Filtered deformations of elliptic algebras

TL;DR

This paper studies filtered deformations of elliptic algebras within noncommutative geometry, providing explicit classifications and connecting them to noncommutative del Pezzo surfaces, while also advancing the understanding of exceptional collections on del Pezzo surfaces.

Contribution

It introduces a detailed analysis and explicit computation of filtered deformations of elliptic algebras, linking them to noncommutative del Pezzo surfaces and classifying exceptional collections.

Findings

Explicit families of filtered deformations computed

Connections established with noncommutative del Pezzo surfaces

New results on classification of exceptional collections

Abstract

One of the difficulties in doing noncommutative projective geometry via explicitly presented graded algebras is that it is usually quite difficult to show flatness, as the Hilbert series is uncomputable in general. If the algebra has a regular central element, one can reduce to understanding the (hopefully more tractable) quotient. If the quotient is particularly nice, one can proceed in reverse and find all algebras of which it is the quotient by a regular central element (the filtered deformations of the quotient). We consider in detail the case that the quotient is an elliptic algebra (the homogeneous endomorphism ring of a vector bundle on an elliptic curve, possibly twisted by translation). We explicitly compute the family of filtered deformations in many cases and give a (conjecturally exhaustive) construction of such deformations from noncommutative del Pezzo surfaces. In the process, we also give a number of results on the classification of exceptional collections on del Pezzo surfaces, which are new even in the commutative case.