Can the excess intersection theory be applied to the following problem:
I have a non-singular irreducible variety $X$ of dimension $k$ and degree $d$ and $k+1$ hyperplane sections of $X$, $H_1,H_2,\ldots,H_{k+1}$ such that $\cap _{j}H_j$ is not empty, as expected, but is a finite set of points $\Sigma$. Is there a way to find the degree of $\Sigma$ (that is the number of points?).