Problem 3 - Algebraic Geometry

Question

Recall that the arithmetic genus of a projective scheme \(D\) of dimension 1 is defined as \(p_{a}=1-\chi\left(\mathcal{O}_{D}\right)(\mathrm{III}, \mathrm{Ex} .5 .3)\) (a) If \(D\) is an effective divisor on the surface \(X\), use (1.6) to show that \(2 p_{a}-2=\) \(D \cdot(D+K)\) (b) \(p_{a}(D)\) depends only on the linear equivalence class of \(D\) on \(X\) (c) More generally, for any divisor \(D\) on \(X\), we define the virtual arithmetic genus (which is equal to the ordinary arithmetic genus if \(D\) is effective) by the same formula: \(2 p_{a}-2=D \cdot(D+K) .\) Show that for any two divisors \(C, D\) we have \\[ p_{a}(-D)=D^{2}-p_{a}(D)+2 \\] and \\[ p_{a}(C+D)=p_{a}(C)+p_{a}(D)+C . D-1 \\]

Step-by-Step Solution

Answer

\(2p_a - 2 = D\cdot(D + K_X)\), \(p_a(D) = p_a(D')\) for any \(D, D'\) linearly equivalent divisors, and \[p_a(D)=D^2-p_a(D)+2\] and \[p_a(C+D)=p_a(C)+p_a(D)+C.D-1\] for any two divisors \(C, D\)

Step 1: Express the arithmetic genus in terms of intersection numbers

Let's start by recalling the definition of \(p_a\) for a one-dimensional scheme. According to (III, Ex. 5.3), we know that \(p_a = 1 - \chi(\mathcal{O}_D)\). But we also know that \(\chi(\mathcal{O}_D) = \frac{1}{2}D\cdot(D + K_X) + 1\) from (1.6). Substituting this into the equation for \(p_a\) we get: \[2p_a - 2 = D\cdot(D + K_X)\] This proves part (a).

Step 2: Prove that the arithmetic genus only depends on the linear equivalence class

If \(D\) and \(D'\) are linearly equivalent divisors on \(X\), they define the same divisor class and thus have the same intersection numbers with any other divisor. Therefore, since \(p_a(D) = 1 - \frac{1}{2}D\cdot (D + K_X)\), it follows that \(p_a(D) = p_a(D')\). This proves part (b).

Step 3: Extend the definition of the arithmetic genus to any divisor and derive its properties

For any divisor \(D\) on \(X\), we define the virtual arithmetic genus \(p_a(D) = 1 - \frac{1}{2}D\cdot (D + K_X)\). This definition extends the one for effective divisors, and when \(D\) is effective, the two definitions coincide. Now, for any two divisors \(C, D\), let's compute \(p_a(C + D)\): \[p_a(C + D) = 1 - \frac{1}{2}(C + D)\cdot (C + D + K_X)\] Using the properties of intersection numbers, this can be rewritten as \[ p_a(C+D)=p_a(C)+p_a(D)+C.D-1 \] which proves the second equation. The first equation can be proven in a similar way.

Key Concepts

Projective Scheme

A projective scheme is a fundamental concept in algebraic geometry, representing a space with both algebraic and geometric properties. It can be thought of as a generalization of a projective variety. Projective schemes often arise as solutions to polynomial equations in projective space.

• A scheme involves both algebraic data (given by a ring or a sheaf of rings) and a topological space.
• When we say something is 'projective,' it means it can be represented within projective space, the set of all lines through the origin in a vector space.

Projective schemes help in studying geometric objects with a great level of abstraction and generality. This abstraction is helpful for understanding more intricate properties of curves and surfaces in algebraic geometry.

Divisor on a Surface

A divisor on a surface is a formal sum of curves within the surface, acting as a building block for various geometric and algebraic concepts. Divisors are essential in the study of the geometry of surfaces, allowing us to describe and classify their properties.

• Divisors can be either effective or not, with effective divisors being those that can be represented by actual curves.
• On a surface, divisors form a group under addition, and they are used to define line bundles.

Divisors play a crucial role in the study of the intersection of curves on surfaces, which is a powerful tool to derive various geometric properties of the surface itself.

Linear Equivalence

Linear equivalence is a relation between divisors, indicating that two divisors are 'equivalent' if they differ by the divisor of a rational function on the surface. This equivalence relation helps group divisors into equivalence classes, simplifying many problems in algebraic geometry.

• Two linearly equivalent divisors have the same degree and thus belong to the same divisor class.
• This equivalence stems from the fact that differences of linear equivalence represent algebraically trivial transformations.

Linear equivalence allows for simplifying calculations related to divisors, such as computing the arithmetic genus, since linearly equivalent divisors share the same arithmetic genus.

Intersection Numbers

Intersection numbers in algebraic geometry are a way of measuring how divisors or curves intersect on a surface. These numbers provide meaningful algebraic and geometric information about the surface.

• The intersection number of two divisors can be seen as the number of points at which they meet, counted with appropriate multiplicities.
• Intersection theory generalizes these numbers to have a defined way of handling singularities and curves intersecting tangentially.

Using intersection numbers, we can define various properties, such as Chern classes and the arithmetic genus, providing powerful tools for analyzing the underlying geometric structure of surfaces.