Let $X \subset \mathbb{P}^N$ be a projective variety of dimension $n$. Let us denote with $TX=\bigcup_{x \in X}\mathbb{T}_xX$ the tangential variety, where $\mathbb{T}_x X$ is the projective tangent space to $X$ at $x$. If $Y \subset X$ is a subvariety, is there a way to compute the linear span of $TX_{|Y}=\bigcup_{x \in Y}\mathbb{T}_xX$ in terms of "natural" data attached to $X,Y$ (clearly dependent on the embedding) and possibly to their tangent bundles?
Thanks in advance
