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 $\dim (Tan(X))<2n$. In this paper we focus on general properties of $\tau (X)$ and of $deg(Tan(X))$. For example $\tau (X)\mathrm{\ne }1$ 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 in the spirit of the paper Ciliberto.Russo.2006. Then we consider the cases in which the previous two invariants attain the lower bounds found here, either in small dimension/codimension and/or under the smoothness assumption. Finally for $N\ge 2n+1$ we consider varieties $X^n\subset {\mathbb{P}}^N$ having $\tau (X)>1$ and provide their classification in small dimension.