Problem 12 - Algebraic Geometry

Question

If \(D\) is an ample divisor on the surface \(X,\) and \(D^{\prime} \equiv D,\) then \(D^{\prime}\) is also ample. Give an example to show, however, that if \(D\) is very ample, \(D^{\prime}\) need not be very ample.

Step-by-Step Solution

Answer

If \(D\) is an ample divisor and \(D'\) is linearly equivalent to \(D\), then \(D'\) is also ample. However, if \(D\) is a very ample divisor, \(D'\) need not be very ample, which can be demonstrated through an example where \(D'\) fails to embed the variety into a projective space unlike \(D\).

Step 1: Proof of Amplitude Transitivity

Given divisors \(D\) and \(D'\) on \(X\) such that \(D'\) is linearly equivalent to \(D\), and given that \(D\) is ample. By definition, an ample divisor stays ample under linear equivalents. Hence, if \(D\) is ample, then \(D'\) equivalent to \(D\) is also ample due to the transitivity of the amplitude property.

Step 2: Counterexample for Very Ampleness Transitivity

The counterexample requires a situation where we have a divisor that is very ample but its equivalent is not. Consider the surface \(X\) to be the complex projective plane \(\mathbb{CP}^2\). Let \(D\) be a hyperplane (very ample divisor by definition) and \(D'\) be the divisor associated with a line in that hyperplane. Even though \(D'\) is linearly equivalent to \(D\) (0-dimensional subvariety of \(D\)), it is not very ample. The hyperplane \(D\) embeds \(\mathbb{CP}^2\) into projective space as a quadratic while \(D'\) cannot embed \(\mathbb{CP}^2\) into a projective space.

Key Concepts

Very Ample Divisor

A very ample divisor on a surface is more than just ample—it has a special role in embedding the surface into a projective space.
An ample divisor ensures a positive degree for curves, but very ample divisors can embed the entire surface.

If a divisor, like a hyperplane on the complex projective plane (denoted by \( \mathbb{CP}^2 \)), is very ample, it means the divisor can embed our surface in such a way that we find our entire space sitting inside the higher-dimensional space. Here are some key aspects of very ample divisors:

• The ability to define a globally generated line bundle.
• Utility in realizing abstract complex surfaces as concrete projective varieties.
• For the complex projective plane, hyperplanes are typical examples of very ample divisors.

However, keep in mind that if two divisors are linearly equivalent, both being ample, they might not both be very ample, as shown with counterexamples such as in the exercise.

Linear Equivalence in Divisors

Linear equivalence is a concept used to relate two divisors on a surface.
Two divisors \( D \) and \( D' \) on a surface \( X \) are linearly equivalent if they differ by a principal divisor.

This means there exists a function on the surface such that the set of zeroes and poles of this function creates a connection between \( D \) and \( D' \).
Some essential details include:

• If \( D \) and \( D' \) are linearly equivalent and one is ample, so is the other, preserving a certain degree of positivity on curves.
• This notion simplifies analyzing complex surfaces by allowing divisors to be grouped into equivalence classes.
• It shows the depth of algebraic structures and functions on these surfaces.

However, as noted in the exercise, even if two divisors share a linear equivalence relation, they may not exhibit identical geometric properties such as very ampleness.

Complex Projective Plane

The complex projective plane, denoted as \( \mathbb{CP}^2 \), serves as a vital setting in algebraic geometry.
This mathematical construct is a two-dimensional complex manifold, which is especially interesting because it can be visualized as a projective space of complex lines.

In essence, \( \mathbb{CP}^2 \) is the projective space of dimension 2 over complex numbers.
This means every point corresponds to a line passing through the origin in \( \mathbb{C}^3 \). Here are some fundamental features of the complex projective plane:

• It's compact without a boundary, making it a model surface in geometry.
• It allows exploration of divisors and line bundles effectively, including ample and very ample divisors.
• Tools like divisors on \( \mathbb{CP}^2 \) help understand key geometric transformations and mappings.

In our exercise, the \( \mathbb{CP}^2 \) framework demonstrates the behavior of ample versus very ample divisors and the subtleties of linear equivalence.