Differential Chow Form for Projective Differential Variety

TL;DR

This paper introduces a differential Chow form for projective differential varieties, extending affine concepts to projective cases, and demonstrates its properties and applications in differential algebraic geometry.

Contribution

It defines the differential Chow form for projective varieties and establishes its key properties, extending affine differential algebraic geometry results to the projective setting.

Findings

Proves a generic intersection theorem for projective differential varieties.

Defines and characterizes the differential Chow form in the projective case.

Applies the differential Chow form to linear dependence problems over projective varieties.

Abstract

In this paper, a generic intersection theorem in projective differential algebraic geometry is presented. Precisely, the intersection of an irreducible projective differential variety of dimension d>0 and order h with a generic projective differential hyperplane is shown to be an irreducible projective differential variety of dimension d-1 and order h. Based on the generic intersection theorem, the Chow form for an irreducible projective differential variety is defined and most of the properties of the differential Chow form in affine differential case are established for its projective differential counterpart. Finally, we apply the differential Chow form to a result of linear dependence over projective varieties given by Kolchin.