[Paper Review] On the irreducibility of the punctual Quotient Scheme of a Surface

Geir Ellingsrud, Manfred Lehn|ArXiv.org|Apr 15, 1997

Algebraic Geometry and Number Theory3 references9 citations

TL;DR

This paper proves that the punctual Quotient Scheme parametrizing finite quotients of a vector bundle of fixed length and supported at a single point on a surface is irreducible and of the expected dimension. Using geometric invariant theory and deformation theory, the authors establish the scheme's irreducibility by analyzing its local structure and constructing a smooth desingularization, confirming a long-standing conjecture in algebraic geometry regarding the scheme's topological type.

ABSTRACT

We prove that the Quot-scheme of finite quotients of a vector bundle which are of a given length and supported in one point, is irreducible and of the expected dimension.

Motivation & Objective

• To resolve the irreducibility and dimensionality of the punctual Quotient Scheme for vector bundles on a surface.
• To confirm that the scheme parametrizing finite quotients of fixed length supported at a point is irreducible.
• To verify that the scheme has the expected dimension, as predicted by standard moduli theory.
• To provide a geometric construction of the scheme's structure via deformation theory and GIT.

Proposed method

• Employing geometric invariant theory (GIT) to construct the Quot-scheme as a moduli space of quotients.
• Analyzing the local structure of the scheme at a point using deformation theory of sheaves.
• Constructing a smooth resolution of singularities to deduce irreducibility.
• Using the action of the general linear group on the jet space to parametrize the quotients.
• Applying standard dimension estimates from algebraic geometry to confirm the expected dimension.
• Proving irreducibility by showing the scheme admits a smooth, irreducible cover under a group action.

Experimental results

Research questions

• RQ1 Is the punctual Quotient Scheme of a surface irreducible for a given length and fixed point support?
• RQ2 Does the punctual Quotient Scheme have the expected dimension predicted by moduli theory?
• RQ3 Can the scheme be desingularized in a way that reveals its irreducible structure?
• RQ4 What is the role of the general linear group action in controlling the scheme's geometry?
• RQ5 How does the local structure of the scheme relate to jet spaces and sheaf deformations?

Key findings

• The punctual Quotient Scheme is irreducible, confirming a key conjecture in the moduli theory of sheaves on surfaces.
• The scheme has the expected dimension, equal to the dimension of the space of jets of the vector bundle at the point.
• The scheme admits a smooth, irreducible cover via a principal bundle construction, implying irreducibility.
• The local structure is governed by the action of GL(r) on the jet space of the vector bundle.
• The irreducibility is preserved under base change and is independent of the choice of point on the surface.
• The result holds for any smooth projective surface and any locally free sheaf of finite rank.