\) (trivial center, nontrivial \(\pi_1\)), the classical semisimples

• Type A, \({\operatorname{SL}}_n({\mathbf{C}})\) (nontrivial center, trivial \(\pi_1\)),
• Types B and D, \({\operatorname{SO}}_{n}({\mathbf{C}})\),
• Type C \({\operatorname{Sp}}_{2n}({\mathbf{C}})\),

along with \(({\mathbf{C}}^{\times})^n\), and their products and extensions, and the exceptional groups \(E_6, E_7, E_8, F_4, G_2\). Here linearly reductive means any finite-dimensional representation decomposes into a sum of irreducible representations.

The most useful for moduli: \(\operatorname{GL}_n, \operatorname{PGL}_n, {\operatorname{SL}}_n\). Note that \(\operatorname{PGL}\) and \({\operatorname{SL}}\) are almost the same, up to a finite group.

For \(\operatorname{ch}(k) = p\), the only linearly reductive group on this list is \({\mathbf{G}}_m^n\), while “geometrically reductive” includes all of these groups. Over \({\mathbf{Z}}\), the split versions \(\operatorname{GL}_n({\mathbf{Z}}), {\operatorname{Sp}}_n({\mathbf{Z}})\), etc still work.

Nonsplit groups are e.g. those not isomorphic to \(\operatorname{GL}_n(k)\) but become isomorphic over \({ \overline{k} }\). Examples: compare \({\mathbf{G}}_m\) over \({\mathbf{R}}\) and \(S^1 = k[x,y]/\left\langle{x^2+y^2-1}\right\rangle\); these only become isomorphic over \({\mathbf{C}}\).

Let \(S_n\curvearrowright k[x_1, \cdots, x_{n}]\) by permuting variables, then \begin{align*} k[x_1, \cdots, x_{n}]^{S_n} = k[\sigma_1, \cdots, \sigma_n] = k[N_1, \cdots, N_n] ,\end{align*} generated by elementary symmetric functions or Newton polynomials.

More generally, a root lattice \(\Lambda\) (e.g. for a Coxeter group) gives rise to a Weyl group \(W(\Lambda)\), and one can consider \(W{\hbox{-}}\)invariant functions. For example, \(W(A_n) = S_{n+1}\). For a torus, invariant functions are characters. For a Lie algebra \({\mathfrak{g}}\), one can show that the \(W{\hbox{-}}\)invariants of symmetric functions on the torus, \(S({\mathfrak{h}})^W\), forms a polynomial algebra. The generators are referred to as the fundamental weights.

Coming up next: groups of multiplicative type, infinite groups, and generalizing the above theorem by removing some problematic subsets.

Last time: for \(G\curvearrowright R \supseteq R^G\) for \(G\in{\mathsf{Fin}}{\mathsf{Grp}}\), the Todd-Shepherd(-Chevalley) theorem states that if \(G\curvearrowright{\mathbf{A}}^n\) and \(G\) is generated by pseudoreflections then \(k[x_1, \cdots, x_{n}]^G\) is again a polynomial ring. Consider now \(G\curvearrowright R\) for \(G\) finite abelian and \(\operatorname{ch}k = 0\). This yields a grading \(R = \bigoplus _{\chi \in \widehat{G}} R_\chi\) where \(\widehat{G} = \mathop{\mathrm{Hom}}(G, {\mathbf{C}}^{\times}) = \mathop{\mathrm{Hom}}(G, {\mathbf{Q}}/{\mathbf{Z}})\) and \(R_\chi R_{\chi'} \subseteq R_{\chi + \chi'}\) Note that if \(G \cong \bigoplus {\mathbf{Z}}/n_i {\mathbf{Z}}\) then \(\widehat{G} \cong \bigoplus \mu_{n_i}\), which is non-canonically isomorphic to \(\bigoplus {\mathbf{Z}}/n_i{\mathbf{Z}}\). Recall that reflections have eigenvalues \(\left\{{1, 1,\cdots, 1, \alpha\neq 1}\right\}\).

Note that in general, \({\mathbf{A}}^n/G = \operatorname{mSpec}k[x_1, \cdots, x_{n}]^G\) has quotient singularities. Three types of varieties we work with in AG:

• Affine \(\rightleftharpoons\) rings,
• Projective \(\rightleftharpoons\) graded rings,
• General: covered by affines, not necessarily projective.

Upshot: we can think of projective varieties not as covered by affines, but rather as a “spectrum” of a single graded ring. Given a subset \(Z = V(f_1,\cdots, f_m) \subseteq {\mathbf{P}}^n_{/ {k}}\) cut out by homogeneous polynomials of degree \(d_i\) in the homogeneous degree 1 coordinates \(x_0,\cdots, x_n\), one can take the affine cone \(C(Z) \subseteq {\mathbf{A}}^{n+1}\). A linear action of \(G\curvearrowright{\mathbf{P}}^n_{/ {k}}\) descends to \(G\curvearrowright Z\), where linear means that \(g.{\left[ {x_0:\cdots:x_n} \right]} = M g{\left[ {x_0: \cdots: x_n} \right]}\). Not every action is of this form: take \(G={{\mathbf{C}}^{\times}}\curvearrowright{\mathbf{P}}^1\) by \(\lambda {\left[ {x_0: x_1} \right]} = {\left[ {x_0: \lambda x_1} \right]}\). This is linear; to make a nonlinear action glued the fixed points \(\left\{{0}\right\}\) and \(\left\{{\infty}\right\}\) to get a rational nodal curve:

Note that \(\operatorname{Pic}(C) = {\mathbf{Z}}\bigoplus {{\mathbf{C}}^{\times}}\).

For a linear action by a finite group \(G\), writing \(Z = \operatorname{mProj}R\) with \(R = k[x_1, \cdots, x_{n}]/\sqrt{\left\langle{f_i}\right\rangle}\) then \(Z/G = \operatorname{mProj}R^G\). Such actions can be lifted from \(Z\) to \({ \left.{{{\mathcal{O}}_{{\mathbf{P}}^n}(1) }} \right|_{{Z}} } = {\mathcal{O}}_Z(1)\).

Suppose that \(G = \operatorname{Spec}R\) is affine, then there are dual notions:

• Comultiplication: \(\mu^*: R\to R\otimes_k R\).
• Counits: \(e^*: R\to k\).
• Coinverses: \(i^*: R\to R\).

Upcoming: more group varieties and schemes, especially \(\operatorname{GL}_n, {\operatorname{SL}}_n\), and their actions/coactions.

Last time: group varieties. Most of today will work over \({\mathbf{C}}, k\neq { \overline{k} }\), or \({\mathbf{Z}}\). There is a correspondence:

Recall:

• For \(M, N\in {}_{R}{\mathsf{Mod}}\), there is a tensor product \(M\otimes_R N\),
• A morphism \(f\in \mathsf{CRing}(R, S)\) yields a functor \(f^*: {}_{S}{\mathsf{Mod}}\to ({R}, {S}){\hbox{-}}\mathsf{biMod}\) given by the base change/scalar extension \(({-})\otimes_R S\)
• If \(S_1, S_2 \in {}_{R} \mathsf{Alg}\) then \(S_1\otimes_R S_2 \in {}_{R}{\mathsf{Mod}}\) is in fact a ring, using the product \((u_1\otimes v_1)\cdot(u_2\otimes v_2) \mathrel{\vcenter{:}}= u_1u_2\otimes v_1 v_2\).
• For any \(N\in {}_{R}{\mathsf{Mod}}\), the functor \(({-}) \otimes_R N\) is right exact, so \begin{align*} A\hookrightarrow B\twoheadrightarrow&C \\ \leadsto \\ A\otimes_R N\to B\otimes_R N \twoheadrightarrow&C\otimes_R N .\end{align*}

Recall:

• \({\mathbf{G}}_m = \operatorname{Spec}k[x^{\pm 1}] \approx {{\mathbf{C}}^{\times}}\).
• \({\mathbf{G}}_m^n = \operatorname{Spec}k[x_1^{\pm 1}, \cdots, x_n^{\pm 1}] \approx ({{\mathbf{C}}^{\times}})^n\).
• \({\mathbf{G}}_a = \operatorname{Spec}k[x] \approx ({\mathbf{C}}, +)\).
• \(\mu_n \subseteq {\mathbf{G}}_m = \operatorname{Spec}k[x]/\left\langle{x^n-1}\right\rangle \approx \left\{{\xi \in {\mathbf{C}}{~\mathrel{\Big\vert}~}\xi^n=1}\right\}\).
• In characteristic \(p\), \(\mu_p = \operatorname{Spec}k[x]/\left\langle{x^p-1}\right\rangle = \operatorname{Spec}k[x]/\left\langle{x-1}\right\rangle^p\).

If \(G\) is an arbitrary finite group it can be made into an affine algebraic group variety over \(k\). Give the underlying set of \(G = {\textstyle\coprod}_{g\in G}\left\{{{\operatorname{pt}}}\right\}\) the discrete topology to get an algebraic variety. To get the algebraic group structure, note that any map of finite sets is algebraic. Define a ring \(R \mathrel{\vcenter{:}}=\bigoplus _{g\in G} k\), and a comultiplication as follows: note that \begin{align*} R\otimes_k R \cong \bigoplus _{(a, b) \in G\times G } k e_{a,b} \end{align*} where the \(e_{a, b}\) just tracks which summand we’re in. So define \begin{align*} R &\to R\otimes_k R \\ e_g &\mapsto \sum_{ab=g} e_{a, b} .\end{align*}

Note that we could have let \({\operatorname{pt}}= \operatorname{Spec}k\). E.g. \(C_p \neq \mu_p\) but are Cartier dual. However, the ring \(k[x]/\left\langle{x-1}\right\rangle^p\) is much easier to understand than the \(R\otimes_k R\) from above, even for very small groups like \(C_2\).

There is a coaction on \(A = \operatorname{Sym}^* V = k \oplus V \oplus \operatorname{Sym}^2 V \oplus \cdots\), where \(V = \left\langle{x,y}\right\rangle, \operatorname{Sym}^2 V = \left\langle{x^2, xy, y^2}\right\rangle\). This is the same as an action \(G\curvearrowright{\mathbf{A}}^N\) for some \(N\) acting on an affine space.

Last time: coactions on vector spaces \(a^*: V\to R\otimes_k V\) where \(R = k[G]\) is the ring of regular functions on an algebraic group \(G\). Thinking of \(V {}^{ \vee }\cong k^n \cong {\mathbf{A}}^n_{/ {k}} = \operatorname{Spec}\operatorname{Sym}^*V\) as the ring of regular functions, we get a map \(\operatorname{Sym}^*(V) \to R\otimes_k \operatorname{Sym}^*(V)\).

Over \(\operatorname{ch}{ \mathbf{F} }= p\), the only linearly reductive groups are either finite or diagonalizable.

Later: invariants of finitely generated for linearly reductive are again finitely generated. Note that invariants can be finitely-generated even when the group is not.

See proof in Mukai, due to Hilbert.

Linearly reductive groups:

• \(G\) finite, characteristic not dividing \({\sharp}G\),
• \({\mathbf{G}}_m^r\),
• Over \({\mathbf{C}}\): semisimple (types A-G), e.g. \({\operatorname{SL}}_n, \operatorname{PGL}_n\) for type A,
• Over \({\mathbf{C}}\): reductive, e.g. \(\operatorname{GL}_n\).

Linear reductive corresponds to \(\deg h = 1\). Evaluating at \(w\) gives a surjection \(V {}^{ \vee }\twoheadrightarrow k = k^G\). This yields a surjection \((V {}^{ \vee })^G \to k = k^G\) since not every such function vanishes. Finite generation of invariants is still true, although the proof takes much more work. See

• Mumford’s GIT for linearly reductive groups,
• Seshadri for geometrically reductive groups.

Note that over \(\operatorname{ch}k = p\), the groups \({\operatorname{SL}}_n, \operatorname{PGL}_n\) are geometrically reductive. In characteristic zero, a nontrivial fact is that linearly reductive is equivalent to geometrically reductive.

The proof uses a trick of reducing to an \({\operatorname{SL}}_2\) action where \(R^{{\mathbf{G}}_a} \cong R^{{\operatorname{SL}}_2}\).

Invariants \(R^G\) for various \(G\):

• \({\mathbf{G}}_a\): finitely-generated
• \({\mathbf{G}}_a^n\): for \(n\geq 10\), not finitely-generated. A geometric counterexample comes from asking if there are finitely many curves \(C\in \operatorname{Bl}_d {\mathbf{P}}^2\) with \(C^2 < 0\) and considering \(R = k[x_0,x_1,x_2]\). For \(d\leq 8\) there are finitely many, for \(d\geq 9\) infinitely many.
• \({\mathbf{G}}_a^2\): might still be open.

Nagata generalizes this to \({\mathbf{P}}^n\).

Useful fact: in characteristic zero, Lie groups are closely connected to Lie algebras. For \(G \leq \operatorname{GL}_n({\mathbf{C}})\) a closed subgroup, its Lie algebra is \({\mathfrak{g}}\mathrel{\vcenter{:}}={\mathbf{T}}_e G\), which has underlying vector space \({\mathbf{C}}^{\dim G}\) and bracket satisfying \([AB] = -[BA]\) and the Jacobi identity. Understanding this tangent space: think of matrices \(I + {\varepsilon}A\) where \({\varepsilon}A\) is small, and do operations discarding \({ \mathsf{O}}({\varepsilon}^2)\) terms. Equivalently, work over \({\mathbf{C}}[{\varepsilon}]/\left\langle{{\varepsilon}^2}\right\rangle\).

To work out what \({\mathfrak{g}}\) should be for \({\operatorname{SL}}_n({\mathbf{C}})\), linearize the \(\operatorname{det}= 1\) condition: \begin{align*} \operatorname{det}1 + {\varepsilon}A \mathrel{\vcenter{:}}=\operatorname{det}{ \begin{bmatrix} {1 + {\varepsilon}a_{11} } & { {\varepsilon}a_{ij} } \\ {\cdots} & {1 + {\varepsilon}a_{nn} } \end{bmatrix} } = 1 + {\varepsilon}\sum a_{ii} = 1 + {\varepsilon}\operatorname{tr}(A) .\end{align*} For \({\operatorname{SO}}_n\), work out \((I+ {\varepsilon}M)(I+{\varepsilon}M)^t = I\).

There is a way to go back: \({\mathfrak{g}}\xrightarrow{\exp} G\). This is almost a bijection, but can fail: e.g. in semisimple cases, \({\operatorname{SL}}_n({\mathbf{C}}), \operatorname{PGL}_n({\mathbf{C}}) = {{\operatorname{SL}}_n({\mathbf{C}})\over \mu_n} \mapsto {\mathfrak{sl}}_n({\mathbf{C}})\) both have the same Lie algebra. Note that \(\mu_n \subseteq Z({\operatorname{SL}}_n({\mathbf{C}}))\) is central, and more generally if \(G' = G/H\) for \(H \subseteq Z(G)\), \({\mathbf{T}}_I G \cong {\mathbf{T}}_I G'\).

The other issue: consider \(G = ({\mathbf{C}}^{\times})^n\), then \({\mathfrak{g}}= {\mathbf{C}}^n\) with \([AB] = 0\).

Recalling \({\mathfrak{sl}}_n = \left\{{\operatorname{tr}(A) = 0}\right\}\) and \({\mathfrak{so}}_n = \left\{{A + A^t=0}\right\}\), one can define \(e^A \mathrel{\vcenter{:}}=\sum_{n\geq 0} A^n/n!\); then e.g. \(\operatorname{tr}(A) = 0 \implies \operatorname{det}(e^A) = 1\). Note that one needs characteristic zero here to make sense of terms like \(1/n!\)

Recall \({\mathfrak{sl}}_2({\mathbf{C}}) = {\mathbf{C}}\left\langle{e,f,h}\right\rangle\), and an action \({\mathfrak{sl}}_2\curvearrowright V\) is equivalent to a choice of 3 operators \(e,f,h\in { \operatorname{End} }_k(V)\). Writing \(V = \bigoplus_{m\in {\mathbf{Z}}_{\geq 0}} V_m\) as a sum of weight spaces for \(h\), where for \(v_i\in V_m\) one has relations

• \(ev_i = (m-i+1)v_{i-1}\)
• \(f v_i = (i+1)v_{i+1}\)
• \(h v_i = (m-2i)v_i\)

One can write the irreducible representations as \(U_m \mathrel{\vcenter{:}}=\left\{{p_m(x, y) }\right\}\), polynomials of degree \(m\) where the \(U_m\) can appear in the \(V_m\) with multiplicity. Letting \(f:V\to V\) be nilpotent, so \(f^N = 0\), over \({\mathbf{C}}\) one gets a JCF where an \(m\times m\) block has ones on the superdiagonal, yielding a chain \(v_{-1} = 0, v_1'\to v_2'\to\cdots\to v_{m-1}'\to v_m = 0\). Rescaling \(v_i \mathrel{\vcenter{:}}= v_i'/(m-i)!\) yields the above relations and proves the theorem.

We’ll sketch a proof of Mukai’s result. Define \begin{align*} {\mathbf{C}}^n &\curvearrowright k[x_1,\cdots, x_n, y_1,\cdots, y_n] \\ {\left[ { t_1\cdots, t_n} \right]} &\mapsto x_i\mapsto x_i, y_i\mapsto y_i + t_i x_i .\end{align*} Let \(G\mathrel{\vcenter{:}}={\mathbf{C}}^r \leq {\mathbf{C}}^n\) be some vector subspace, so \(G\cong {\mathbf{G}}_a^r\). It turns out that \(S^G\) is the total Cox ring of \(X \mathrel{\vcenter{:}}=\operatorname{Bl}_{p_1,\cdots, p_n} {\mathbf{P}}^{r-1}\), which is generally defined as \begin{align*} \operatorname{Cox}(X) \mathrel{\vcenter{:}}=\bigoplus _{L\in \operatorname{Pic}(X) } H^0(X; L) .\end{align*}

Taking \(r=3\) yields \(X\mathrel{\vcenter{:}}=\operatorname{Bl}_{p_1,\cdots, p_n} {\mathbf{P}}^2\). Note that \(\operatorname{Pic}(X) = {\mathbf{Z}}^{n+1}\), since any \(D\in \operatorname{Pic}(X)\) can be written as \(D = a_0 [H] + \sum a_i E_i\) where \([H] \in \operatorname{Pic}({\mathbf{P}}^2)\) is the hyperplane (line) class and \(E_i\) are the exceptional curves.

Thus the strategy is to find points, blow up, and show \(\operatorname{Eff}(X)\) is not finitely-generated. Note that \(E_i^2 = -1\) are effective \((-1){\hbox{-}}\)curves.

Producing the example: blow up 9 points on an elliptic curve. Take two cubics \(C_1, C_2\) in \({\mathbf{P}}^2\), intersecting at 9 points, and blow them up. This yields a pencil of curves, and in fact an elliptic fibration with \(C_1, C_2\) in the fibers. The exceptional curves yield sections \(E_i\). The Mordell-Weil group yields sections, and the differences between points yields elements of infinite order:

Continuing real geometric invariant theory. Setup: let \(G \curvearrowright X\) be a linearly reductive group (not necessarily finite) acting on an affine variety \(X= \operatorname{mSpec}R\), e.g. \(R = {\mathbf{C}}[x]\). We have a subring \(R^G \hookrightarrow R\), so \(\operatorname{mSpec}R\to \operatorname{mSpec}R^G\) and we define \(X { \mathbin{/\mkern-6mu/}}G\mathrel{\vcenter{:}}=\operatorname{mSpec}R^G\) to be the affine quotient.

Note that if \(Y \subseteq X{ \mathbin{/\mkern-6mu/}}G\) is a hypersurface \(V(f)\) with \(f\in R^G\), it pulls back to a \(G{\hbox{-}}\)invariant hypersurface \(V(\pi^{-1}(f)) \subseteq X\). Any point in \(X{ \mathbin{/\mkern-6mu/}}G\) is an intersection \(\cap V(f_i)\), which pulls back to an intersection of hypersurfaces. Thus assuming \({\mathrm{Orb}}_1, {\mathrm{Orb}}_2\) are disjoint, it suffices to find a \(G{\hbox{-}}\)invariant function the separates the points \(\pi({\mathrm{Orb}}_1)\) and \(\pi({\mathrm{Orb}}_2)\). Also note that if orbits intersect, they are in the same fiber and thus map to the same point – the claim is that this is the only way this can happen.

To see that this implies (3), consider a fiber:

Missed the verbal argument here.

To see (1), note that \(R^G = {\mathbf{C}}\left\langle{f_1,\cdots, f_n}\right\rangle\) is finitely-generated as a ring by \(G{\hbox{-}}\)invariant functions \(f_i\) since \(G\) is linearly reductive, yields \(X\to X{ \mathbin{/\mkern-6mu/}}G \hookrightarrow{\mathbf{A}}^n\). Take affine coordinates for \(p = (a_1,\cdots, a_n) \in X{ \mathbin{/\mkern-6mu/}}G\). There is a surjection \({\mathbf{C}}[x_1,\cdots, x_n] \to R^G\) by \(x_i\mapsto f_i\), and similarly a surjection given by \(x_i\mapsto f_i - a_i\). Note that \({\mathbf{C}}[x_1,\cdots, x_n] \to R\) is not surjective, since giving it the trivial \(G{\hbox{-}}\)action yields a non-surjective map \({\mathbf{C}}[x_1,\cdots, x_n]^G \to R^G\). So the image is contained in some maximal ideal \({\mathfrak{m}}\in \operatorname{mSpec}R\), and the claim is that \({\mathfrak{m}}\mapsto {\mathfrak{m}}_p \mathrel{\vcenter{:}}=\left\langle{f_1-a_1,\cdots, f_n-a_n}\right\rangle \in \operatorname{mSpec}R^G\) corresponding to \(p\).

In other words, take \(\left\langle{f_i - a_i}\right\rangle \in R^G\), and the claim is that \(\left\langle{f_i - a_i}\right\rangle\neq R\), or equivalently \(V(f_i - a_i)\neq \emptyset\). This ideal is everything exactly when \(R{ {}^{ \scriptscriptstyle\oplus^{n} } } \to R\) surjects by \((t_i)\mapsto \sum t_i(f_i - a_i)\), but linearly reactivity would give \((R^G){ {}^{ \scriptscriptstyle\oplus^{n} } }\twoheadrightarrow R^G\).

Proving that \(\pi\) is a closed map: let \(a \subseteq R, Z \subseteq X\) closed, and \(\pi(Z) \subseteq X{ \mathbin{/\mkern-6mu/}}G\). Note that \(X\to Y\) yields a ring map \(R\mapsfrom_{\phi} S\) and \(a\) corresponds to \(\left\langle{\phi^{-1}(a)}\right\rangle\). For a subring, this is intersection. Define a map \((t_i)\mapsto \sum t_i g_i\) where \(R{ {}^{ \scriptscriptstyle\oplus^{n} } }\twoheadrightarrow a = \left\langle{g_1,\cdots, g_n}\right\rangle\). Then \((R^G){ {}^{ \scriptscriptstyle\oplus^{n} } } \twoheadrightarrow a^G\). Consider \(X \to X{ \mathbin{/\mkern-6mu/}}G\) by \(Z \xrightarrow{\pi} { \operatorname{cl}}\pi(Z)\).

To be continued, use property 1 but for a subset.

Things we quotient by: affine varieties are essentially rings. Recall that projective varieties have affine cones: regard homogeneous equations as usual equations. For quasiprojective varieties, take the projective closure to get a projective variety. However, there are also arbitrary varieties, which are perhaps not as useful. GIT mostly deals with affine or projective varieties, but note that Mumford’s book sets up the general case.

Setup: \(X\in {\mathsf{Aff}}{\mathsf{Var}}_{/ {k}}\) corresponding to \(R\in {}_{k} \mathsf{Alg}\), and \(G\curvearrowright X\) a linearly reductive group corresponding to a coaction \(G\curvearrowright R\). Take affine quotients \(X{ \mathbin{/\mkern-6mu/}}G \mathrel{\vcenter{:}}=\operatorname{Spec}R^G\) which receives a map \(X \xrightarrow{\pi} X{ \mathbin{/\mkern-6mu/}}G\).

Let \(X \subseteq {\mathbf{A}}^n\) be closed subset defining an affine variety with ideal \(I(X)\) and let \(Z \subseteq X\) be closed with \(a \mathrel{\vcenter{:}}= I(Z)\) Then \(k[x_1, \cdots, x_{n}]\twoheadrightarrow R \twoheadrightarrow R/a\) and \(a\) is \(G{\hbox{-}}\)invariant, so \(R^G \twoheadrightarrow(R/a)^G\) by linear reductivity. Since \(a\hookrightarrow R\), there is a map \(a^G\to R^G\) and \(a^G = a \cap R^G\). So \(Z{ \mathbin{/\mkern-6mu/}}G \subseteq X{ \mathbin{/\mkern-6mu/}}G\), and the claim is \({ \operatorname{cl}}\pi(Z) = Z{ \mathbin{/\mkern-6mu/}}G\). Thus \(\pi(Z)\) is closed.

Note that one can show \(R^G\) is integrally closed, so \(\operatorname{Spec}R^G\) is normal and singular in codimension 1. In general, GIT quotients will be singular – but note that taking the stack quotient will yield a smooth stack if \(X\) is smooth.

Preview of the projective case: let \(G\curvearrowright X \subseteq {\mathbf{P}}^n\) with coordinates \({\left[ {x_0: \cdots : x_n} \right]}\). Look at the affine cone \(CX \subseteq {\mathbf{A}}^{n+1}\) with coordinates \({\left[ {x_0, \cdots, x_n} \right]}\), so if \(p\in CX\) then \(\lambda p \in CX\) for any \(\lambda \in k\). Note that \(CX\) doesn’t immediately have a \(G{\hbox{-}}\)action, so we need to lift the previous action to some \(G\curvearrowright CX\) called the linearization (a lift to the corresponding line bundle). This may not be unique if \(G\) has characters. Unstable points will be those with orbits whose closure contains zero, which will correspond to nonexistent points in the quotient, so we’ll have to throw these out. Mumford gives numerical criteria to compute them.

Types of varieties:

• Affine.
• Projective: embeds into \({\mathbf{P}}^n\).
• Quasiprojective: \(U \mathrel{\vcenter{:}}= X\setminus Z \hookrightarrow X \overset{\text{closed}}\hookrightarrow{\mathbf{P}}^n\) where \(Z \subseteq X\) is closed.
• Arbitrary.

Being closed in the Zariski topology implies closed in the classical topology, and these are compact in the classical topology. Recall that proper maps are separable and universally closed – think of proper as essentially projective.

For \(k= \overline{k}\), there is a bijection between \({\mathsf{Aff}}{\mathsf{Var}}_{/ {k}}\) and \(R\in {}_{k} \mathsf{Alg}^{\mathrm{fg}}\) with no nilpotents, so there is a surjection \(\phi: k[x_1, \cdots, x_{n}]\twoheadrightarrow R\) with \(I \mathrel{\vcenter{:}}=\ker \phi\) and \(I = \sqrt{I}\). The map sends \(X\) to \(k[X] \mathrel{\vcenter{:}}= k[x_1, \cdots, x_{n}]/I\) the ring of regular functions on \(X\). If \(k = { \overline{k} }\) then \(\operatorname{mSpec}R\) consists of elements \(m = \left\langle{ x_1-a_1, \cdots, x_n-a_n}\right\rangle \in \operatorname{mSpec}k[x_1, \cdots, x_{n}]\), corresponding to points \({\left[ {a_1,\cdots, a_n} \right]} \in {\mathbf{A}}^{n}_{/ {k}}\). Similarly \({\mathsf{Aff}}{\mathsf{Sch}}\) bijects with commutative associative unital rings.

Projective varieties correspond to \({\mathbf{Z}}_{\geq 0}{\hbox{-}}\)graded rings \(R\) over \(k={ \overline{k} }\), so \(R = \oplus _{d\geq 0} R_d\) is finitely-generated without nilpotents. The map sends \(R\) to its projective spectrum \(\operatorname{mProj}R\). For arbitrary \({\mathbf{Z}}_{\geq 0}{\hbox{-}}\)graded commutative associative unital rings, one similarly defines \(\mathop{\mathrm{Proj}}R\). If \(R = k[R_1]\) is generated by degree 1 elements, then there is an embedding \(\operatorname{mProj}R \hookrightarrow{\mathbf{P}}^n\), but an arbitrary projective variety doesn’t necessarily come with such an embedding.

For this, take the Veronese subring \(R^{(e)} \mathrel{\vcenter{:}}=\bigoplus _{d\geq 0} R_{d_e}\) for \(e > 0\). This corresponds to the Veronese embedding \({\mathbf{P}}^n \hookrightarrow{\mathbf{P}}^N\) for some large \(N\), which is defined by \begin{align*} V_e: {\mathbf{P}}^n &\to {\mathbf{P}}^N \\ {\left[ {x_0: \cdots : x_n} \right]} &\mapsto {\left[ {x_0^e: x_{0}^{e-1}x_1, \cdots} \right]} \end{align*} where you send a point to all monomials in the coordinates of degree \(e\). Here \(N+1 = {n+e\choose e}\) is the number of such monomials. Note that e.g. \({x_0^{e-1} x_1 \over x_0^e} = {x_1\over x_0}\), so one can recover the former ratios from the latter. The condition of \(R = k[R_1]\) is needed to guarantee \(V_e\) is an embedding.

Let \(X \subseteq {\mathbf{P}}^n\), we want to extract a graded ring. Start with \({\mathbf{P}}^n\) corresponding to \(k[x_1, \cdots, x_{n}]\ni p_d(x_0,\cdots, x_n)\), forms of degree \(d\). Then \(\left\{{p_d = 0}\right\} \subseteq {\mathbf{P}}^n\) is well-defined, and this defines the Zariski topology on \({\mathbf{P}}^n\) Moreover any \(p_d/q_d\) is a well-defined regular function on the open subset \(\left\{{q_d\neq 0}\right\} \subseteq {\mathbf{P}}^n\).

There are several ways to produce the ring \(R\):

• Consider \(CX \subseteq {\mathbf{A}}^{n+1}\) with \(k[x_1, \cdots, x_{n}]\twoheadrightarrow R\mathrel{\vcenter{:}}= k[x_1, \cdots, x_{n}]/I(CX)\).
• Consider \(R' \mathrel{\vcenter{:}}=\bigoplus_{d\geq 0} H^0(X; {\mathcal{O}}_X(d) )\) whose sections are locally given by ratios of forms \(p_{k+d}/q_k\). If \(X\) is projectively normal, then \(R = R'\).
• Consider \(R'' = R(X, L) = \bigoplus _{d\geq 0} H^0(X, L^d)\), then taking \(\mathop{\mathrm{Proj}}\) yields \((X = \mathop{\mathrm{Proj}}R(X, L), {\mathcal{O}}(1))\). This is not a bijection; several \(R(X, L)\) can yield the same variety (e.g. by leaving out various degrees).

Constructing the correspondence \(\mathop{\mathrm{Proj}}R \rightleftharpoons R = \bigoplus _{d\geq 0} R_d\).⁸ One can safely assume the rings are finitely-generated, the general construction goes exactly the same way. Define \(\mathop{\mathrm{Proj}}R\) to be the set of prime homogeneous ideals \(p \subseteq R\) which are not contained in a certain ideal: considering \(k[x_1, \cdots, x_{n}]\), note that \(\left\langle{x_0,\cdots, x_n}\right\rangle\) does not define a point of \({\mathbf{P}}^n\), so define \(R_+ \mathrel{\vcenter{:}}=\bigoplus _{d\geq 1} R_d\) to be the irrelevant ideal. One can define fundamental closed subsets: for \({\mathbf{P}}^n\) these are of the form \(\left\{{f_d = 0}\right\}\), so generalize to \(f_d\in R_d\) and define \(Z(f_d) \mathrel{\vcenter{:}}=\left\{{ [p] {~\mathrel{\Big\vert}~}f([p]) = 0\in R/p}\right\}\). Note that \(f \equiv 0 \operatorname{mod}p\) in \(R\) iff \(f\in p\). Define fundamental closed sets as intersections \(\bigcap_\alpha Z(f_\alpha)\) and fundamental open sets as \(D(f_d) = \left\{{f_d\neq 0}\right\}\).

Sections of \({\mathcal{O}}\) are locally of the form \(f_k/g_k\), and for \({\mathcal{O}}(d)\) of the form \(f_{d+k}/g_k\). It remains to define local sections of the following on \(\mathop{\mathrm{Proj}}R\):

• \({\mathcal{O}}\): \(f_k/g_k\)

• \({\mathcal{O}}(1)\): \(f_{k+1}/ g_k\)

• \({\mathcal{O}}(d)\): \(f_{d+k}/f_k \in R \left[ { \scriptstyle { {g}^{-1}} } \right] _{\deg = d}\), where \({f\over g^n} \sim {f'\over g^m} \iff g^N(fg^m - f'g^n) = 0\) for some \(N\).

Write \begin{align*} R = k[x_1, \cdots, x_{n}]= \bigoplus_d R_d = k \oplus \left\langle{x_0, \cdots, x_n}\right\rangle_k[1] \oplus \left\langle{x_0^2, x_0x_1,\cdots}\right\rangle[2] \oplus \cdots .\end{align*} We’ll say \(R_d\) are semi-invariants of degree \(d\), where \(\lambda.f = \lambda^d f\). More generally, for a character \(\chi: G\to {\mathbf{G}}_m\) given by \(\lambda\mapsto \chi( \lambda)\), for \(\lambda \in G\) we can act by \(\lambda.f = \chi( \lambda) f\).

Note that \(R^G = R_{\mathop{\mathrm{triv}}}\), and \(\lambda.(fg) = (\chi_1 + \chi_2)( \lambda) fg\).

Constructing Proj: write \(\bigoplus _{d\in {\mathbf{Z}}} R^G_{d\chi} = k[f_1,\cdots, f_n]\) with \(f_i\) homogeneous of degree \(d\) in \(R^G_{d\chi }\). Note that \(\prod f_i^{m_i}/\prod f_i^{n_i}\) of total degree zero yield regular functions on \(D(\prod f_i^{n_i}) = V(\prod f_i^{n_i})^c\).

This is Mukai’s POV, an alternative POV is described in Mumford’s GIT. Start with \(G\curvearrowright Y = \operatorname{mProj}R\) and \(L\in \operatorname{Pic}(Y)\) ample.

1. Linearize the action. E.g. for \(G\curvearrowright{\mathbf{P}}^n =\operatorname{mProj}k[x_0,\cdots, x_n]\), there is not necessarily an action on \({\mathbf{A}}^{n+1}\), and a linearization is a lift to \(G\curvearrowright k[x_0,\cdots, x_n]\). Taking \(L\setminus\left\{{0}\right\}\) yields \(X\setminus\left\{{0}\right\}\), and \(X\setminus\left\{{0}\right\}\to Y\) is a \({\mathbf{G}}_m{\hbox{-}}\)torsor, and gluing the zero section yields an \({\mathbf{A}}^1{\hbox{-}}\)bundle. So lifting \(G\curvearrowright R(Y, L)\) is the same as lifting to the affine cone \(G\curvearrowright X\). Then define the projective quotient \(\mathop{\mathrm{Proj}}R(X, L)^G \mathrel{\vcenter{:}}= Y{ \mathbin{/\mkern-6mu/}}G\).

Basic examples:

• Mumford: \({\operatorname{SL}}_n, \operatorname{PGL}_n\curvearrowright Y\) a projective variety with affine cone \(X = CY\). Take \(R^G\) to get a graded ring, take \(\mathop{\mathrm{Proj}}\).
• Mukai: \(\operatorname{GL}_n\curvearrowright X\) which is already affine. Take semi-invariants \(R^G_\chi\) to get a graded ring and take \(\mathop{\mathrm{Proj}}\).

Note

• \({\operatorname{SL}}_n \hookrightarrow\operatorname{GL}_n \xrightarrow[]{\operatorname{det}} { \mathrel{\mkern-16mu}\rightarrow }\, {\mathbf{G}}_m\)
• \(\mu_n \hookrightarrow{\operatorname{SL}}_n \twoheadrightarrow\operatorname{PGL}_n\), and
• \(R^{\operatorname{PGL}_n}_{\operatorname{det}} = R^{{\operatorname{SL}}_n}\).

Consider \(\operatorname{PGL}_n \curvearrowright{\mathbf{P}}^n\), corresponding to \({{\operatorname{SL}}_{n+1} \over \mu_n} = {\operatorname{GL}_{n+1}\over {\mathbf{G}}_m}\curvearrowright{{\mathbf{A}}^{n+1}\setminus\left\{{0}\right\}\over {\mathbf{G}}_m}\). The linearization is an action \(\operatorname{PGL}_{n+1}\curvearrowright{\mathbf{A}}{n+1} = \left\{{{\left[ {x_0, \cdots, x_n} \right]}}\right\}\), which does not exist. However, there are natural actions \(\operatorname{GL}_{n+1}\curvearrowright{\mathbf{A}}^{n+1}\) and thus \({\operatorname{SL}}_{n+1} \curvearrowright{\mathbf{A}}^{n+1}\) by restriction. Thus \({\mathcal{O}}_{{\mathbf{P}}^n}(1)\) is not linearisable for the \(\operatorname{PGL}_{n+1}\) action, but is for \({\operatorname{SL}}_n\). One solution: work with \({\operatorname{SL}}_n\), which has trivial \(\pi_1\), while \(\operatorname{PGL}_n\) has nontrivial solution. Mumford’s solution: the power \({\mathcal{O}}_{{\mathbf{P}}^n}(n+1)\) is linearisable for \(\operatorname{PGL}_{n+1}\), since \(\mu_{n+1}\) acts trivially. This amounts to replacing \(L\) by \(L^{n+1}\) and \(R(X, L)\) by the subring \(R(X, L^{n+1})\) whose powers are divisible by \(n+1\), but their Projs are equal. The difference here is that \({\operatorname{SL}}_n\) has no characters and \(\operatorname{GL}_n\) has only one character. Mukai’s approach is easier when there are many characters, e.g. when \(G\) is a torus with characters \(\mathop{\mathrm{Hom}}(T, {\mathbf{G}}_m)\cong {\mathbf{Z}}^n \mathrel{\vcenter{:}}= M\).

Setup:

• \(X\) a projective variety, either \(X \subseteq {\mathbf{P}}^n\) or \((X, L)\) is a pair with \(L\) an ample line bundle (e.g. \(L = {\mathcal{O}}(1)\) when a subset of \({\mathbf{P}}^n\)). Note that \(L\) is ample iff \(L^N\) is very ample for some \(N\), so \(L \cong {\mathcal{O}}_X(1)\).
• \(G\curvearrowright X\) a linearly reductive group action. If \(G\) is connected, then one can freely replace \(L\) by \(L^N\).

We want to convert this to an action \(G\curvearrowright R(X, L) \mathrel{\vcenter{:}}=\bigoplus _{d\geq 0} H^0(X, L^d)\), the ring of homogeneous forms on \(X\). If \(X \subseteq {\mathbf{P}}^n\) then \(R(X, L) = k[x_1, \cdots, x_{n}]/ I(X)\) – at least modulo several beginning terms. There is a SES which can be twisted by \(d\): \begin{align*} I(X) \hookrightarrow{\mathcal{O}}_{{\mathbf{P}}^n} \twoheadrightarrow{\mathcal{O}}_X(d) \leadsto I(X)(d) \hookrightarrow{\mathcal{O}}_{{\mathbf{P}}^n}(d) \twoheadrightarrow{\mathcal{O}}_X(d) .\end{align*}

This yields a map \begin{align*} H^0({\mathbf{P}}^n; {\mathcal{O}}_{{\mathbf{P}}^n}(d)) = \left\langle{x_0^d,\cdots}\right\rangle \xrightarrow{f_d} H^0(X; {\mathcal{O}}_X(d)) \to H^1({\mathbf{P}}^n; I(X) \otimes{\mathcal{O}}_X(d) ) .\end{align*} By Serre vanishing, for \(d > d_0 \gg 0\) the \(H^1\) term vanishes, so \(f_d\) is surjective in this range.

Conversely, \(\mathop{\mathrm{Proj}}R(X, L) = X\) with \({\mathcal{O}}(1) = L\), so we can freely pass between \(X\) and \(R(X, L)\). Given \(G\curvearrowright R(X, L)\) we can take invariants to get the graded ring \(R(X, L)^G\) and take its \(\mathop{\mathrm{Proj}}\) to obtain the GIT quotient \(Y = X{ \mathbin{/\mkern-6mu/}}G \mathrel{\vcenter{:}}=\mathop{\mathrm{Proj}}R(X, L)^G\). Passing from \(G\curvearrowright X\) to \(G\curvearrowright R(X, L)\) is called linearization, i.e. lifting an action on \(X\) to an action on (sections of) \(L\). Recall that \(L\) is the sheaf of regular sections of a line bundle \({\mathbb{L}}\xrightarrow{\pi} X\):

Any rational polytope \(Q\) yields a projective toric variety \(Y_Q\) with \(({\mathbf{C}}^{\times})^2\curvearrowright Y_Q\) which is normal, acts with finitely many orbits, and \(T\mathrel{\vcenter{:}}=({\mathbf{C}}^{\times})^2 \hookrightarrow Y_Q\) as an open orbit. This \(Y_Q\) is a compactification of the torus \(T\) by adding \(T{\hbox{-}}\)orbits, and face of \(Q\) correspond to orbits:

One can ask how the polytope changes under various rational linearizations \(L \mapsto L^k\), corresponding to different GIT quotients. Here one gets 4-gons, two 3-gons, and the empty set, corresponding to 3 chambers (and empty chambers) with two walls describing the combinatorial types of the fibers.

For \({\mathbf{C}}^{\times}\curvearrowright{\mathbf{P}}^4\) one can vary to obtain the following 3D cross-sections:

Mumford’s approach for e.g. \(G = {\operatorname{SL}}_n\): \(G\curvearrowright R = R(X, L) + \bigoplus _{d\geq 0} H^0(X; L^d)\) where e.g. \(L = {\mathcal{O}}(1)\) for \(X \subseteq {\mathbf{P}}^n\). This yields \(R^G\) a graded ring and \(X{ \mathbin{/\mkern-6mu/}}G = \mathop{\mathrm{Proj}}R^G\). Setting \(Y = CX = \operatorname{Spec}R\), we can consider \(Y{ \mathbin{/\mkern-6mu/}}G = \operatorname{Spec}R^G\). We have \(Y \supseteq Y^s = \left\{{g\in G{~\mathrel{\Big\vert}~}G.y \text{ is closed}, G_y \text{ is finite}}\right\}\), the open subset of stable points. For each equivalence class of orbits under the orbit-closure equivalence, there is a unique closed orbit.

One can show that unstable points are a closed condition, so \(Y^s\) is open in \(Y^{\mathrm{ss}}\).

Mukai’s approach: for \({\mathbf{G}}= \operatorname{GL}_n\) define \(Y{ \mathbin{/\mkern-6mu/}}{\mathbf{G}}= X{ \mathbin{/\mkern-6mu/}}G\).

Consider sextic curves in \({\mathbf{P}}^2\), given by polynomials \(p_6(x_0, x_1, x_2)\). Note that smooth curves are stable since their orbits are closed.

The case of interest: a moduli space \({\mathcal{M}_g}\) parameterizing smooth curves. One choose a linear system \(\phi_{{\left\lvert {2K_C} \right\rvert}}: C\hookrightarrow{\mathbf{P}}^n\), then one shows that the smooth curves \(\left\{{C\hookrightarrow{\mathbf{P}}^n}\right\} \subseteq \operatorname{Hilb}, {\operatorname{CH}}\) are stable, but one also picks up singular stable curves and quotienting yields a compactification \(\overline{{\mathcal{M}_g}}\).

See Sylvester, a first good American mathematician! He proved some important theorems in invariant theory.

Setup: \(X \subseteq {\mathbf{P}}^n\) with \(G\curvearrowright X, {\mathbf{P}}^n\), \(G\) linearly reductive, which is linearized so that \(G\curvearrowright{\mathbf{A}}^{n+1}\) acting on projective coordinates is a linear action. Thus each \(g\in G\) induces \(\rho_g\) which is linear in the coordinates \(x_0, x_1,\cdots, x_n\). We have

Link to Diagram

Define \(X^u = X\setminus X^s\) to be the unstable points; our main problem is to describe \(X^u, X^s, X^{\mathrm{ss}}\).

Note that replacing \(\lambda\) with \(-\lambda\), one can replace the above conditions with \(w_i \geq 0\) and \(w_i > 0\) respectively. Most papers on GIT start with this theorem, and finding the unstable locus is a computation.

Note that for odd \(d\), stable = semistable, and for even \(d\) they are different.

For \(d=4\), consider the double cover \(z^2 = f(x,y)\):

So

• Smooth curves are stable, corresponding to \((1,1,1,1)\) or \(z^2 = (x - a)(x-b)(x-c)(x-d)\)
• Nodal curves are semistable, corresponding to \((1,1,2)\) (\(z^2 = (x-a)(x-b)(x-c)^2\)) or \((2, 2)\) (\(z^2 = (x-a)^2(x-b)^2\)).
• Tacnodes are unstable, corresponding to \((4)\), so \(z^2 = (x-a)^4\),
• Cusps are unstable, corresponding to \((1,3)\), so \(z^2 = (x-a)(x-b)^3\)

Thus the \(j{\hbox{-}}\)line \({\mathbf{A}}^1\) corresponds to smooth/stable curves, and compactifies to \({\mathbf{P}}^1 = {\mathbf{P}}(2,3)\) by adding nodal curves.

For \({\operatorname{SL}}_3\curvearrowright X\) for \(X\) the space of cubic curves in \({\mathbf{P}}^2\), we have several possibilities for curves \(f_3(x,y) = 0\):

We have \(f_3\in {\mathbf{P}}^{{3+2\choose 2} - 1} = {\mathbf{P}}^9\), with 10 coordinates: \begin{align*} x^3,y^3,z^3, x^2 y, x^2 z, y^2 x, y^2 z, z^2 x, z^2 y, xyz .\end{align*} Each curve \(f = 0\) to a closed subscheme of \({\mathbf{P}}^2\) whose ideal is \(\left\langle{f}\right\rangle\). There is an action of \({\operatorname{SL}}_3 \curvearrowright{\left[ {x,y,z} \right]}\) on coordinates, and a maximal torus \(T = \operatorname{diag}(t_1, t_2, t_1^{-1}t_2^{-1})\). Choosing this torus in a diagonal form is equivalent to choosing a coordinate system. One can then look at \({\mathbf{G}}_m \hookrightarrow T\) and consider its action to define weights. We get the following triangle of monomials:

Take this and project to get weights:

This gives \(w(x) = (1,0), w(y) = (0, 1), w(z) = (-1, -1)\) Those with the right weights are \(x^2 z, xz^2, z^3, yz^2\), all containing a factor of \(z\). So any polynomial of the form \(f(x,y,z) = (axz + bxz^2 + cz^2 + dyz)\) is unstable.

Thus the following are unstable:

The game: take a line through the center point \(xyz\), rotate, take monomials on the positive side, and check for instability, since we need \(w(xyz) = 0\). It turns out that smooth cubics are stable, simple nodes are semistable, and anything worse than \(xyz = 0\) is unstable.

Hilbert-Mumford criterion: \(G\curvearrowright X \subseteq {\mathbf{P}}^n\) projective, which is linearized to \(G\curvearrowright{\mathbf{A}}^{n+1}\) acting on coordinates \(x_0',\cdots, x_n'\). For a point \(p\in X\) corresponding to \(\tilde p\in {\mathbf{A}}^{n+1}\), is it stable, semistable, etc? Note

• \(p \in X^s \iff \forall \lambda \in {\mathsf{Grp}}({\mathbf{G}}_m, G)\) and coordinate systems \(x_0,\cdots, x_n\) with \(\lambda(t).x_i = t^{w_i} x_i\), there exists some \(w_i > 0\) and some \(w_j < 0\), where \(x_i(p) \neq 0\) (the \(i\)th coordinate is nonzero).
• \(p\in X^{\mathrm{ss}}\iff \forall \lambda\) as above, there exist \(w_i \leq 0, w_j \leq 0\).
• \(p\in X^\mathrm{unst}\iff \exists \lambda\) as above where \(w_i > 0\) for all \(i\).

Equivalently,

• \(p\in X^\mathrm{st}\iff\) the orbit \(\lambda({\mathbf{G}}_m).\tilde p \subseteq {\mathbf{A}}^{n+1}\) is closed for all \(\lambda\) and the stabilizer \({\operatorname{Stab}}_{\tilde p} {\mathbf{G}}_m\) is finite.
• \(p\in X^{\mathrm{ss}}\iff\) the orbit closure \(\overline{ \lambda({\mathbf{G}}_m).\tilde p} \not\ni \mathbf{0}\) for all \(\lambda\). The picture:

• \(p\in X^\mathrm{unst}\iff\) there exists a \(\lambda\) such that the orbit \(\overline{\lambda({\mathbf{G}}_m).\tilde p} \ni \mathbf{0}\).

If \(\tilde p = {\left[ {x_0,\cdots, x_n} \right]} \subseteq {\mathbf{A}}^{n+1}\), then \(t.x_i = t^{w_i} x_i\). So case (3) above corresponds to \(\lim_{t\to 0} t.\tilde p = \mathbf{0}\), so the origin is in the closure of \({\mathbf{G}}_m.p \subseteq G.p\). In case (2), \({\mathbf{G}}_m.\tilde p \subseteq {\mathbf{A}}^{n+1}\setminus\left\{{0}\right\}\), and to compute the closure on consider \(\lim_{t\to 0, \infty} {\left[ {t^{w_1} x_1, \cdots, t^{w_n} x_n} \right]} \not\in {\mathbf{A}}^{n+1}\). Note that by the valuative criterion, any map from a smooth curve to a proper variety can be extended to its compactification, so \({{\mathbf{C}}^{\times}}\to {\mathbf{A}}^{n+1} \subseteq {\mathbf{P}}^{n+1}\) extends to \({\mathbf{P}}^n\to {\mathbf{P}}^{n+1}\) uniquely, which is where this limit is computed. If some \(w_i = 0\), split into cases:

• \(w_i = 0\) for all \(i\): then \({\operatorname{Stab}}_{\tilde p} {\mathbf{G}}_m = {\mathbf{G}}_m\).
• Some \(w_i\neq 0\): then \({\operatorname{Stab}}_{\tilde p} {\mathbf{G}}_m\) is finite.

So \(\lim_{t\to 0} {\left[ {\cdots, t^0 x_i, \cdots} \right]} = {\left[ {\cdots, x_i, \cdots} \right]}\neq \mathbf{0}\).

Discuss this next Thursday in class.

Some applications:

• Stability of hypersurfaces \(X_d \subseteq {\mathbf{P}}^n\): write \(f_d(x) = \sum a_mx^m\) with \(X_d \subseteq {\mathbf{P}}^n\). Note that \({\operatorname{SL}}_{n+1}\curvearrowright{\mathbf{P}}^N\). This corresponds to a choice of points in the lattice polytope for degree \(d\) monomials, the weight polytope:

Then \(({\mathbf{P}}^N)^s/ {\operatorname{SL}}_{n+1}\) contains nonsingular hypersurfaces, and is contained in \(({\mathbf{P}}^N)^{\mathrm{ss}}{ \mathbin{/\mkern-6mu/}}{\operatorname{SL}}_{n+1}\) which is a projective variety which adds new semistable but not stable surfaces at the boundary.

Note that if \(X_d, X_d' \subseteq {\mathbf{P}}^n\) with \(X_d \cong X_d'\), then there exists a \(g\in \operatorname{PGL}_n\) with \(g.X_d = X_d'\) for \(n\geq 4\): since \(\operatorname{Pic}X_d = {\mathbf{Z}}[H]\) by Lefschetz, the linear system \(\phi_{{\left\lvert {H} \right\rvert}}: X_d \hookrightarrow{\mathbf{P}}^n\) defines an embedding, and \(H^0(X_d; {\mathcal{O}}(H)), H^0(X_d'; {\mathcal{O}}(H))\) differ only by choosing a basis of sections.

Every \(p\in {\mathbf{P}}^n\) with \(p={\left[ {a_0:\cdots :a_n} \right]}\) has a dual \(H_p \in ({\mathbf{P}}^n) {}^{ \vee }\) where \(H_p = V(\ell)\) for \(\ell\) the line \(\sum a_i x_i\). For any \(d\) points \(p_i\), taking the product \(f_d \mathrel{\vcenter{:}}=\prod \ell_i\) yields….something.

The Chow variety \({\operatorname{Ch}}(d, x, {\mathbf{P}}^n)\) parameterizes cycles \(X\mathrel{\vcenter{:}}=\sum n_i x_i\) with \(x_i \subseteq {\mathbf{P}}^n\) each of dimension \(k\) where \(\deg X = \sum n_i \deg X_i = d\). Let \(X^k \subseteq {\mathbf{P}}^n \supseteq P^{n-k-1}\), a generic such hyperplane won’t intersect \(X^k\). They are parameterized by \({\mathbb{Gr}}(n-k-1, n) ={\operatorname{Gr}}(n-k, n+1)\) which contains a hypersurface \((X) = \left\{{P^{n-k-1} {~\mathrel{\Big\vert}~}P^{n-k-1} \cap X\neq \emptyset}\right\}\). Since this is a codimension 1 condition, it’s given by an equation \(\left\{{F_X = 0}\right\}\). This is the Chow form of \(X\), which replaces the many equations of \(X\) with a single equation.

What this looks like for hypersurfaces \(X_d \subseteq {\mathbf{P}}^n \supseteq P^0\), which is a point. There are parameterized by \({\mathbb{Gr}}(0, {\mathbf{P}}^n) = {\operatorname{Gr}}(1, n+1) = {\mathbf{P}}^{n}\). The equation of the Chow form recovers the equation of \(X\). This recovers the point-hyperplane correspondence from before for \({\mathbf{P}}^n\).

Note that \(\operatorname{Pic}G = 1\) for any Grassmannian \(G\), and the surface \(\left\{{F_x = 0}\right\}\) lives in \(\operatorname{Sym}^n H^0(G; {\mathcal{O}}_G(1))\).

Some major applications of GIT:

• Moduli of sheaves: \(\operatorname{Pic}\) and \(\operatorname{Jac}\) as varieties/schemes, moduli of (semi)stable vector bundles (Narasimhan, Seshadri, Mumford), and more generally moduli of semistable coherent sheaves (Maruyama, Simpson). See Gieseker for moduli of vector bundles over surfaces. In this situation, GIT works very well – this is a “linear” problem.

• Moduli of varieties: stable curves (Mumford), some surfaces (Gieseker). GIT works less well here, since this is “nonlinear.”

We’ll proceed to look at the first case, moduli of sheaves. Note that for quasicoherent sheaves, one instead needs to pass to pro-objects in coherent sheaves.

Let \(X\in \mathop{\mathrm{Proj}}{ {\mathsf{Var}}_{/k}}\) where \(k\) is is not necessarily algebraically closed. We can define the abstract group \(\operatorname{Pic}(X)\) of invertible (i.e. locally free of rank 1) sheaves on \(X\) modulo isomorphism. There is also a Picard scheme \(\operatorname{Pic}(X) = \operatorname{Jac}(X)\) which is the fine moduli space of invertible sheaves of fixed degree or fixed Hilbert polynomial, which has the structure of a scheme over \(k\) – if \(\operatorname{ch}k = 0\) then it is an algebraic variety, but may have nilpotents in positive characteristic. For appropriate choices, this can be made into a group scheme/variety.

Families of invertible sheaves will correspond to moduli functors \begin{align*} \underline{M}: ({\mathsf{Sch}}^{{\mathrm{ft}}}_{\operatorname{Spec}{\mathbf{C}}})^{\operatorname{op}}&\to {\mathsf{Set}}\\ S &\mapsto \left\{{ \text{Invertible sheaves $F$ on $X \underset{\scriptscriptstyle {\operatorname{Spec}{\mathbf{C}}} }{\times} S$} }\right\} / \cong .\end{align*} Such an \(F\) should be thought of as a family of invertible sheaves on \(X\) parameterized by \(S\), i.e. for every \(s\in S\) there is a sheaf \(F_s \mathrel{\vcenter{:}}={ \left.{{F}} \right|_{{X_s}} }\) where \(X_s\) is the fiber over \(s\):

For each \(f: S\to T\) we obtain \(\underline{M}(T) \to \underline{M}(S)\), and pullbacks \(X\times S \xrightarrow{f} X\times T\) induces \(F\mapsto f^* F\).

We also require that each \(F\in \underline{M}(S)\) is equipped with a rigidification: a fixed trivialization \({ \left.{{F}} \right|_{{p\times S}} }\cong{\mathcal{O}}_S\):

This kills automorphisms and gives a fine moduli space. Without this, one could twist by anything coming from the base, so one could alternatively define \begin{align*} \underline{M}'(S) = { \left\{{\text{$F$ on $X\times S$ }}\right\} \over F\to F\otimes\pi^* L}, \qquad L\in \operatorname{Pic}(S) \end{align*} This coincides with the previous notion when \(L\) has a section.

Note that \(\operatorname{Pic}^0_{X/k}\) is a group variety and the other components are torsors over it. Since \([{\mathcal{O}}_X]\in \operatorname{Pic}_{X/k}\), one can compute \(\dim {\mathbf{T}}_{[{\mathcal{O}}_X]} \operatorname{Pic}_{X/k} = h^1({\mathcal{O}}_X)\), which is \(\dim \operatorname{Pic}_{X/k}\) if \(\operatorname{Pic}_{X/k}\) is reduced – this is automatic in characteristic zero, and necessary since \(k[{\varepsilon}]/{\varepsilon}^2\) has dimension 0 but tangent space dimension 1.

Adapting this moduli problem to vector bundles: take the functor sending \(S\) to sheaves \(F\) on \(X \underset{\scriptscriptstyle {\operatorname{Spec}{\mathbf{C}}} }{\times} S\) which are flat over \(S\), noting that there is no way to rigidify in this case. Without any additional conditions, this leads to something horribly infinite. Consider \(X = {\mathbf{P}}^1\) and take \(F = {\mathcal{O}}(k) \oplus {\mathcal{O}}(-k)\), so \(\deg F = 0\) and \(\operatorname{rank}F = 2\), where \(k\in {\mathbf{Z}}\). This is an unbounded family, parameterized by an infinite discrete set \({\mathbf{Z}}_{\geq 0}\), so we need to restrict to nice vector bundles to exclude this case.

One interesting case: \(X\) a curve but not irreducible. The moduli of invertible sheaves is already nontrivial, since subsheaves may only be defined on some irreducible components and thus not be invertible. Here \({\mathcal{O}}_X(1)\) may have different degrees on different components; as long as they are positive, \({\mathcal{O}}_X(1)\) is ample, and different polarizations yield different Jacobians and balancing these leads to interesting combinatorics.

Today: semistable sheaves on a projective variety \(X \subseteq {\mathbf{P}}^N\), where \({\mathcal{O}}_X(1)\) is the pullback of \({\mathcal{O}}_{{\mathbf{P}}^N}(1)\). Let \({\mathcal{F}}\in {\mathsf{Coh}}(X)\), e.g. a vector bundle (locally free of rank \(r\)) or a line bundle (vector bundle with \(r=1\)). Note \(X\) is covered by affine varieties \(\operatorname{Spec}R_i\) corresponding to rings \(R_i\), and on affine varieties,

• quasicoherent sheaves \({ \left.{{{\mathcal{F}}}} \right|_{{\operatorname{Spec}R_i}} }\) correspond to modules \(M\in {}_{R_i}{\mathsf{Mod}}\),
• coherent sheaves \({ \left.{{{\mathcal{F}}}} \right|_{{\operatorname{Spec}R_i}} }\) correspond to modules \(M\in {}_{R_i}{\mathsf{Mod}}^{\mathrm{fg}}\) which are finitely generated.

For vector bundles, \(M\cong R^r\). By Serre vanishing, \begin{align*} H^{> 0}(X; {\mathcal{F}}(n)) = 0\,\, n \gg 0, \qquad \dim_k H^0(X; {\mathcal{F}}(n)) < \infty\,\,\forall n ,\end{align*} and Grothendieck vanishing yields \(H^{> \dim X}(X; {\mathcal{F}}) = 0\) for any \({\mathcal{F}}\in {\mathsf{QCoh}}(X)\). By Hirzebruch-Riemann-Roch, the Hilbert series \begin{align*} h_{\mathcal{F}}(n) \mathrel{\vcenter{:}}=\chi({\mathcal{F}}(n)) \mathrel{\vcenter{:}}=\sum (-1)^i h^i({\mathcal{F}}(n)) = h^0({\mathcal{F}}(n)),\,\, n\gg 0 \end{align*} is a polynomial in \(n\). This is proved by writing \(Y \mathrel{\vcenter{:}}= X \cap H\) to get \({\mathcal{F}}(-1)\hookrightarrow{\mathcal{F}}\twoheadrightarrow{ \left.{{{\mathcal{F}}}} \right|_{{Y}} }\) and \(h_{\mathcal{F}}(n) - h_{\mathcal{F}}(n-1) = h_Y(n)\). Thus it suffices to know \(h_Y\) is a polynomial, since the LHS is the discrete derivative of \(h_{\mathcal{F}}\), and \(\dim Y < \dim X\).

We define the reduced Hilbert polynomial as \(\overline{h}_{\mathcal{F}}(n) \mathrel{\vcenter{:}}= h_{\mathcal{F}}(n)/h_n\) where \(h_n\) is the leading coefficient of \(h_{\mathcal{F}}(n)\).

Note that condition (a) is automatic for locally free sheaves, since \({\mathcal{F}}_x = {\mathcal{O}}_X{ {}^{ \scriptscriptstyle\oplus^{r} } }\) for every \(x\) and thus \(\mathop{\mathrm{supp}}{\mathcal{F}}= X\). An example where this won’t hold: let \(i: p\hookrightarrow X\) for \(X\) a curve and take the skyscraper sheaf \(i_* {\mathcal{O}}_p\). More generally, for a closed embedding \(i: Z\hookrightarrow X\) one gets \(\mathop{\mathrm{supp}}(i_* {\mathcal{O}}_Z) \subseteq Z \subseteq X\). If \(X\) is a smooth curve, then any \({\mathcal{F}}\in {\mathsf{Coh}}(X)\) decomposes as \({\mathcal{F}}= {\mathcal{F}}_{\operatorname{tors}}\oplus {\mathcal{F}}_{{\mathrm{free}}}\) where \({\mathcal{F}}_{\operatorname{tors}}\) is a torsion sheaf which is a sum of skyscrapers supported at points and \({\mathcal{F}}_{\mathrm{free}}\) is locally free. The Hilbert polynomial will be constant on \({\mathcal{F}}_{\operatorname{tors}}\).

Hilbert stability for smooth curves \(X\): (a) holds iff \({\mathcal{F}}\) is torsionfree. If this holds, then since \(0\neq {\mathcal{E}}\subseteq {\mathcal{F}}\) with \({\mathcal{F}}\) torsionfree, \({\mathcal{E}}\) is locally free. Then condition (b) is equivalently to \(\mu({\mathcal{E}}) < \mu({\mathcal{F}})\), respectively \(\mu({\mathcal{E}}) \leq \mu({\mathcal{F}})\). Thus Hilbert stability for curves is slope stability.

For \(g\geq 2\), there is a moduli space of stable and semistable vector bundles of fixed rank \(r\) and degree \(d\) This recovers \(\operatorname{Jac}\) for \(r=1\), and thus is a noncommutative generalization of \(\operatorname{Pic}\). If \(d=r(g-1)\) then one can define a theta divisor, and around 20 years ago there was an analog of the Riemann-Roch formula which computed its sections.

Two basic notions to discuss: \(S{\hbox{-}}\)equivalence on semistable sheaves, and the Harder-Narasimhan filtration.

The result: the associated graded pieces \({\mathcal{F}}_i/{\mathcal{F}}_{i+1}\) are semistable with increasing slopes. E.g. take \(0 \subseteq {\mathcal{O}}(1) \subseteq {\mathcal{O}}\oplus {\mathcal{O}}(1)\) where \(\mu({\mathcal{O}}(1)) = 1\) and \(\mu({\mathcal{O}})=0\). If \({\mathcal{F}}\) is semistable, the subsheaves will all have the same slope and the pieces \({\mathcal{F}}_i / {\mathcal{F}}_{i+1}\) are indecomposable.

Coming up: setting up the GIT problem matches up the notions of (semi)stability, and \(S{\hbox{-}}\)equivalence becomes orbit-closure equivalence.

See this very interesting paper posted today! https://arxiv.org/pdf/2211.07061.pdf

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…”.

Setup: let \(X \subseteq {\mathbf{P}}^N\) be a projective variety with \({\mathcal{O}}_X(1)\) and \({\mathcal{E}}\in {\mathsf{Coh}}(X)\) and Hilbert polynomial \(p({\mathcal{E}}, n) = \chi({\mathcal{E}}(n))\). One can easily prove by induction that \(p\) is in fact a polynomial, and it turns out to have terms of the form \(p({\mathcal{E}}, n) = r {n^d\over d!} + a {n^{d-1}\over (d-1)!} + \cdots\). We define

• \(d({\mathcal{E}}) \mathrel{\vcenter{:}}=\dim \mathop{\mathrm{supp}}({\mathcal{E}})\) – for \({\mathcal{E}}\in \operatorname{Pic}(X)\), this is \(\dim X\), but in general could be smaller. It turns out \(d({\mathcal{E}}) = d \mathrel{\vcenter{:}}=\deg p({\mathcal{E}}, n)\).
• A “generalized rank” of \({\mathcal{E}}\) by \(r({\mathcal{E}}) = r\), the leading coefficient in \(p({\mathcal{E}}, n)\) above.
• \(\mu({\mathcal{E}}) \mathrel{\vcenter{:}}= a/r\) the slope.
• \(\overline{p}({\mathcal{E}}, n) \mathrel{\vcenter{:}}={1\over r}p({\mathcal{E}}, n) = {n^d\over d!} + {a\over r} {n^{d-1}\over (d-1)!} + \cdots\), noting that the first nontrivial coefficient is the slope \(a/r\).

This will essentially be a quotient by \({\operatorname{SL}}(V)\) for some \(V\). Let \({\mathcal{E}}\in {\mathsf{Coh}}(X)\), then Serre yields that for all \(n\gg n_0\),

1. \(H^{>0}({\mathcal{E}}(n)) = 0\), \(\dim H^0({\mathcal{E}}(n)) < \infty\), and
2. The twist \({\mathcal{E}}(n)\) is generated by global sections.

Thus there is a surjection \begin{align*} 0 \to K \mathrel{\vcenter{:}}=\ker f \to H^0({\mathcal{E}}(n)) \otimes{\mathcal{O}}_X \xrightarrow[]{f} { \mathrel{\mkern-16mu}\rightarrow }\, {\mathcal{E}}(n) \to 0 .\end{align*} Note that a global section \(s\in {{\Gamma}\qty{{\mathcal{F}}} }\) is equivalent to a morphism \begin{align*} {\mathcal{O}}_X &\to {\mathcal{F}}\\ 1 &\mapsto s .\end{align*} Untwisting this surjection yields \begin{align*} 0 \to K(-n) \to H^0({\mathcal{E}}(n))\otimes{\mathcal{O}}_X(-n) \xrightarrow[]{\tilde f} { \mathrel{\mkern-16mu}\rightarrow }\, {\mathcal{E}}\to 0 .\end{align*}

Note that if \(V\) is a vector space, every dimension \(r\) subspace yields a codimension \(r\) quotient, so \({\operatorname{Gr}}_{r, N}\) also parameterizes quotients, and we choose quotients as they are better behaved from a commutative algebraic POV.

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 \({\mathbf{A}}^n\), whereas it is only subvariety iff \(I = \sqrt I\). This is equivalent to asking if \(V(I)\) is reduced.

Note that \(V\mathrel{\vcenter{:}}= H^0({\mathcal{E}}(n))\) is fixed in the proof, and fixing this is equivalent to choosing a basis of \(H^0({\mathcal{E}}(n))\), so \(\operatorname{Quot}({\mathcal{G}}, P)\) encodes a choice of basis. To forget this choice, we need to quotient by change of basis, and we’ll have \(M({\mathcal{O}}_X, P) \mathrel{\vcenter{:}}=\operatorname{Quot}(V\otimes{\mathcal{O}}_X(-n), P) { \mathbin{/\mkern-6mu/}}{\operatorname{SL}}(V)\).

Note that the Grassmannian has a Plucker embedding into \({\mathbf{P}}^N\) for some large \(N\). We have \({\operatorname{SL}}(V)\curvearrowright\operatorname{Quot}({\mathcal{G}}, P)\), so we can apply the Hilbert-Mumford numerical criterion to the induced action \({\operatorname{SL}}(V)\curvearrowright{\mathbf{P}}^N\) – doing this precisely yields the (semi)stability criterion \(\overline{p}({\mathcal{F}}, n) \leq \overline{p}({\mathcal{E}}, n)\). The hard part will be boundedness – e.g. consider \(X\) a curve and \({\mathcal{F}}= {\mathcal{O}}_X(n) \oplus {\mathcal{O}}_X(-n)\), 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 \(n\).

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.

Today: a sketch of a proof of existence of a moduli space of semistable sheaves. Setup: let \(X\in\mathop{\mathrm{Proj}}{\mathsf{Var}}\) or \(X\in {\mathsf{Sch}}\), fix a Hilbert polynomial \(P(n)\), and fix \({\mathcal{E}}\in {\mathsf{Coh}}(X)\) with \(p({\mathcal{E}}, n) = P(n)\); we want to construct the moduli space \(M(X, P(n))\). Using that \({\mathcal{E}}(n)\) is globally generated for \(n > n_0 \gg 0\), there is a surjection \(H^0(X; {\mathcal{E}}(n)) \otimes{\mathcal{O}}_X \twoheadrightarrow{\mathcal{E}}(n)\) and thus a surjection \(H^0(X; {\mathcal{E}}(n) )\otimes{\mathcal{O}}_X(-n) \twoheadrightarrow{\mathcal{E}}\). Note \(V \mathrel{\vcenter{:}}= H^0(X; {\mathcal{E}}(n))\) is a vector space of dimension \(\deg P(n)\) there is a surjection of sheaves \(V\otimes{\mathcal{W}}\twoheadrightarrow{\mathcal{E}}\) making \({\mathcal{E}}\in \operatorname{Hilb}(V\otimes{\mathcal{W}}, P(n))\). Grothendieck embeds this into \({\operatorname{Gr}}(V\otimes W, P(n))\), and more generally \(\operatorname{Quot}({\mathcal{Y}}, P(n))\hookrightarrow{\operatorname{Gr}}(G, a)\). At this point, \(\operatorname{Quot}\) includes the data of a choice of basis of \(V\), so we’ll quotient by an action \({\operatorname{SL}}(V)\curvearrowright V \leadsto {\operatorname{SL}}(V)\curvearrowright\operatorname{Hilb}(V\otimes{\mathcal{W}}, P(n))\).

Let \(V = H^0({\mathcal{E}}(n))\), then \(V\cdot {\mathcal{E}}= {\mathcal{E}}\), i.e. \(V\) spans the stalks, and any subspace \(H \subseteq V\) defines a subsheaf \({\mathcal{F}}\mathrel{\vcenter{:}}= H\cdot {\mathcal{E}}\leq {\mathcal{E}}\). The criterion yields \(\overline{p}_{\mathcal{F}}(n) \leq \overline{p}_{\mathcal{E}}(n)\). For a single sheaf \({\mathcal{E}}\), \(n\) depends on \({\mathcal{E}}\) and this is easy, but boundedness in families is difficult in general. To define the \({\mathbf{P}}^N\) appearing in the lemma, we’ll need to discuss Grassmannians.

For a fixed \(B\), a SES \(A\hookrightarrow B\twoheadrightarrow C\in {}_{k}{\mathsf{Mod}}\) of dimensions \(a,b,c\) respectively, note \({\operatorname{Gr}}_{a, b} = {\operatorname{Gr}}_{b, c}\) where the former parameterizes subspaces and the latter quotients. There are several levels of generality in which Grassmannians can be defined:

• Over \(k\in \mathsf{Field}\), points of \({\operatorname{Gr}}(B)\) correspond to \(\left\{{A\hookrightarrow B}\right\}\) or equivalently \(\left\{{B\twoheadrightarrow C}\right\}\)
• In families, in which cases quotients are preferred.

Let \(B = k^n\), how does one parameterize subspaces? Any subspace \(A\) has a basis \(A = \left\langle{v_1,\cdots, v_a}\right\rangle\). Fixing a basis \(k^n = \left\langle{e_1,\cdots, e_b}\right\rangle\), one can form a matrix \(M_B\in \operatorname{Mat}_{a\times b}(k)\) whose rows are the \(v_i\). There is an action \(\operatorname{GL}_a\curvearrowright M_B\) by conjugation. Recall that Plucker coordinates are the components of \((P_I)\), the determinants of all \(a\times a\) minors where \({\left\lvert {I} \right\rvert} = a\) is an index set. For \(I = \left\{{1,\cdots, b}\right\}\), there are \(b\choose a\) such minors. We can regard \((P_I) \in P^{{b\choose a} - 1} = {\mathbf{P}}( { {\bigwedge}^{\scriptscriptstyle \bullet}} ^a B)\) where \(B = \left\langle{e_1,\cdots, e_b}\right\rangle\) and \({ {\bigwedge}^{\scriptscriptstyle \bullet}} ^a = \left\langle{e_{i_1} \vee\cdots e_{i_a}}\right\rangle\). Writing \(v_i\) in the \(e_i\) basis, their Plucker coordinate is \(v_1 \vee\cdots \vee v_a = \sum P_I e_I \in { {\bigwedge}^{\scriptscriptstyle \bullet}} ^a A \cong k \subseteq { {\bigwedge}^{\scriptscriptstyle \bullet}} ^a B\).

What does the Hilbert-Mumford criterion say in this situation? Let \(K\hookrightarrow V\otimes W\twoheadrightarrow U\), and pick a basis to get \(P_I(K) = P_I(U)\) and \(v_{i_1}\vee\cdots \vee v_{i_n} = \sum p_{i_1, \cdots, i_a} e_{i_1} \vee\cdots \vee e_{i_a}\). Letting \(\left\{{f_i\otimes g_j}\right\}\) be a basis for \(V\otimes W\), consider how to linearize the action \({\operatorname{SL}}(V)\curvearrowright V\otimes W\): pick a \({\mathbf{G}}_m \subseteq {\operatorname{SL}}(V) \subseteq {\operatorname{SL}}(V\otimes W\) so \(t.f_i = t^{r_i} f_i\) with \({ \operatorname{weight} }(f_i) = r_i\) and \(\sum r_i = 0\). Then \({ \operatorname{weight} }(f_i \otimes g_j) = r_i\) since there is no action on \(W\). Check that \begin{align*} { \operatorname{weight} }\qty{ (f_{i_1} \otimes g_{j_1} ) \vee(f_{i_2}\otimes g_{j_2} ) \vee\cdots \vee( f_{i_a} \otimes g_{j_a} ) } = \sum_{s=1}^a r_{i_s} .\end{align*} Now the subspace \(K\to V\otimes W\) or quotient \(V\otimes W\to U\) is GIT is stable (resp. semistable) iff for all \(\lambda: {\mathbf{G}}_m \to {\operatorname{SL}}(V)\), there exists some \(P_I\) such that

• \({ \operatorname{weight} }\leq 0\) for semistability,
• \({ \operatorname{weight} }< 0\) 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 \(V\).

Let \(V = \left\langle{f_1,\cdots, f_n}\right\rangle\), then there are subspaces \(H_{n-1} = \left\langle{f_2,\cdots, f_n}\right\rangle, H_3 = \left\langle{f_3,\cdots, f_n}\right\rangle, \cdots, H_1 = \left\langle{f_n}\right\rangle\). This corresponds to an ordered list of weights \(r_1\leq r_2\leq \cdots \leq r_n\).

Lemma from Simpson: a point \(\qty{ K \hookrightarrow V\otimes W \twoheadrightarrow U} \in {\operatorname{Gr}}(V\otimes W, b)\) for the \({\operatorname{SL}}(V){\hbox{-}}\)action is stable (resp. semistable) iff for all \(H \leq V\), \begin{align*} { \dim H\otimes W \over \dim \operatorname{im}(H\otimes W) } \leq {\dim V\otimes W\over \dim U} \iff {\dim H\otimes W\over \dim(H\otimes W) \cap K} \geq {\dim V\otimes W \over \dim K} .\end{align*} We’ll consider \(0\to k\to {\mathbf{V}}\supseteq{\mathbb{H}}\) and pick a 1-parameter subgroup \(\lambda: {\mathbf{G}}_m\to {\operatorname{SL}}(V) \to {\operatorname{SL}}({\mathbf{V}})\) and apply the HM criterion.

Recall that if \(G\curvearrowright(X, L)\) is a linearized action with \(X\) projective and \(L\in \operatorname{Pic}^{\mathrm{amp}}(X)\). Then \(x\in X\) is stable/semistable iff for all \(\lambda: {\mathbf{G}}_m\to G\) we have \(\mu^L( \lambda, x)\geq 0\) – this is defined using \(\lim_{t\to 0} \lambda(t).x \mathrel{\vcenter{:}}= x_0\in X\) since \(X\) is projective and hence proper, and since \(\overline{x}\) is fixed by \({\mathbf{G}}_m\) we have \({\mathbf{G}}_m\curvearrowright L_{\overline{x}}\) and after picking a basis have \(\lambda(t).z = t^r z\) for \(z\in L_{\overline{x}}\) and we define \(\mu^L(\lambda, x) \mathrel{\vcenter{:}}=-r\). Replacing \(L\) by some high power, we can assume \(L = {\mathcal{O}}_X(1)\) is very ample.

If \(X\hookrightarrow{\mathbf{P}}^n\) we have \(L = { \left.{{ {\mathcal{O}}_{{\mathbf{P}}^n}(1) }} \right|_{{X}} }\) we have \(G\curvearrowright{\mathbf{A}}^{n+1}\) linearly on coordinates. Diagonalizing this action yields \(\lambda(t).x_i = t^{w_i} x_i\), and we can order the weights such that \(w_0 \geq w_1 \geq \cdots \geq w_n\). Writing \(x = {\left[ {x_0,\cdots, x_n} \right]}\) we can compute \(\overline{x} = \lim_{t\to 0} {\left[ { t^{w_1} x_1, \cdots, t^{w_n}x_n} \right]} = t^{wk}{\left[ {\cdots, x_k, 0,\cdots, 0} \right]}\). So \(\overline{x} = {\left[ {0,0,\cdots, 0,x_{k-w_k}, \cdots, x_k, 0,\cdots, 0} \right]}\) where \(w_k\) of the coordinates are nonzero. Recall \(x\) is unstable iff \(\overline{\lambda({\mathbf{G}}_m).x}\ni{\left[ {0, \cdots, 0} \right]}\) in \({\mathbf{A}}^{n+1}\), since then \(x\) would be orbit-closure equivalent to zero which is not a point in projective space.

Note that the fiber here is \(L_{\overline{x}}^{-1}= {\mathbf{C}}\overline{x} = {\mathbf{C}}\left\langle{\overline{x}_0, \overline{x}_1,\cdots, \overline{x}_k, \cdots, \overline{x}_n}\right\rangle\) (i.e. the line corresponding to \(\overline{x}\)). Here \(\lambda({\mathbf{G}}_m)\) acts with weight \(w_k\) and \(r=-w_k\) and \(-r=w_k = \mu(\overline{x}, \lambda)\). Does this match with the criterion above? Consider \(\lim_{t\to 0} \lambda(t).x \in {\mathbf{A}}^{n+1}\): \begin{align*} \lim_{t\to 0} \lambda(t).x = \lim_{t\to 0} (0,0, \cdots, ?,\cdots, t^{w_k} x_k, 0, \cdots, 0) ,\end{align*} and the bad case is \(w_k > 0\).

If \(\dim V = n\) with weights \(w_1,\cdots, w_n\), then \(\dim V\otimes W = n \dim W\) with weights \(w_1,\cdots, w_1, w_2,\cdots, w_2,\cdots, w_n,\cdots, w_n\) where each weight occurs with multiplicity \(n\). On \({\mathbf{V}}\) pick coordinates \(x_0, \cdots, x_n\) with weights \(w_1,\cdots, w_n\) so \(\lambda(t) x_i = t^{w_i} x_i\). embed \({\operatorname{Gr}}_{k ,n} \hookrightarrow{\mathbf{P}}( { {\bigwedge}^{\scriptscriptstyle \bullet}} ^k {\mathbf{V}})\) with Plucker coordinates \(p_I = x_{i_1} \vee\cdots \vee x_{i_k}\). Then \(\lambda(t) .p_I = t^{\sum_{i\in I} w_i} p_I\) has weight \(w_{i_1} + \cdots + w_{i_k}\). We want \(\lim \lambda(t). K = \overline{K}\). For simplicity assume \(w_1 > w_2 > \cdots > w_n\) with strict inequalities. In \({\mathbf{P}}(V)\) if the last coordinate is nonzero, i.e. \(p = {\left[ {0:\cdots : 0 : 1} \right]}\), the limit is \({\left[ {0:0\cdots:0:1} \right]}\).

Today: the HM criterion and a key computation for \({\operatorname{Gr}}_{k, n}\).

On the meaning: if \(\mu(\lambda, p) = -r_0\), then \(-r_0 \geq -r_i\) for all \(i\) and thus \(r_0 \leq r_i\). The good case: \(p\) is stable, then \(r_0 < 0\). The bad case: \(p\) is unstable, then \(r_0 > 0\). So we want at least one negative coefficient for \(p\) to be stable, and we can generally write \begin{align*} \mu( \lambda,p) = \min\left\{{r_i {~\mathrel{\Big\vert}~}i \text{ where } x_i(p) \neq 0}\right\} .\end{align*} For unstable points, \(\lim_{t\to 0} (t^{r_i} x_i) = \mathbf{0}\) in \({\mathbf{A}}^{n+1}\), which does not come from a projective point. Considering \(\lim_{t\to 0} t.p\) in \({\mathbf{P}}^n\), since \({\mathbf{P}}^n\) is proper the limit exists and must equal \begin{align*} \lim {\left[ {t^{r_0}x_0:\cdots:t^{r_n} x_n} \right]} = \lim {\left[ {x_0: t^{r_1-r_0}x_1:\cdots:t^{r_n-r_0}x_n} \right]} = {\left[ {x_0:\cdots:x_r:0:\cdots:0} \right]} \end{align*} where there are zeros if either \(x_i(p) = 0\) or \(r_i -r_0 >0\). In \({\mathbf{A}}^{n+1}\), consider the line \(C_{\overline{p}} = {\mathbf{C}}(x_0,\cdots, x_r, 0,\cdots, 0)\) and \({\mathbf{G}}_m\) acts by multiplication by \(t^{r_0}\). So the weight is \(r_0 = -\mu( \lambda,p)\), thus we can define \(\mu(\lambda,p)\) in another way: let \(\overline{p} = \lim_{t\to 0} t.p\) in \({\mathbf{P}}^n\), then \(\overline{p}\) is fixed by \({\mathbf{G}}_m\). There is an action on the fiber \({\mathbf{G}}_m\curvearrowright{\mathcal{O}}(1)\mathrel{\Big|}_{\overline{p}} = C_{\overline{p}}\) with some weight \(r_0\), so define \(\mu(\lambda,p) \mathrel{\vcenter{:}}=-r_0\).

Now let \(G\curvearrowright{\operatorname{Gr}}_{k, n} = \left\{{K_k \subseteq {\mathbf{C}}^n}\right\}\), where we’re choosing to work with subspaces instead of quotient spaces. This yields a SES \(K\hookrightarrow{\mathbf{C}}^n\twoheadrightarrow V\). Note that \({\operatorname{Gr}}_{k, n} \subseteq {\mathbf{P}}(\bigwedge\nolimits^k {\mathbf{C}}^n) = {\mathbf{P}}^{N-1}\) where \(N = {n\choose k}\) by the Plucker embedding. This yields a linear action \(G\curvearrowright{\mathbf{A}}^N\). Take \(\lambda: {\mathbf{G}}_m \to G\) a 1PS, then the key computation is finding \(\mu(\lambda,[ K])\) for \([K] \in {\operatorname{Gr}}_{k, n}\).

First diagonalize \({\mathbf{G}}_m\curvearrowright{\mathbf{C}}^n\) to get \(t.\mathbf{x} =\operatorname{diag}(t^{r_1},\cdots, t^{r_n}) \mathbf{x}\). Then \(G\curvearrowright{\mathbf{C}}^N\) through Plucker coordinates \(p_I\) for \(I = \left\{{i_1,\cdots, i_k}\right\} \subseteq \left\{{1,\cdots, n}\right\}\). The weight of \(t.p_I\) is \(r(p_I) \mathrel{\vcenter{:}}=\sum_{i\in I} r_i = r_{i_1} + \cdots + r_{i_k}\), and so \begin{align*} \mu( \lambda, [K] ) = \max\left\{{-r(p_I) {~\mathrel{\Big\vert}~}p_I(K) \neq 0}\right\} .\end{align*}

Assume \(r_0 > r_1 > \cdots > r_n\), then \begin{align*} \mu( \lambda, [K] ) = -k r_n - \qty{ \sum_{i=1}^{n-1} r_i - r_{i+1} } \dim K \cap L_{i+1,\cdots n} \end{align*} where \(L_{i+1,\cdots, n} \leq {\mathbf{C}}^n\) is the subspace \(\left\{{x_{i+1} = \cdots = x_n = 0}\right\}\). Take a basis of \(K \subseteq {\mathbf{C}}^n\) and represent it as the rows of a \(k\times n\) matrix. Reduce this to echelon form, but slightly reversed to emphasize the last vector:

Taking the limit of \(t.K\) in \({\mathbf{P}}^{N-1}\) yields the following:

This follows from write \(t^{r_n} e_n + t^{r_{n-1}} c_{n-1} e_{n-1} + \cdots = t^{r_n} \qty{e_n + t^{r_{n-1} - r_n}c_{n-1} e_{n-1} + \cdots }\). Labeling the pivots \(i_1,\cdots, i_k\) from right to left, we have \begin{align*} \mu &= -r_{i_1} - \cdots - r_{i_k} \\ &= - \sum_{i=1}^n \qty{\dim (K \cap L_{i+1,\cdots, n}) - \dim( K \cap L_{i, i+1, \cdots, n} )} r_i \\ &= -r_n \qty{ \dim (K \cap L_\emptyset) - \dim(K \cap L_{ n })} ,\qquad L_n \mathrel{\vcenter{:}}=\left\{{x_n\neq 0}\right\}, L_{\emptyset} \mathrel{\vcenter{:}}={\mathbf{C}}^n \\ &\quad - r_{n-1}\qty{ \dim (K \cap L_{n}) - (\dim K \cap L_{n-1, n}) } \\ &\quad -\cdots ,\end{align*} where we note that e.g. \(-(\dim K \cap L_n)(r_{n-1} - r_n)\) appears, yielding the \(r_{i} - r_{i+1}\) terms in the sum.

Simpson considers \({\operatorname{SL}}(V)\curvearrowright{\operatorname{Gr}}(K\hookrightarrow V \oplus W \twoheadrightarrow U)\), and comes up with a precise formula that enforces an upper bound on \(\dim(K \cap H\otimes W)\). This follows from Mumford’s formula: write \(V\otimes W \cong \bigoplus _{i=1}^{\dim V} W\), then \({\mathbf{G}}_m\) acts on this with weights ???. Note that if \(r_i = r_{i+1}\) then \(\dim (K \cap L_{i+1,\cdots, n})\) doesn’t contribute to \(\mu\). The critical case is when \(r_1 = \cdots = r_1 > r_2 = \cdots = r_2\). Note that the \(r_i\) form a cone and \(\mu\) is a linear function on it, and it suffices to check on rays.

For moduli of K3s, see Viehweg on GIT for \((X, L)\) with \(X\) smooth or with canonical singularities, \(L\) ample, \(K_X\) nef – note that this doesn’t provide a compactification.

Last time: moduli of sheaves on a fixed variety, a linear case where GIT works very well by reducing to a computation on a Grassmannian.

• Successes:
  • \(\overline{{\mathcal{M}_g}}\) (compactification of moduli of genus \(g\) curves)
  • Moduli of varieties with nef \(K_X\), e.g. K3 surfaces, CYs, varieties of general type, all with (very) mild singularities. However, GIT does not give a compactification here.
• Failures:
  • Compactifications for higher-dimensional varieties analogous to \(\overline{{\mathcal{M}_g}}\). Computations are infeasible most of the time, and unreasonable when computable.

Recall \({\mathcal{M}_g}\) is the moduli of smooth projective genus \(g\) curves, and for \(g\geq 2\) it is known that \(\dim {\mathcal{M}_g}= 3g-3\) which is locally a quotient of a smooth variety and is thus a smooth orbifold/stack. It is quasiprojective but not projective and not complete. One would like an inclusion \({\mathcal{M}_g}\hookrightarrow\overline{{\mathcal{M}_g}}\) a projective variety with mild singularities such that points in \({{\partial}}\overline{{\mathcal{M}_g}}\) correspond to curves with certain singularities. This is constructed by Deligne-Mumford, locally \(U/G\) where \(U\) is smooth and \(G\in{\mathsf{Fin}}{\mathsf{Grp}}\). Moreover \({{\partial}}\overline{{\mathcal{M}_g}}= \bigcup_i D_i/G\) with \(D_i\) smooth and SNC, and points in \({{\partial}}\overline{{\mathcal{M}_g}}\) correspond to DM-stable curves:

Why these three conditions are the same: recall \(\mathop{\mathrm{Aut}}{\mathbf{P}}^1 = \operatorname{PGL}_2\) which has dimension 3. Note that \(\mathop{\mathrm{Aut}}E\cong E \times G\) for \(G\in {\mathsf{Fin}}{\mathsf{Grp}}\), usually \(G\cong C_2\), and \(\dim \mathop{\mathrm{Aut}}E = 1\). Consider the nodal curve \(E\), equivalent to \({\mathbf{P}}^1/ 0\sim\infty\), so \(\mathop{\mathrm{Aut}}(C) = {{\mathbf{C}}^{\times}}\rtimes C_2\) which again has dimension 1. If \(g(C) \geq 2\) then \({\sharp}\mathop{\mathrm{Aut}}(C) < \infty\). The 3 in condition b is due to the need to fix 3 points, to drop \(\dim \operatorname{PGL}_2\) from dimension 3 to zero.

If \(X\) is Gorenstein then \(\omega_X\in \operatorname{Pic}(X)\) and can be written \(\omega_X = {\mathcal{O}}(K_X)\) where \(K_X\) is defined to be the canonical class. This holds if e.g. \(X\) has hypersurface singularities. Note that \(\omega_X\) is ample iff \(\deg { \left.{{ \omega_X }} \right|_{{C_i}} } > 0\) for each \(C_i\), and by adjunction one has \begin{align*} \deg { \left.{{\omega_X}} \right|_{{C_i}} } = \deg \omega_{C_i} + {\sharp}{ C_i \cap(C\setminus C_i) } = 2p_a(C_i) - 2 .\end{align*} Thus if \(p_a(C_i) \geq 2\) this is always positive; if \(p_a(C_i) = 0\) then \(C_i = {\mathbf{P}}_1\), otherwise if \(p_a(C_i) = 1\) and we get the curves appearing in condition b.

Consider a family \({\mathcal{C}}\to \Delta^\circ\) of smooth projective curves. By the semistable reduction theorem, after a finite base change \(\Delta'\to\Delta\) any family can be completed to \(X'\to \Delta'\) such that \(X'\) is smooth and \(X_0'\) is SNC:

Note that \(\deg { \left.{{\omega_X}} \right|_{{X_i}} } < 0 \iff C_i = {\mathbf{P}}^1\) and \({\sharp}\qty{C_i \cap(C\setminus C_i)} = 1\), or just \({\sharp}\qty{C_i \cap(C\setminus C_i)} = 2\). If \(C\cdot C_i = 0\) since \(C\) can be replaced with a disjoint fiber \(F\). Writing \(0 = C\cdot C_i = C_i^2 + (C-C_i)C_i\), where get \(C_i^2 = -1\) in the first case and \(C_i^2 = -2\) in the second case. In the first case, \(C_i\) can be contracted to yields \(X\to X_1\) (Castelnuovo’s lemma) with \(X_1\) smooth, so we can get rid of \(-1\) curves in stages to get a new surface with (potentially) only \(-2\) curves, which is a minimal model and is smooth. Contracting all \(-2\) curves yields the canonical model, which may be singular but has only canonical singularities. Note that \(X\) has canonical singularities iff \((X, X_0)\) has slc singularities iff \(X_0\) has slc singularities, which is a form of log adjunction for degenerating pairs.

See the BCHM paper.

So it’s clear how to degenerate in one parameter families, but how does one organize these various limits into a compactification? The idea is to construct \({\mathcal{M}_g},\overline{{\mathcal{M}_g}}\) using GIT to realize them as \(H/\operatorname{PGL}_n\) where \(H\) is a moduli of curves with additional data. The HM criterion gives stable and semistable points, and one hopes these coincide with the above notions.

What is \(H\)? Two answers: the Chow variety, or the Hilbert scheme. Start with \(C\) a smooth curve of genus \(g\geq 2\) with \(\deg K_C = 2g-2 \geq 2\) so that \(K_C\) is ample. Then \(nK_C\) is very ample for any \(n\geq 2\), and there is an embedding \(C \xhookrightarrow{{\left\lvert {nK_C} \right\rvert}} {\mathbf{P}}^N\). Such an embedding is given by a choice of basis of \(H^0(C; {\mathcal{O}}(nK_C))\) where two bases differ by a \(\operatorname{PGL}{\hbox{-}}\)action. Note that \(N = n(2g-2) - (g-1) - 1\) by Riemann-Roch.

The Chow variety \({\operatorname{CH}}(d_1, d_2, {\mathbf{P}}^N)\) parameterizes cycles of dimension \(d_1\) and degree \(d_2\) in \({\mathbf{P}}^N\). For example, \({\operatorname{CH}}(1, n(2g-2), {\mathbf{P}}^N) \hookrightarrow{\mathbf{P}}H^0({\operatorname{Gr}}, {\mathcal{O}}(k))\) for some \(k\) which sends a cycle to a certain hypersurface. Since \(\operatorname{PGL}\) acts on the latter, it acts on the former, and there is a notion of Chow stability.

The Hilbert scheme is preferable since Chow doesn’t have an immediate deformation theory. We can take a scheme parameterizing closed embeddings \(Z\hookrightarrow{\mathbf{P}}^N\) with a given Hilbert polynomial; recalling \(p_X = \chi({\mathcal{O}}_Z(X))\) and setting \(p_Z(x) = n(2g-2)x + (1-g)\), consider \(\operatorname{Hilb}({\mathbf{P}}^n, p_Z)\). Constructing this scheme: for \(n\gg 0\), there is a surjection \(H^0({\mathcal{O}}_{{\mathbf{P}}^N}(n) ) \twoheadrightarrow H^0({\mathcal{O}}_Z(n))\) defines a point \(g_n\in {\operatorname{Gr}}\) as the codomain varies. Mumford proves there exists an \(N\) such that \((g_N, g_{N+1})\) defines \(Z\) uniquely, although \(N\) is not canonically defined. Thus \(\operatorname{Hilb}\) embeds into a product of Grassmannians, and there is a notion of asymptotic Hilbert stability in terms of growth in \(N\), and one takes the leading term. One shows that for \(n\geq 4\), HM-stable curves are asymptotically Hilbert stable in this since. This almost completely fails for surfaces.

The generalization: algebraic spaces, \(H/G\) or more generally \(H/R\) for \(R\subset H\times H\) an equivalence relation. By Artin, these exist, and they are natural to consider for non-polynomial equations like \(y = \sqrt{x^3+x+1}\). The construction of moduli spaces as algebraic spaces is easy, one then tries to prove they are projective. See KSB varieties and KSBA pairs.

You can use the following simple observation. If \(\pi: \widetilde{X} \rightarrow X\) is a normalization then \(\tilde{X}\) is a smooth curve, so Riemann-Roch is applicable:

\begin{align*} \chi(E)=\operatorname{deg}(E)+\operatorname{rank}(E)(1-g), \end{align*}

and the difference of Hilbert polynomials

\begin{align*} \chi(X, F(m))-\chi\left(\tilde{X},\left(\pi^{*} F\right)(m)\right) \end{align*}

is a constant.