Examples of non--effective rays at the boundary of the Mori cone of blow--ups of the plane

TL;DR

This paper proves that certain linear systems of plane curves with specified multiplicities and points are not effective, providing insights into the structure of the Mori cone boundary for blow-ups of the plane.

Contribution

It establishes non-effectiveness of specific linear systems, advancing understanding of the Mori cone boundary in algebraic geometry.

Findings

No multiple of the given linear system is effective for the specified parameters.

The result constrains the shape of the Mori cone boundary.

Provides new examples of non-effective rays at the boundary.

Abstract

In this paper we prove that no multiple of the linear system of plane curves of degree $d\geq 4$ with a point of multiplicity $d-m$ (with $2 \leq m \leq d$) and $m(2d-m)$ simple general points is effective.