January 30, 2013

The Nakai-Moishezon criterion

Posted by Akhil Mathew under algebraic geometry | Tags: , , , |
Leave a Comment 

Let {S} be a smooth, projective surface over the algebraically closed field {k}. Previous posts have set up an intersection theory

\displaystyle \mathrm{Pic}(S) \times \mathrm{Pic}(S) \rightarrow \mathbb{Z}

on {S} with very convenient formal properties. We also described a historically important use of this machinery: the Weil bound on points on a smooth curve over a finite field. The purpose of this post is to prove an entirely numerical criterion for ampleness of a line bundle on a surface, due to Nakai and Moishezon.

Let {D} be a very ample divisor on {S}. Then we have:

  • {D.C > 0} for all curves (i.e., strictly effective divisors) on {S}. In fact, if {D} defines an imbedding {S \hookrightarrow \mathbb{P}^M}, then the degree of {C} under this imbedding is {D.C}.
  • As a special case of this, {D.D > 0}. In fact, {D} must be effective.

Since a power of an ample divisor is very ample, the same is true for an ample divisor.

The purpose of this post is to prove the converse:

 

Theorem 1 (Nakai-Moishezon) Let {S} be a smooth projective surface as above. If {D} is a divisor on {S} (not necessarily effective!) satisfying {D.D>0} and {D.C > 0} for all curves on {S}, then {D} is ample. In particular, ampleness depends only on the numerical equivalence class of {D}.

Once again, the source for this material is Hartshorne’s Algebraic geometry. The goal is to get to some computations and examples as soon as possible.   (more…)

 

January 29, 2013

The Weil bound for curves over finite fields

Posted by Akhil Mathew under algebraic geometry, number theory | Tags: , , , |
Leave a Comment 

The purpose of this post is to describe an application of the general intersection theory machinery (for curves on surfaces) developed in the previous posts: the Weil bound on points on a curve over a finite field.

1. Statement of the Weil bound

Let {C} be a smooth, projective, geometrically irreducible curve over {\mathbb{F}_q} of genus g. Then the Weil bound states that:

\displaystyle |C(\mathbb{F}_q) - q - 1 | \leq 2 g \sqrt{q}.

Weil’s proof of this bound is based on intersection theory on the surface {C \times C}. More precisely, let

\displaystyle \overline{C} = C \times_{\mathbb{F}_q} \overline{\mathbb{F}_q},

so that {\overline{C}} is a smooth, connected, projective curve. It comes with a Frobenius map

\displaystyle F: \overline{C} \rightarrow \overline{C}

of {\overline{\mathbb{F}_q}}-varieties: in projective coordinates the Frobenius runs

\displaystyle [x_0: \dots : x_n] \mapsto [x_0^q: \dots : x_n^q].

In particular, the map has degree {q}. One has

\displaystyle C( \mathbb{F}_q) = \mathrm{Fix}(F, \overline{C}(\overline{\mathbb{F}}_q))

representing the {\mathbb{F}_q}-valued points of {C} as the fixed points of the Frobenius (Galois) action on the {\overline{\mathbb{F}_q}}-valued points. So the strategy is to count fixed points, using intersection theory.

Using the (later) theory of {l}-adic cohomology, one represents the number of fixed points of the Frobenius as the Lefschetz number of {F}: the action of {F} on {H^0} and {H^2} give the terms {q+1}. The fact that (remaining) action of {F} on the {2g}-dimensional vector space {H^1} can be bounded is one of the Weil conjectures, proved by Deligne for general varieties: here it states that {F} has eigenvalues which are algebraic integers all of whose conjugates have absolute value {\sqrt{q}}. (more…)

 

January 28, 2013

Intersection theory on surfaces II

Posted by Akhil Mathew under algebraic geometry | Tags: , , , |
[2] Comments 

Let {S} be a smooth, projective surface over an algebraically closed field {k} and let {C, D \subset S} be curves (subschemes pure of codimension one) on {S}. In the previous post, we discussed what a good theory of intersections {C.D} would look like. We wanted to be able to define the intersection {C.D} in such a manner that:

  • If {C, D} intersect transversely, then {C.D = |C \cap D|}.
  • The intersection product is additive. That is, given curves {C_1, C_2, D}, we have

    \displaystyle (C_1 + C_2). D = C_1.D + C_2.D,

    where {C_1+C_2} is treated as an effective Cartier divisor.

  • The intersection product is invariant under linear equivalence and descends to a pairing on the Picard group.

1. Definition of the intersection product

In the previous post, we saw that any intersection theory as above was necessarily unique, and suggested that the Euler characteristic formula

\displaystyle C.D \stackrel{\mathrm{def}}{=} \chi( \mathcal{O}_C \stackrel{\mathbb{L}}{\otimes}_{\mathcal{O}_S} \mathcal{O}_D ) = \sum_i (-1)^i \mathbb{H}^i( \mathcal{O}_C \stackrel{\mathbb{L}}{\otimes}_{\mathcal{O}_S} \mathcal{O}_D) \ \ \ \ \ (1)

would be a good definition: i.e., that the failure of

\displaystyle C.D = |C \cap D|

in general was due to two factors: the existence of nilpotents in the (scheme-theoretic as opposed to set-theoretic) intersection {C \cap D} and higher homotopy groups in the (derived as opposed to scheme-theoretic) intersection {C \stackrel{h}{\cap} D}. The main goal of this post is to prove that (1)does give a good theory. That is, we would like to prove:

Theorem 1 The definition of {C.D} in (1) satisfies the conditions desired of an intersection product. (more…)

 

January 27, 2013

Intersection theory on surfaces

Posted by Akhil Mathew under algebraic geometry | Tags: , , , |
[2] Comments 

The purpose of this post and the next is to work through a basic example of intersection theory: intersections of curves on a surface. This is a fundamental and basic example in algebraic geometry, and since I’ve never studied intersection theory, it like seems a reasonable place to start. The references here are chapter 5 of Hartshorne’s Algebraic geometry and Mumford’s Lectures on curves on an algebraic surface.

1. Curves on surfaces

The subject of “curves on a surface” is the subject of Mumford’s book mentioned above; the purpose of this section is simply to set down the definitions.

Let {k} be an algebraically closed field. A surface {S} is a smooth projective surface over {k}. There is a classification of surfaces, but let’s just list a couple of basic examples: {\mathbb{P}^2, \mathbb{P}^1 \times \mathbb{P}^1}, (smooth) hypersurfaces in {\mathbb{P}^3}, and ruled surfaces.

Definition 1 A curve on a surface {S} is an (effective) divisor on {S}. Equivalently, it is a subscheme {C \subset S} pure of codimension one, so locally cut out by one equation. (But {C} is not necessarily smooth, or even reduced.)

The goal of this post and the next is to set up a basic intersection theory for curves on surfaces. Given two curves {C, D \subset S}, we’d like to define the intersection product {C.D}. There is one case where it is easy: suppose {C} and {D} meet only transversely. In other words, for each {p \in C \cap D}, we choose local equations {f,g \in \mathfrak{m}_{S, p} \subset\mathcal{O}_{S, p}} for the subschemes {C, D}, and

\displaystyle (f,g) = \mathfrak{m}_{S, p}.

In particular, this implies that {C, D} are nonsingular at all points of intersection. In this case, we would like to require

\displaystyle C.D = \sum_{p \in C \cap D} 1 \quad (\text{if transverse intersection}). \ \ \ \ \ (1)

Once we require the above condition and two more natural conditions, we will prove that the intersection product is uniquely determined:

  • The equation (1) holds under transversality assumptions and if {C, D} are smooth.
  • The intersection product is additive. That is, given curves {C_1, C_2, D}, we have

    \displaystyle (C_1 + C_2). D = C_1.D + C_2.D,

    where {C_1+C_2} is treated as an effective Cartier divisor.

  • The intersection product is invariant under linear equivalence. If {C, C'} are linearly equivalent curves, we want

    \displaystyle C. D = C'.D,

    so that the intersection product is invariant under deformation. In particular, this and the previous item show that the intersection product only depends on the line bundle associated to a divisor (and can make sense for any divisor, not necessarily effective).

Our goal is to prove:

Theorem 2 There is a unique pairing

\displaystyle \mathrm{Pic}(S) \times \mathrm{Pic}(S) \rightarrow \mathbb{Z}

satisfying the above three conditions. (more…)