# The Degree of the Tangent and Secant Variety to a Projective Surface

## TL;DR

This paper introduces a method to compute the degrees of tangent and secant varieties of smooth projective surfaces using the Hilbert scheme of length-2 subschemes, assuming 3-very ampleness.

## Contribution

It provides a novel approach linking secant and tangent varieties to Hilbert schemes for surfaces embedded with 3-very ample divisors.

## Key findings

- Method computes degrees of secant and tangent varieties.
- Applicable to surfaces with 3-very ample embeddings.
- Establishes a link between these varieties and Hilbert schemes.

## Abstract

In this paper we present a way of computing the degree of the secant (resp., tangent) variety of a smooth projective surface, under the assumption that the divisor giving the embedding in the projective space is $3$-very ample. This method exploits the link between these varieties and the Hilbert scheme $0$-dimensional subschemes of length $2$ of the surface.