27 lines on a cubic surface

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 ; 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 is a smooth surface, imbedded in some projective space, and consider the scheme of lines in .

Fix a line in . In this case, the normal sheaf is actually a vector bundle of normal vector fields, given by the adjunction formula

In particular, is a line bundle on and has a well-defined degree. This degree is in fact the self-intersection of , considered as a divisor on the smooth surface .

 

Fano schemes

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

imbedded in via the Segre embedding. For a nonsingular cubic surface in , 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 be a variety. There is a natural subset of the Grassmannian of -planes in (i.e., -dimensional subspaces of ) that parametrizes those -planes which happen to be contained in . 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 itself has a moduli interpretation: it is the Hilbert scheme of -dimensional subschemes of consisting of subschemes whose Hilbert polynomial is given by ; 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  of is the subscheme of parametrizing subschemes whose Hilbert polynomial is .

 

Dual curves, bitangents, and jet bundles

Let be a smooth degree curve. Then there is a dual curve

which sends , to the (projectivized) tangent line at . Such lines live in the dual projective space  of lines in . We will denote the image by ; it is another irreducible curve, birational to .

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

 

Jet bundles and flexes

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

It is a classical fact that there are bitangents on a smooth plane quartic. In other words, of the theta characteristics, exactly of them are effective. A bitangent here will mean a line such that the intersection is a divisor of the form for points, not necessarily distinct. So a line intersecting 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 flex lines. This is a special case of the Plücker formulas, and this post will describe a couple of the relevant ideas. 

 

Theta characteristics and framings

Let be an algebraic curve over . A theta characteristic on is a (holomorphic or algebraic) square root of the canonical line bundle , i.e. a line bundle such that

Since the degree of is even, such theta characteristics exist, and in fact form a torsor over the 2-torsion in the Jacobian , which is isomorphic to .

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 . On a framed manifolds, there is a canonical choice of quadratic refinement on the middle-dimensional mod 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 function on the theta characteristics is precisely this invariant. In other words, theta characteristics give a purely algebraic (valid in all characteristics, at least ) 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.

 

Genus two curves

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

Since has degree two, the map is a two-fold cover: that is, is a hyperelliptic curve. In particular, as with any two-fold cover, there is a canonical involution of the cover , 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 curve has no automorphisms.

A count using Riemann-Hurwitz shows that the canonical map must be branched at precisely six points, which we can assume are . There is no further monodromy data to give for the cover , since it is a two-fold cover; it follows that is exhibited as the Riemann surface associated to the equation

More precisely, the curve is cut out in weighted projective space  by the homogenized form of the above equation,

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 , modulo automorphisms of . In other words, one takes an open subset , and quotients by the action of . In fact, this is a description of the coarse moduli space of genus two curves: that is, it is a variety whose complex points parametrize precisely genus two curves, and which is “topologized” such that any family of genus two curves over a base gives a map . Moreover, is initial with respect to this property.

It can sometimes simplify things to assume that three of the branch points in are given by , which rigidifies most of the action of ; then one simply has to choose three (unordered) distinct points on modulo action of the group consisting of automorphisms of that preserve . In other words,

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 with .

A striking feature here is that the moduli space 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 as the as vary) such that every genus two curve occurs in the family (albeit more than once).

The simplicity of , 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 , and one can thus parametrize most genus four curves by a rational variety.) As , the variety parametrizing genus curves is known to be of general type, by a theorem of Harris and Mumford.

 

The homology of tmf

I’ve just uploaded to arXiv my paper “The homology of ,” 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 cohomology of the spectrum of (connective) topological modular forms, as a module over the Steenrod algebra: one has

where is the Steenrod algebra and is the 64-dimensional subalgebra generated by and . This computation means that the Adams spectral sequence can be used to compute the homotopy groups of ; one has a spectral sequence

Since is finite-dimensional, the entire  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 page for 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 in this paper is based on a certain eight-cell (2-local) complex , with the property that

where 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 is a complex-orientable ring spectrum, so that computations with it (instead of ) become much simpler. In particular, one can compute the cohomology of (e.g., from the cohomology of ), and one finds that it is cyclic over the Steenrod algebra. One can then try to “descend” to the cohomology of . 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.

 

Topological modular forms

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: is a higher analog of –theory (or rather, connective -theory).

1. What is ?

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

that is, one uses the class of the line bundle itself in (modulo a normalization) to define.

The formal multiplicative group has the property that it is Landweber-exact: that is, the map classifying ,

from to the moduli stack of formal groups , is a flat morphism.

 

Projective normality and independent conditions

Let be a (smooth) curve in projective space of some degree . We will assume that is nondegenerate: that is, that is not contained in a hyperplane. In other words, one has an abstract algebraic curve , and the data of a line bundle of degree on , and a subspace of dimension such that the sections in have no common zeros in .

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 inside . The curve is defined by a homogeneous ideal (consisting of all homogeneous polynomials whose vanishing locus contains ). The homogeneous coordinate ring of is defined via

it is an integral domain. Equivalently, it can be defined as the image of in . But that in turn suggests another natural ring associated to , which only depends on the line bundle and not the projective imbedding: that is the ring

where the multiplication comes from the natural maps for line bundles on . One has a natural map

which is injective by construction. Moreover, since higher cohomology always vanishes after enough twisting, the map is surjective in all large dimensions.

Definition 1 The curve is said to be projectively normal if the map is an isomorphism.

 

Hyperelliptic and trigonal curves

Let be a genus curve over the field of complex numbers. I’ve been trying to understand a little about special linear series on : that is, low degree maps , or equivalently divisors on that move in a pencil. Once the degree is at least , any divisor will produce a map to (in fact, many maps), and these fit into nice families. In degrees , maps are harder to write down, and the families they form (for fixed ) 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 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 curve has a map to of degree at most , but for degrees below that, the “general” genus 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.