Noncommutative Blowups of Elliptic Algebras

TL;DR

This paper introduces a ring-theoretic method for constructing blowups of noncommutative elliptic surfaces, demonstrating that these blowups preserve key algebraic properties and have a rigid ideal structure.

Contribution

It develops a new approach to blowups in noncommutative projective geometry, extending properties of elliptic algebras and enabling classification of orders in Sklyanin algebras.

Findings

Constructed blowups T(d) of elliptic algebras T at divisors d.

Proved T(d) retains elliptic algebra properties, including being strongly noetherian and Auslander-Gorenstein.

Showed the ideal structure of T(d) is highly rigid.

Abstract

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element g of degree 1, T/gT is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T(d) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex. We also show that the ideal structure of T(d) is quite rigid. Our results generalise those of the first author. In the companion paper "Classifying Orders in the Sklyanin Algebra", we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra.