Problem 11 - Algebraic Curves: An Introduction to Algebraic Geometry

Question

Let \(D\) be a divisor. Show that \(l(D)>0\) if and only if \(D\) is linearly equivalent to an effective divisor.

Step-by-Step Solution

Answer

Question: Show that a divisor \(D\) satisfies the condition \(l(D) > 0\) if and only if \(D\) is linearly equivalent to an effective divisor. Answer: A divisor \(D\) satisfies the condition \(l(D) > 0\) if and only if it is linearly equivalent to an effective divisor. This means that there exists a rational function that connects \(D\) with an effective divisor. To prove this, we proved both directions of the equivalence: firstly, when \(D\) is linearly equivalent to an effective divisor, \(l(D) > 0\); secondly, when \(l(D) > 0\), \(D\) is linearly equivalent to an effective divisor.

Step 1: Definition of Divisors, Linear Equivalence, and Effective Divisors

Divisors on a curve are formal sums of points with integer coefficients. An effective divisor is a divisor whose coefficients are nonnegative integers. Two divisors are said to be linearly equivalent if their difference is the principal divisor of a rational function on the curve.

Step 1: Show that if D is linearly equivalent to an effective divisor, then l(D) > 0

Assume \(D\) is linearly equivalent to an effective divisor, say \(D'\). By definition of linear equivalence, there exists a rational function \(f\) on the curve \(C\) such that \(D - D' = div(f)\). Now, consider the rational function \(f\). Since \(D'\) is effective, the coefficient of every term in \(D - D'\) is non-positive. Thus, for any point \(Q\) on \(C\), we have, \(ord_Q(f) \ge 0\); which means, \(f\) is a meromorphic function on \(C\) that is regular at every point of \(C\). So, \(f \in L(D)\), and as a result, we have \(l(D) > 0\).

Step 2: Show that if l(D) > 0, then D is linearly equivalent to an effective divisor

If \(l(D) > 0\), there exists at least one rational function, say, \(f \in L(D)\). Since \(f \in L(D)\), we have \(ord_P(f) \ge -n_P\) for any point \(P\) and an integer \(n_P\) from \(D\). Define a new divisor \(D'\) by: \(D' = \sum_P(n_P + ord_P(f))P\). This divisor is effective, since for each point \(P\), we have \(n_P + ord_P(f) \ge 0\). Also, we have \(D - D' = div(f)\), which shows that \(D\) and \(D'\) are linearly equivalent. Thus, we've shown that \(l(D) > 0\) implies \(D\) is linearly equivalent to an effective divisor.

Step 4: Conclusion

We have shown that \(l(D) > 0\) if and only if \(D\) is linearly equivalent to an effective divisor. We demonstrated this by proving both directions of the equivalence: firstly, by showing that if \(D\) is linearly equivalent to an effective divisor, then \(l(D) > 0\), and secondly, by showing that if \(l(D) > 0\), then \(D\) is linearly equivalent to an effective divisor.

Key Concepts

Algebraic Curves

Algebraic curves are among the central objects studied in algebraic geometry. Think of them as smooth, one-dimensional figures, like elongated circles or ovals, but they can have far more complex shapes. They are defined as the set of solutions to polynomials in two variables over a given field.

For example, the equation \( y^2 = x^3 - x \) describes what's known as an elliptic curve. This type of curve has special properties and plays a significant role in number theory and cryptography. Algebraic curves are not confined to the real numbers; they also exist over complex numbers or any other field, leading to rich geometrical properties and open a gateway to understanding higher-dimensional objects.

Effective Divisors

In the world of algebraic geometry, divisors signify a formal way to keep track of zeroes and poles of functions along a curve. Particularly, an effective divisor is a collection of points on a curve, each tagged with a nonnegative integer. These integers can be envisioned as weights, representing how many times the curve 'visits' a particular point.

Imagine you're given a polynomial function, and it intersects an algebraic curve at specific points. If you assign a weight to these points that equals the intersection multiplicity—which tells you how many times the curve and the function 'touch'—you've essentially determined an effective divisor. Because all coefficients are nonnegative, effective divisors can be thought of as purely 'adding' something to the curve, similar to how temperature readings don't go below absolute zero.

Rational Functions

Within mathematics, rational functions are ratios of two polynomials. For instance, the formula \( f(x) = \frac{x^2 - 1}{x - 1} \) is a rational function. They're vital in algebraic geometry because they describe relationships between variables that aren't strictly linear.

When studying algebraic curves, rational functions serve as the 'glue' between divisors. They can create new divisors, called principal divisors, from the zeroes and poles of the function on the curve. These relations are foundational when considering linear equivalence of divisors, as they provide the mechanism to move from one divisor to another without changing the larger geometric structure.

Divisors in Algebraic Geometry

Divisors in algebraic geometry are essential tools that abstractly encode the zeroes and poles of meromorphic (which can loosely be seen as the algebraic version of 'rational') functions. In simpler terms, they are a way to catalog which points on a curve are 'special' in some way and how special they are.

To understand divisors, picture a curve with certain points highlighted. Each of these points has an integer attached to it, which could be positive, negative, or zero. These integers indicate zeroes (positive) and poles (negative) of some related meromorphic functions. As we saw in the exercise, the concept of linear equivalence comes into play when two divisors differ by a principal divisor, reflecting an inset pathway of transformation between divisors that doesn't change their overall 'value' or interaction with the curve.