Fourier-Mukai transform on Weierstrass cubics and commuting differential operators

TL;DR

This paper studies the spectral sheaves of commuting differential operators of genus one with singular spectral curves, classifies semi-stable sheaves on cuspidal Weierstrass cubics, and addresses a problem posed by Previato and Wilson.

Contribution

It provides a detailed description of spectral sheaves for certain commuting differential operators and classifies semi-stable sheaves on a cuspidal Weierstrass cubic, solving an open problem.

Findings

Spectral sheaves of genus one, rank two commuting differential operators with singular spectral curves are characterized.

All indecomposable semi-stable sheaves of slope one and ranks two or three on a cuspidal Weierstrass cubic are classified.

The work addresses and resolves a problem posed by Previato and Wilson.

Abstract

In this article, we describe the spectral sheaves of algebras of commuting differential operators of genus one and rank two with singular spectral curve, solving a problem posed by Previato and Wilson. We also classify all indecomposable semi-stable sheaves of slope one and ranks two or three on a cuspidal Weierstrass cubic.