I’ve been trying to learn a little about algebraic curves lately, and genus two is a nice starting point where the general features don’t get too unmanageable, but plenty of interesting phenomena still arise.

0. Introduction

Every genus two curve ${C}$ is hyperelliptic in a natural manner. As with any curve, the canonical line bundle ${K_C}$ is generated by global sections. Since there are two linearly independent holomorphic differentials on ${C}$, one gets a map

$\displaystyle \phi: C \rightarrow \mathbb{P}^1.$

Since ${K_C}$ has degree two, the map ${\phi}$ is a two-fold cover: that is, ${C}$ is a hyperelliptic curve. In particular, as with any two-fold cover, there is a canonical involution ${\iota}$ of the cover ${\phi: C \rightarrow \mathbb{P}^1}$, the hyperelliptic involution. That is, every genus two curve has a nontrivial automorphism group. This is in contrast to the situation for higher genus: the general genus ${g \geq 3}$ curve has no automorphisms.

A count using Riemann-Hurwitz shows that the canonical map ${\phi: C \rightarrow \mathbb{P}^1}$ must be branched at precisely six points, which we can assume are ${x_1, \dots, x_6 \in \mathbb{C}}$. There is no further monodromy data to give for the cover ${C \rightarrow \mathbb{P}^1}$, since it is a two-fold cover; it follows that ${C}$ is exhibited as the Riemann surface associated to the equation

$\displaystyle y^2 = \prod_{i=1}^6 (x - x_i).$

More precisely, the curve ${C}$ is cut out in weighted projective space ${\mathbb{P}(3, 1, 1)}$ by the homogenized form of the above equation,

$\displaystyle Y^2 = \prod_{i = 1}^6 ( X - x_i Z).$

1. Moduli of genus two curves

It follows that genus two curves can be classified, or at least parametrized. That is, an isomorphism class of a genus two curve is precisely given by six distinct (unordered) points on ${\mathbb{P}^1}$, modulo automorphisms of ${\mathbb{P}^1}$. In other words, one takes an open subset ${U \subset (\mathbb{P}^1)^6/\Sigma_6 \simeq \mathbb{P}^6}$, and quotients by the action of ${PGL_2(\mathbb{C})}$. In fact, this is a description of the coarse moduli space of genus two curves: that is, it is a variety ${M_2}$ whose complex points parametrize precisely genus two curves, and which is “topologized” such that any family of genus two curves over a base ${B}$ gives a map ${B \rightarrow M_2}$. Moreover, ${M_2}$ is initial with respect to this property.

It can sometimes simplify things to assume that three of the branch points in ${\mathbb{P}^1}$ are given by ${\left\{0, 1, \infty\right\}}$, which rigidifies most of the action of ${PGL_2(\mathbb{C})}$; then one simply has to choose three (unordered) distinct points on ${\mathbb{P}^1 \setminus \left\{0, 1, \infty\right\}}$ modulo action of the group ${S_3 \subset PGL_2(\mathbb{C})}$ consisting of automorphisms of ${\mathbb{P}^1}$ that preserve ${\left\{0, 1, \infty\right\}}$. In other words,

$\displaystyle M_2 = \left( \mathrm{Sym}^3 \mathbb{P}^1 \setminus \left\{0, 1, \infty\right\} \setminus \left\{\mathrm{diagonals}\right\}\right)/S_3.$

Observe that the moduli space is three-dimensional, as predicted by a deformation theoretic calculation that identifies the tangent space to the moduli space (or rather, the moduli stack) at a curve ${C}$ with ${H^1(T_C)}$.

A striking feature here is that the moduli space ${M_2}$ is unirational: that is, it admits a dominant rational map from a projective space. In fact, one even has a little more: one has a family of genus curves over an open subset in projective space (given by the family ${y^2 = \prod (x - x_i)}$ as the ${\left\{x_i\right\}}$ as vary) such that every genus two curve occurs in the family (albeit more than once).

The simplicity of ${M_2}$, and in particular the parametrization of genus two curves by points in a projective space, is a low genus phenomenon, although similar “classifications” can be made in a few higher genera. (For example, a general genus four curve is an intersection of a quadric and cubic in ${\mathbb{P}^3}$, and one can thus parametrize most genus three curves by a rational variety.) As ${g \rightarrow \infty}$, the variety ${M_g}$ parametrizing genus ${g}$ curves is known to be of general type, by a theorem of Harris and Mumford. (more…)

I’ve just uploaded to arXiv my paper “The homology of ${\mathrm{tmf}}$,” which is an outgrowth of a project I was working on last summer. The main result of the paper is a description, well-known in the field but never written down in detail, of the mod ${2}$ cohomology of the spectrum ${\mathrm{tmf}}$ of (connective) topological modular forms, as a module over the Steenrod algebra: one has

$\displaystyle H^*(\mathrm{tmf}; \mathbb{Z}/2) \simeq \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2,$

where ${\mathcal{A}}$ is the Steenrod algebra and ${\mathcal{A}(2) \subset \mathcal{A}}$ is the 64-dimensional subalgebra generated by ${\mathrm{Sq}^1, \mathrm{Sq}^2,}$ and ${ \mathrm{Sq}^4}$. This computation means that the Adams spectral sequence can be used to compute the homotopy groups of ${\mathrm{tmf}}$; one has a spectral sequence

$\displaystyle \mathrm{Ext}^{s,t}( \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2, \mathbb{Z}/2) \simeq \mathrm{Ext}^{s,t}_{\mathcal{A}(2)}(\mathbb{Z}/2, \mathbb{Z}/2) \implies \pi_{t-s} \mathrm{tmf} \otimes \widehat{\mathbb{Z}_2}.$

Since ${\mathcal{A}(2) \subset \mathcal{A}}$ is finite-dimensional, the entire ${E_2}$ page of the ASS can be computed, although the result is quite complicated. Christian Nassau has developed software to do these calculations, and a picture of the ${E_2}$ page for ${\mathrm{tmf}}$ is in the notes from André Henriques‘s 2007 talk at the Talbot workshop. (Of course, the determination of the differentials remains.)

The approach to the calculation of ${H^*(\mathrm{tmf}; \mathbb{Z}/2)}$ in this paper is based on a certain eight-cell (2-local) complex ${DA(1)}$, with the property that

$\displaystyle \mathrm{tmf} \wedge DA(1) \simeq BP\left \langle 2\right\rangle,$

where ${BP\left \langle 2\right\rangle = BP/(v_3, v_4, \dots, )}$ is a quotient of the classical Brown-Peterson spectrum by a regular sequence. The usefulness of this equivalence, a folk theorem that is proved in the paper, is that the spectrum ${BP\left \langle 2\right\rangle}$ is a complex-orientable ring spectrum, so that computations with it (instead of ${\mathrm{tmf}}$) become much simpler. In particular, one can compute the cohomology of ${BP\left \langle 2\right\rangle}$ (e.g., from the cohomology of ${BP}$), and one finds that it is cyclic over the Steenrod algebra. One can then try to “descend” to the cohomology of ${\mathrm{tmf}}$. This “descent” procedure is made much simpler by a battery of techniques from Hopf algebra theory: the cohomologies in question are graded, connected Hopf algebras. (more…)

The topic of topological modular forms is a very broad one, and a single blog post cannot do justice to the whole theory. In this section, I’ll try to answer the question as follows: ${\mathrm{tmf}}$ is a higher analog of ${KO}$-theory (or rather, connective ${KO}$-theory).

1. What is ${\mathrm{tmf}}$?

The spectrum of (real) ${KO}$-theory is usually thought of geometrically, but it’s also possible to give a purely homotopy-theoretic description. First, one has complex ${K}$-theory. As a ring spectrum, ${K}$ is complex orientable, and it corresponds to the formal group ${\hat{\mathbb{G}_m}}$: the formal multiplicative group. Along with ${\hat{\mathbb{G}_a}}$, the formal multiplicative group ${\hat{\mathbb{G}_m}}$ is one of the few “tautological” formal groups, and it is not surprising that ${K}$-theory has a “tautological” formal group because the Chern classes of a line bundle ${\mathcal{L}}$ (over a topological space ${X}$) in ${K}$-theory are defined by

$\displaystyle c_1( \mathcal{L}) = [\mathcal{L}] - [\mathbf{1}];$

that is, one uses the class of the line bundle ${\mathcal{L}}$ itself in ${K^0(X)}$ (modulo a normalization) to define.

The formal multiplicative group has the property that it is Landweber-exact: that is, the map classifying ${\hat{\mathbb{G}_m}}$,

$\displaystyle \mathrm{Spec} \mathbb{Z} \rightarrow M_{FG},$

from ${\mathrm{Spec} \mathbb{Z}}$ to the moduli stack of formal groups ${M_{FG}}$, is a flat morphism. (more…)

Let ${C \subset \mathbb{P}^r}$ be a (smooth) curve in projective space of some degree ${d}$. We will assume that ${C}$ is nondegenerate: that is, that ${C}$ is not contained in a hyperplane. In other words, one has an abstract algebraic curve ${C}$, and the data of a line bundle ${\mathcal{L} = \mathcal{O}_C(1)}$ of degree ${d}$ on ${C}$, and a subspace ${V \subset H^0( \mathcal{L})}$ of dimension ${r+1}$ such that the sections in ${V}$ have no common zeros in ${C}$.

In this post, I’d like to discuss a useful condition on such an imbedding, and some of the geometry that it leads to. Most of this material is, once again, from ACGH’s book Geometry of algebraic curves.

1. Projective normality

In general, there are two natural commutative graded rings one can associate to this data. First, one has the homogeneous coordinate ring of ${C}$ inside ${\mathbb{P}^r}$. The curve ${C \subset \mathbb{P}^r}$ is defined by a homogeneous ideal ${I \subset k[x_0, \dots, x_r]}$ (consisting of all homogeneous polynomials whose vanishing locus contains ${C}$). The homogeneous coordinate ring of ${C}$ is defined via

$\displaystyle S = k[x_0, \dots, x_r]/I;$

it is an integral domain. Equivalently, it can be defined as the image of ${k[x_0, \dots, x_r] = \bigoplus_{n = 0}^\infty H^0( \mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(n))}$ in ${\bigoplus_{n = 0}^\infty H^0( C, \mathcal{O}_C(n))}$. But that in turn suggests another natural ring associated to ${C}$, which only depends on the line bundle ${\mathcal{L}}$ and not the projective imbedding: that is the ring

$\displaystyle \widetilde{S} = \bigoplus_{n = 0}^\infty H^0( C, \mathcal{O}_C(n)),$

where the multiplication comes from the natural maps ${H^0(\mathcal{M}) \otimes H^0(\mathcal{N}) \rightarrow H^0( \mathcal{M} \otimes \mathcal{N})}$ for line bundles ${\mathcal{M}, \mathcal{N}}$ on ${C}$. One has a natural map

$\displaystyle S \hookrightarrow \widetilde{S},$

which is injective by construction. Moreover, since higher cohomology always vanishes after enough twisting, the map ${S \rightarrow \widetilde{S}}$ is surjective in all large dimensions.

Definition 1 The curve ${C \subset \mathbb{P}^r}$ is said to be projectively normal if the map ${S \hookrightarrow \widetilde{S}}$ is an isomorphism. (more…)

Let ${C}$ be a genus ${g}$ curve over the field ${\mathbb{C}}$ of complex numbers. I’ve been trying to understand a little about special linear series on ${C}$: that is, low degree maps ${C \rightarrow \mathbb{P}^1}$, or equivalently divisors on ${C}$ that move in a pencil. Once the degree is at least ${2g + 1}$, any divisor will produce a map to ${\mathbb{P}^1}$ (in fact, many maps), and these fit into nice families. In degrees ${\leq 2g-2}$, maps ${C \rightarrow \mathbb{P}^1}$ are harder to write down, and the families they form (for fixed $C$) aren’t quite as nice.

However, it turns out that there are varieties of special linear series—that is, varieties parametrizing line bundles of degree ${\leq 2g-2}$ with a certain number of sections, and techniques from deformation theory and intersection theory can be used to bound below and predict their dimensions (the predictions will turn out to be accurate for a general curve). For instance, one can show that any genus ${g}$ curve has a map to ${\mathbb{P}^1}$ of degree at most ${\sim \frac{g}{2}}$, but for degrees below that, the “general” genus ${g}$ curve does not admit such a map. This is the subject of the Brill-Noether theory.

In this post, I’d just like to do a couple of low-degree examples, to warm up for more general results. Most of this material is from Arbarello-Cornalba-Griffiths-Harris’s book Geometry of algebraic curves.  (more…)

The purpose of this post, the third in a series on deformation theory and DGLAs, is to describe the obstruction theory for a formal moduli problem associated to a DGLA.

1. Tangent-obstruction theories

Standard problems in classical deformation theory usually have a “tangent-obstruction theory” parametrized by certain successive cohomology groups. For example, let’s consider the problem of deformations of a smooth variety ${X}$ over an algebraically closed field ${k}$, over finite-dimensional local ${k}$-algebras. Then:

• The “infinitesimal automorphisms” of ${X}$—that is, automorphisms of the trivial deformation over ${k[\epsilon]/\epsilon^2}$—are given by ${H^0( X, T_X)}$ where ${T_X}$ is the tangent bundle (i.e., vector fields).
• The isomorphism classes of deformations of ${X}$ over the dual numbers ${k[\epsilon]/\epsilon^2}$ are given by ${H^1(X, T_X)}$.
• There is an obstruction theory with ${H^1, H^2}$. Specifically, given a square-zero extension of finite-dimensional local ${k}$-algebras

$\displaystyle 0 \rightarrow I \rightarrow A' \rightarrow A \rightarrow 0,$

and given a deformation ${\xi}$ of ${X}$ over ${\mathrm{Spec} A}$, there is a functorial obstruction in ${H^2(X, T_X) \otimes_k I}$ to extending the deformation over the inclusion ${\mathrm{Spec} A \hookrightarrow \mathrm{Spec} A'}$.

• In the previous item, if the obstruction vanishes, then the isomorphism classes of extensions of ${\xi}$ over ${\mathrm{Spec} A'}$ are a torsor for ${H^1(X, T_X) \otimes_k I}$.

One has a similar picture for other deformation problems, for example deformations of vector bundles or closed subschemes. The “derived” approach to deformation theory provides (at least in characteristic zero) a general explanation for this phenomenon. (more…)

Let ${k}$ be a field of characteristic zero. In the previous post, we introduced the category (i.e., ${\infty}$-category) ${\mathrm{Moduli}_k}$ of formal moduli problems over ${k}$. A formal moduli problem over ${k}$ is a moduli problem, taking values in spaces, that can be evaluated on the class of “derived” artinian ${k}$-algebras with residue field ${k}$: this was the category ${\mathrm{CAlg}_{sm}}$ introduced in the previous post.

In other words, a formal moduli problem was a functor

$\displaystyle F: \mathrm{CAlg}_{sm} \rightarrow \mathcal{S} \ (= \text{spaces}),$

which was required to send ${k}$ itself to a point, and satisfy a certain cohesiveness condition: ${F}$ respects certain pullbacks in ${\mathrm{CAlg}_{sm}}$ (which corresponded geometrically to pushouts of schemes).

The main goal of the series of posts was to sketch a proof of (and define everything in) the following result:

Theorem 7 (Lurie; Pridham) There is an equivalence of categories between ${\mathrm{Moduli}_k}$ and the ${\infty}$-category ${\mathrm{dgLie}}$ of DGLAs over ${k}$.

4. Overview

Here’s a rough sketch of the idea. Given a formal moduli problem ${F}$, we should think of ${F}$ as something like a small space, concentrated at a point but with lots of “infinitesimal” thickening. (Something like a ${\mathrm{Spf}}$.) Moreover, ${F}$ has a canonical basepoint corresponding to the “trivial deformation.” That is, we can think of ${F}$ as taking values in pointed spaces rather than spaces.

It follows that we can form the loop space ${\Omega F = \ast \times_F \ast}$ of ${F}$, which is a new formal moduli problem. However, ${\Omega F}$ has more structure: it’s a group object in the category of formal moduli problems — that is, it’s some sort of derived formal Lie group. Moreover, knowledge of the original ${F}$ is equivalent to knowledge of ${\Omega F}$ together with its group structure: we can recover ${F}$ as ${B \Omega F}$ (modulo connectivity issues that end up not being a problem). This relation between ordinary objects and group objects (via ${B, \Omega}$) is something very specific to the derived or homotopy world, and it’s what leads to phenomena such as Koszul duality. (more…)