January 30, 2013

The Nakai-Moishezon criterion

Posted by Akhil Mathew under algebraic geometry | Tags: , , , |
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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 28, 2013

The Riemann-Roch and Hodge index theorems on surfaces

Posted by Akhil Mathew under algebraic geometry, differential geometry | Tags: , , |
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As in the previous two posts, let {S/k} be a smooth, projective surface over an algebraically closed field {k}. In the previous posts, we set up an intersection theory for divisors, which was a symmetric bilinear form

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

that gave the “natural” answer for the intersection of two transversely intersecting curves. Specifically, we had

\displaystyle \mathcal{L} . \mathcal{L}' = \chi(\mathcal{O}_S) - \chi(\mathcal{L}^{-1}) - \chi(\mathcal{L}'^{-1}) + \chi( \mathcal{L}^{-1} \otimes \mathcal{L}'^{-1});

the bilinearity of this map had to do with the fact that the Euler characteristic was a quadratic function on the Picard group. The purpose of this post is to prove a few more general and classical facts about this intersection pairing. As usual, Hartshorne’s Algebraic geometry and Mumford’s Lectures on curves on an algebraic surface are very helpful sources for this material; I also found Abhinav Kumar’s lecture notes useful.

1. The Riemann-Roch theorem

The Euler characteristic of a line bundle {\mathcal{L}} on {S} is a “topological” invariant: it is unchanged under deformations. Given an algebraic family of line bundles {\mathcal{L}_t} on {S} — in other words, a scheme {T} and a line bundle on {S \times_{k} T} which restricts on the fibers to {\mathcal{L}_t} — the Euler characteristics {\chi(\mathcal{L}_t)} are constant. This is one of the parts of the semicontinuity theorem on the cohomology of a flat family of sheaves. Over the complex numbers, one can see this by observing that the Euler characteristic of a line bundle is the index of an elliptic operator — more specifically, the index of the Dolbeault complex associated to {\mathcal{L}} — and can therefore be computed in purely topological terms via the Hirzebruch-Riemann-Roch formula.

In algebraic geometry, the fact that the Euler characteristic is a topological invariant is reflected in the following result, which computes it solely in terms of intersection numbers:

Theorem 1 Let {\mathcal{L}} be a line bundle on {S}. Then

\displaystyle \chi(\mathcal{L}) = \frac{1}{2} \mathcal{L}.( \mathcal{L} - K) + \chi(\mathcal{O}_S), \ \ \ \ \ (1)

where {K} is the canonical divisor on {S}. (more…)

 

January 28, 2013

Intersection theory on surfaces II

Posted by Akhil Mathew under algebraic geometry | Tags: , , , |
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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: , , , |
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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…)