Euler sequence for complete smooth k*-surfaces

TL;DR

This paper generalizes the Euler sequence from toric varieties to complete smooth k*-surfaces without elliptic points, establishing a criterion for rigidity based on the Fano property.

Contribution

It introduces exact sheaf sequences for complexity one surfaces, extending the classical Euler sequence concept beyond toric varieties.

Findings

A new exact sequence of sheaves for k*-surfaces

Rigidity characterized by the Fano condition

Extension of Euler sequence to non-toric surfaces

Abstract

In this note we introduce exact sequences of sheaves on a complete smooth k*-surface without elliptic points. These sequences are an attempt to generalize the Euler sequence for a toric variety to complexity one surfaces. As an application we show that such a surface is rigid if and only if it is Fano.