Problem 9 - Algebraic Geometry

Question

(a) If \(H\) is an ample divisor on the surface \(X\), and if \(D\) is any divisor, show that \\[ \left(D^{2}\right)\left(H^{2}\right) \leqslant(D . H)^{2} \\] (b) Now let \(X\) be a product of two curves \(X=C \times C^{\prime} .\) Let \(l=C \times p t,\) and \(m=\mathrm{pt} \times C^{\prime} .\) For any divisor \(D\) on \(X,\) let \(a=D . l, b=D . m .\) Then we say \(D\) has type \((a, b) .\) If \(D\) has type \((a, b),\) with \(a, b \in \mathbf{Z},\) show that \\[ D^{2} \leqslant 2 a b \\] and equality holds if and only if \(D \equiv b l+a m .\) [Hint: Show that \(H=l+m\) is ample, let \(E=l-m\), let \(D^{\prime}=\left(H^{2}\right)\left(E^{2}\right) D-\left(E^{2}\right)(D . H) H-\left(H^{2}\right)(D . E) E,\) and apply (1.9). This inequality is due to Castelnuovo and Severi. See Grothendieck \([2] .]\)

Step-by-Step Solution

Answer

The given inequalities hold due to the properties of intersection numbers of divisors and properties of ample divisors on algebraic surfaces, as well as on the product of two curves. For a general divisor \(D\) on a surface with an ample divisor \(H\), we have \((D^2)(H^2) \leq (D.H)^2\). On the product of two curves, for a divisor \(D\) of type \((a, b)\), we have \(D^2 \leq 2ab\), with equality if and only if \(D = bl + am\).

Step 1: Analyzing the first inequality

Recall that an ample divisor \(H\) on a surface \(X\) is one such that some multiple of \(H\) is very ample, i.e., it allows to embed \(X\) into some projective space. The given inequality involves the square of the intersection number of \(H\) (which must be positive by definition of ampleness), the square of the intersection number of \(D\), and the square of the intersection number of \(H\) and \(D\). To show this inequality, one can rely on the property of intersection numbers that if \(D\) is any divisor and \(H\) is an ample divisor, then \(D.H\) is non-negative.

Step 2: Utilizing the properties of intersection numbers

Suppose by contradiction that \(D^2.H^2 > (D.H)^2\), then, defining \(t = D.H / H^2\), we would have \(D^2 > t^2.H^2. Since t^2.H^2 = (tH)^2 > 0\), this implies that \(D^2 > 0\), but we also know that \(D.H \geq 0\), so we would have \(D^2 > D.H\), which contradicts the fact that if an ample divisor \(H\) and any divisor \(D\) satisfy \(D.H \geq 0\), then \(D^2 \leq D.H\). Hence the given inequality must hold.

Step 3: Analyzing the second inequality

Remember that the product of two curves, \(X = C \times C'\), naturally has two divisors: \(l\) and \(m\), such that any divisor \(D\) on \(X\) can be written as a combination \(D = al + bm\) for some integers \(a\) and \(b\). The task here is to show that \(D^2 \leq 2ab\) for any such combination. The hint suggests introducing an ample divisor \(H = l + m\) and applying the inequality shown in the first part of the task.

Step 4: Utilize the properties of divisors and intersection numbers on the product of curves

First, let's note that \(H\) is indeed an ample divisor, as the sum of two very ample divisors is ample. Next, introducing the divisors \(E = l - m\) and \(D' = H^2.E^2.D - E^2.(D.H).H - H^2.(D.E).E\), it follows by the properties of intersection numbers that \(D'.H = 0\) and \(D'.E = 0\). This means that \(D'\), \(H\) and \(E\) form a direct system of divisors and \(D'.H.E = D'.E.H = 0\). Now apply the inequality from the first part, noting that \(D'.^2 = D^2 - 2ab\), which gives that \(D'^2 \leq 0\), whence it immediately follows that \(D^2 \leq 2ab\).

Step 5: Characterizing the case of equality

In the inequality \(D^2 \leq 2ab\) equality holds if and only if \(D'\) is effective, that is, if \(D'\) can be written as a sum of irreducible effective divisors. But by construction, \(D' = D - (bl + am)\), so \(D'\) is effective if and only if \(D = bl+am\), which gives exactly the condition in the task.

Key Concepts

Intersection Number

In algebraic geometry, an intersection number is a crucial concept that measures how different geometric objects, like curves, intersect with each other on a surface. Think of an intersection as the way two roads meet at a crossing. Similarly, if you have two divisors like a line and a circle on a plane, the intersection number would tell you how many points they share.
To determine the intersection number, you would generally see how many points the divisors share, count them, and this count, sometimes adjusted for multiplicity, becomes the intersection number. For example, if two curves on a plane meet at one point, their intersection number could be one. If they are tangent to each other and meet at one point with tangents also aligning, you might count more based on complexities.
This concept is very helpful because it allows mathematicians to understand properties of divisors on surfaces. For example, if certain divisors have non-zero intersection numbers, it tells us they do, in fact, intersect. This can reveal a lot about their behavior and characteristics in various scenarios.

Product of Curves

When we talk about the product of curves in algebraic geometry, we are considering a new geometric object constructed from two given curves. Imagine taking curve \(C\) and another curve \(C'\), and together they create a new surface \(X = C \times C'\). This new surface is essentially a combination of all points \((x, y)\) where \(x\) comes from \(C\) and \(y\) comes from \(C'\).
In our scenario from the original exercise, there are specific divisors on this product surface \(X\), noted as \(l = C \times pt\) and \(m = pt \times C'\). These divisors represent fixed directions on the surface, like principal streets in a city. Any other divisor \(D\) on this surface can be expressed as a combination \(D = a \cdot l + b \cdot m\). You can think of this as a way to represent locations on the surface in terms of these two basic directions.
Understanding the product of curves becomes crucial when proving inequalities, like showing that \(D^2 \leq 2ab\), because it relies on how \(D\) can be expressed in terms of \(l\) and \(m\). Being able to decompose the surface this way offers insights into its geometry.

Castelnuovo-Severi Inequality

The Castelnuovo-Severi Inequality is a specific result in algebraic geometry that provides an upper bound for squares of divisors. Just like any mathematical inequality, it essentially tells us that something is smaller, or at most equal, to something else. Here, it tells us that the self-intersection number, \(D^2\), of a divisor \(D\) cannot exceed a certain bound.
In our context, if \(D\) is a divisor of type \((a, b)\) on a surface that results from the product of two curves, then the inequality specifies \(D^2 \leq 2ab\). Why two times \(ab\)? The factors \(a\) and \(b\) represent the contributions of divisors \(l\) and \(m\) which gives an idea of how the divisor \(D\) covers the surface.
The interesting part of the Castelnuovo-Severi Inequality is understanding when the equality holds. This happens exactly when \(D\) behaves nicely, aligning with the divisors \(bl + am\). Such results are not only elegant but also powerful tools for understanding geometric relationships, helping to predict the behavior of intricate surfaces.

Divisor Types

Divisors in algebraic geometry are fundamental in analyzing and interpreting the geometrical structure of a surface. They can be thought of as formal sums of subvarieties of codimension one, often curves. The type of a divisor provides a way of expressing it in terms of basic, fundamental divisors on a complex surface.
In the context of a product of curves like \(X = C \times C'\), a divisor \(D\) is said to be of type \((a, b)\). This means it can be written in the form \(D = a \, \cdot l + b \, \cdot m\). Thus, the type describes the proportional influence of each basic divisor type forming \(X\).
Understanding divisor types allows one to delve into their interaction properties, like their intersection numbers, and assists in executing further computations and proofs, as demonstrated by the inequality \(D^2 \leq 2ab\). Exploring these types offers insights into how complex surfaces can be dissected into simpler, manageable parts.