On the tangent degree and the degree of the tangent variety of a projective variety

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 $Tan(X)\subseteq\mathbb P^N$, if positive and finite; it is equal to zero if $\dim(Tan(X))$ < $2n$.

We shall focus on general properties of $\tau(X)$ and of $\deg(Tan(X))$. For example $\tau(X)\neq 1$ if $N=2n$ and, as soon as $Tan(X)$ does not coincide with the secant variety, we prove a linear lower bound for the degree of $Tan(X)$ in terms of its codimension, generalizing the classical Bertini-del Pezzo lower bound for arbitrary varieties.

Then we shall present some results about the cases in which the previous two invariants attain the lower bounds present, either in small dimension/codimension and/or under the smoothness assumption.

Finally, if time allows, for $N\geq 2n+1$ we consider varieties $X^n\subset\mathbb P^N$ having $\tau(X)$ > $1$ and provide their classification in small dimension.

This is joint work with Jordi Hernandez Gomez.