A note on rational normal curves totally tangent to a Hermitian variety

TL;DR

This paper counts rational normal curves tangent to a Hermitian variety at q+1 points, extending Segre's classical result on Hermitian curves and conics, and provides explicit enumeration in higher dimensions.

Contribution

It generalizes Segre's result by counting rational normal curves tangent to Hermitian varieties at multiple points in higher-dimensional projective spaces.

Findings

Explicit count of tangent rational normal curves to Hermitian varieties.

Extension of classical results from curves to higher-dimensional varieties.

Provides formulas for intersection multiplicities in complex projective spaces.

Abstract

Let q be a power of a prime integer p, and let X be a Hermitian variety of degree q+1 in the n-dimensional projective space. We count the number of rational normal curves that are tangent to X at distinct q+1 points with intersection multiplicity n. This generalizes a result of B. Segre on the permutable pairs of a Hermitian curve and a smooth conic.