Francesco Russo

ABSTRACT

The tangent degree $\tau(X)$ of a projective variety $X^n\subset\mathbb{P}^N$ is the number of tangent spaces to $X$ at smooth points passing through a general point of the tangent variety $\text{Tan}(X)\subseteq\mathbb{P}^N$ if $\dim(\text{Tan}(X))=2n$; it is equal to zero if $\dim(\text{Tan}(X))<2n$. After showing that $\tau(X)\neq 1$ when $N=2n$ we shall describe some classification results for $N=2n$ and $\tau(X)=2$ either in small dimension and/or under the smoothness assumption. Finally for $N\geq 2n+1$ we shall consider varieties $X^n\subset\mathbb{P}^N$ having $\tau(X)>1$ (unexpected behaviour), provide their classification for $n=2$ and discuss the case $n\geq 3$.

This is joint work with Jordi Hernandez Gomez.