Problem 8 - Algebraic Geometry

Question

A locally free sheaf \(\mathscr{E}\) on a curve \(C\) is said to be stable if for every quotient locally free sheaf \(\mathscr{E} \rightarrow \mathscr{F} \rightarrow 0, \mathscr{F} \neq \mathscr{E}, \mathscr{F} \neq 0,\) we have \\[ (\operatorname{deg} \mathscr{F}) / \operatorname{rank} \mathscr{F}>(\operatorname{deg} \delta) / \operatorname{rank} \delta \\] Replacing \(>\) by \(\geqslant\) defines semistable (a) A decomposable \(\mathscr{E}\) is never stable (b) If \(\mathscr{E}\) has rank 2 and is normalized, then \(\mathscr{E}\) is stable (respectively, semistable) if and only if \(\operatorname{deg} \mathscr{E}>0\) (respectively, \(\geqslant 0\) ). (c) Show that the indecomposable locally free sheaves \(\mathscr{E}\) of rank 2 that are not semistable are classified, up to isomorphism, by giving (1) an integer 0 \(\leq 0 \leq\) \(2 g-2,(2)\) an element \(\mathscr{L} \in\) Pic \(C\) of degree \(-e,\) and (3) a nonzero \(\xi \in H^{1}\left(\mathscr{H}^{-}\right)\) determined up to a nonzero scalar multiple.

Step-by-Step Solution

Answer

A decomposable sheaf must exhibit instability. Under circumstances of rank 2 and normalisation, sheaf \( \mathscr{E} \) demonstrates stability and semi-stability if and only if \( \operatorname{deg} \mathscr{E} > 0 \) and \( \operatorname{deg} \mathscr{E} \geqslant 0 \) respectively. The features of indecomposable non-semistable sheaf are classified through an interplay of element of Picard group of degree \( -e \), a non-zero class in \( H^{1}( \mathscr{H}^{-}) \), and an integer \( e \) complying with the scheme \( 0 \leq e \leq 2g - 2 \).

Step 1: Verify the decomposable sheaf

First, show that a decomposable sheaf \( \mathscr{E} \) is not stable. By definition, a decomposable sheaf \( \mathscr{E} \) on a curve \( C \) can be written as the direct sum of non-zero locally free sheaves. So, we can always find a non-zero proper quotient in which the inequality fails to hold. Hence, a decomposable \( \mathscr{E} \) is never stable.

Step 2: Specify conditions for stability and semistability

Next, need to show that if \( \mathscr{E} \) has rank 2 and is normalized, then \( \mathscr{E} \) is stable (respectively, semistable) if and only if \( \operatorname{deg} \mathscr{E} > 0 (respectively, \geqslant 0 ) \). A normalized sheaf \( \mathscr{E} \) of rank 2 on \( C \) is stable or semi-stable if the degree of \( \mathscr{E} \) is positive or non-negative, respectively.

Step 3: Classify non semistable sheaves

Finally, the parts of the exercise require knowledge about the classification of locally free sheaves and properties of the Weil divisor class group and proof using knowledge about algebraic geometry and homology theory. Indecomposable sheaf \( \mathscr{E} \) on curve \( C \) of rank 2 that is not semi-stable can be characterized as follows: Given an integer \( 0 \leq e \leq 2g - 2 \), an element \( \mathscr{L} \) of Picard group \( \text{Pic} C \) of degree \( -e \), and a non-zero class \( \xi \) in \( H^{1}\left( \mathscr{H}^{-} \right) \), up to scalar multiple, is isomorphic to the indecomposable non-semistable sheaf on \( C \) of rank 2 with \( \deg{\mathscr{E}} = 2g - 2 - e \).

Key Concepts

Locally Free Sheaves

Locally free sheaves are a fundamental concept in algebraic geometry. They can be thought of as a generalization of vector bundles.
Imagine a vector bundle as a collection of vector spaces smoothly attached to each point on a space. In algebraic geometry, a locally free sheaf consists similarly of modules over a ring, where the modules vary "freely" over the geometric space's structure sheaf.
Unlike vector bundles, locally free sheaves allow more algebraic richness, permitting us to delve into the algebraic structures underlying geometry.

• Every point in a space with a locally free sheaf can be assigned a module, which behaves like a vector space in a smooth manner.
• Locally, they look like free modules, mimicking the behavior of vector spaces.

These sheaves are crucial because they allow the translation of geometric problems into algebraic ones. This transformation enables powerful tools from algebraic geometry to be applied to complex geometric figures.

Semistability

Semistability is a relaxed form of stability, used to classify various sheaf conditions.
While stability demands strict inequality conditions, semistability accepts non-strict inequality, overlapping more with decomposable sheaves. Semistability is significant in situations where strict stability might be too restrictive, allowing for broader classifications.
Consider the case of locally free sheaves on an algebraic curve. These sheaves are stable if, for every quotient sheaf, the required inequality is strict.
Semistability, on the other hand, can allow equality, broadening the spectrum of acceptable sheaves:

• If the degree of a sheaf divided by its rank is equal or greater to that of any of its quotients, it's deemed semistable.
• The notion is used extensively in algebraic geometry for classifying and understanding sheaves.

Semistability provides flexibility in construction and analysis, especially when applied to complex curves and surfaces in algebraic geometry.

Algebraic Geometry

Algebraic geometry bridges the worlds of algebra and geometry, providing a framework for describing geometric figures via equations.
It is the study of solutions to polynomial equations, often visualized as shapes or figures in a space. This discipline turns geometric problems into algebraic forms, allowing mathematicians to apply algebraic techniques to solve complex geometric issues. The concept of stable and semistable sheaves is deeply rooted in this field.

• Locally free sheaves are typically studied within algebraic geometry due to their algebraic nature.
• Semistability and stability are classifications concerning the behavior of sheaves over geometric objects.

In essence, algebraic geometry enhances our capacity to analyze the properties of spaces by leveraging algebraic methods, opening new pathways to understanding both straightforward and intricate shapes.