Torsion of a finite base locus

TL;DR

This paper explores the geometric interpretation of torsion in the symmetric algebra of an ideal sheaf, generalizes a formula for rational transformations with finite base loci, and constructs a specific example of a homaloidal curve in characteristic 3.

Contribution

It provides a geometric interpretation of torsion in symmetric algebras and extends a formula to positive characteristic, with applications to explicit curve construction.

Findings

Generalized a formula for rational transformations in positive characteristic.

Constructed a homaloidal curve of degree 5 in characteristic 3.

Answered a question on the existence of certain homaloidal curves.

Abstract

We interpret geometrically the torsion of the symmetric algebra of the ideal sheaf of a zero-dimensional scheme Z defined by $n+1$ equations in an $n$-dimensional variety. This leads us to generalise a formula of A.Dimca and S.Papadima in positive characteristic for a rational transformation with finite base locus. Among other applications, we construct an explicit example of a homaloidal curve of degree $5$ in characteristic $3$, answering negatively a question of A.V.D\'oria, S.H.Hassanzadeh and A.Simis.