Non-existence of phantoms on some non-generic blowups of the projective plane

TL;DR

This paper proves that certain blowups of the projective plane at points on a smooth cubic curve do not contain phantoms when the points are in very general position, clarifying the structure of these algebraic surfaces.

Contribution

It establishes the non-existence of phantoms on specific non-generic blowups of the projective plane, a result previously unknown.

Findings

Blowups at points on a smooth cubic curve lack phantoms in very general position.

The result applies to non-generic configurations of points on the cubic curve.

Provides new insights into the structure of algebraic surfaces obtained by such blowups.

Abstract

We show that blowups of the projective plane at points lying on a smooth cubic curve do not contain phantoms, provided the points are chosen in very general position on this curve.