Birational modifications of surfaces via unprojections

TL;DR

This paper introduces a new approach to transforming minimal models of rational surfaces using unprojections that extend beyond traditional frameworks, requiring additional data for their definition.

Contribution

It presents a novel method for birational modifications of surfaces via unprojections that are not limited to projectively Gorenstein varieties and depend on extra divisor data.

Findings

Describes elementary transformations between minimal models of rational surfaces.

Extends unprojection techniques beyond Kustin-Miller framework.

Requires additional divisor data for unprojection construction.

Abstract

We describe elementary transformations between minimal models of rational surfaces in terms of unprojections. These do not fit into the framework of Kustin-Miller unprojections as introduced by Papadakis and Reid, since we have to leave the world of projectively Gorenstein varieties. Also, our unprojections do not depend on the choice of the unprojection locus only, but need extra data corresponding to the choice of a divisor on this unprojection locus.