algebraic geometry


In the previous post, we introduced the Fano scheme of a subscheme of projective space, as the Hilbert scheme of planes of a certain dimension on that subscheme. In this post, I’d like to work out an explicit example, of the 27 lines on a smooth cubic surface in \mathbb{P}^3; as we’ll see, the Fano scheme is 27 reduced points, and the count can be made with a little calculation on the Grassmannian. Although the calculation is elementary, I found it worthwhile to work carefully through it, not only for its intrinsic interest but also as motivation for the study of intersection theory on moduli spaces in general. Once again, most of this material is from Eisenbud-Harris’s draft book 3264 and All That.

1. The normal bundle as self-intersection

Suppose {X = S} is a smooth surface, imbedded in some projective space, and consider the scheme {F_1 S} of lines in {S}.

Fix a line {L} in S. In this case, the normal sheaf {N_{S/L}} is actually a vector bundle of normal vector fields, given by the adjunction formula

\displaystyle N_{S/L} = \left(\mathcal{I}_L/\mathcal{I}_L^2\right)^{\vee} = \left(\mathcal{O}_S(-L)/\mathcal{O}_S(-2L)\right)^{\vee} = \mathcal{O}_L(L).

In particular, {N_{S/L}} is a line bundle on {L \simeq \mathbb{P}^1} and has a well-defined degree. This degree is in fact the self-intersection {L.L} of {L}, considered as a divisor on the smooth surface {S}. (more…)

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Let {X \subset \mathbb{P}^r} be a subvariety (or scheme). A natural question one might ask is whether {X} contains lines, or more generally, planes {\mathbb{P}^{k} \subset X \subset \mathbb{P}^r} and, if so, what the family of such look like. For example, if {Q \subset \mathbb{P}^3} is a nonsingular quadric surface, then {Q} has two families of lines (or “rulings”) that sweep out {Q}; this corresponds to the expression

\displaystyle Q \simeq \mathbb{P}^1 \times \mathbb{P}^1,

imbedded in {\mathbb{P}^3} via the Segre embedding. For a nonsingular cubic surface in {\mathbb{P}^3}, it is a famous and classical result of Cayley and Salmon that there are twenty-seven lines. In this post and the next, I’d like to discuss this result and more generally the question of planes in hypersurfaces.

Most of this material is classical; I recently learned it from Eisenbud-Harris’s (very enjoyable) draft textbook 3264 and All That.

1. Varieties of planes

Let {X \subset \mathbb{P}^r} be a variety. There is a natural subset of the Grassmannian {\mathbb{G}(k, r)} of {k}-planes in {\mathbb{P}^r} (i.e., {k+1}-dimensional subspaces of {\mathbb{C}^{r+1}}) that parametrizes those {k}-planes which happen to be contained in {X}. This is called the Fano variety.

However, the Fano variety has a natural (and possibly nonreduced) subscheme structure that arises from its interpretation as the solution to a moduli problem, so perhaps it should be called a Fano scheme. The first observation is that the {\mathbb{G}(k, r)} itself has a moduli interpretation: it is the Hilbert scheme of {k}-dimensional subschemes of {\mathbb{P}^r} consisting of subschemes whose Hilbert polynomial is given by {n \mapsto \binom{n+k}{k}}; such a subscheme is necessarily a linear subspace.

This suggests that we should think of the Fano scheme as a Hilbert scheme.

Definition 1 The Fano scheme {F_k X} of {X} is the subscheme of {\mathrm{Hilb}_X} parametrizing subschemes {L \subset X} whose Hilbert polynomial is {n \mapsto \binom{n+k}{k}}. (more…)

Let {C \subset \mathbb{P}^2} be a smooth degree {d} curve. Then there is a dual curve

\displaystyle C \rightarrow (\mathbb{P}^2)^*,

which sends {p \in C \mapsto \mathbb{T}_p C}, to the (projectivized) tangent line at {p \in C}. Such lines live in the dual projective space {(\mathbb{P}^2)^*} of lines in {\mathbb{P}^2}. We will denote the image by {C^* \subset \mathbb{P}^2}; it is another irreducible curve, birational to {C}.

This map is naturally of interest to us, because, for example, it lets us count bitangents. A bitangent to {C} will correspond to a node of the image of the dual curve, or equivalently it will be a point in {(\mathbb{P}^2)^*} where the dual map {C \rightarrow (\mathbb{P}^2)^*} fails to be one-to-one. In fact, if {C} is general, then {C^*} will have only nodal and cuspidal singularities, and we we will be able to work out the degree of {C^*}. By the genus formula, this will determine the number of nodes in {C^*} and let us count bitangents.

The purpose of this post is to describe this, and to discuss this map from the point of view of jet bundles, discussed in the previous post. (more…)

Let {C \subset \mathbb{P}^2} be a smooth plane quartic, so that {C} is a nonhyperelliptic genus 3 curve imbedded canonically. In the previous post, we saw that bitangent lines to {C} were in natural bijection with effective theta characteristics on {C}, or equivalently spin structures (or framings) of the underlying smooth manifold.

It is a classical fact that there are {28} bitangents on a smooth plane quartic. In other words, of the {64} theta characteristics, exactly {28} of them are effective. A bitangent here will mean a line {L \subset \mathbb{P}^2} such that the intersection {L \cap C} is a divisor of the form {2(p + q)} for {p, q \in C} points, not necessarily distinct. So a line intersecting {C} in a single point (with contact necessarily to order four) is counted as a bitangent line. In this post, I’d like to discuss a proof of a closely related claim, that there are {24} flex lines. This is a special case of the Plücker formulas, and this post will describe a couple of the relevant ideas.  (more…)

Let {C} be an algebraic curve over {\mathbb{C}}. A theta characteristic on {C} is a (holomorphic or algebraic) square root of the canonical line bundle {K_C}, i.e. a line bundle {L \in \mathrm{Pic}(C)} such that

\displaystyle L^{\otimes 2} \simeq K_C.

Since the degree of {K_C} is even, such theta characteristics exist, and in fact form a torsor over the 2-torsion in the Jacobian {J(C) = \mathrm{Pic}^0(C)}, which is isomorphic to {H^1(C; \mathbb{Z}/2\mathbb{Z}) \simeq (\mathbb{Z}/2\mathbb{Z})^{2g}}.

One piece of geometric motivation for theta characteristics comes from the following observation: theta characteristics form an algebro-geometric approach to framings. By a theorem of Atiyah, holomorphic square roots of the canonical bundle on a compact complex manifold are equivalent to spin structures. In complex dimension one, a choice of a spin structure is equivalent to a framing of {M}. On a framed manifolds, there is a canonical choice of quadratic refinement on the middle-dimensional mod {2} homology (with its intersection pairing), which gives an important invariant of the framed manifold known as the Kervaire invariant. (See for instance this post on the paper of Kervaire that introduced it.)

It turns out that the mod {2} function {L \mapsto \dim H^0(C, L)} on the theta characteristics is precisely this invariant. In other words, theta characteristics give a purely algebraic (valid in all characteristics, at least {\neq 2}) approach to the Kervaire invariant, for surfaces!

Most of the material in this post is from two papers: Atiyah’s Riemann surfaces and spin structures and Mumford’s Theta characteristics of an algebraic curve. (more…)

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 four 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…)

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