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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?).

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  • $\begingroup$ I doubt it. What can you tell us about your variety $X$? Is it linearly nondegenerate? $\endgroup$ – Jason Starr Aug 31 '14 at 14:51
  • $\begingroup$ $X$ is the Segre variety product of $k$ lines. $\endgroup$ – user46071 Aug 31 '14 at 14:55
  • $\begingroup$ For your variety, the number of points contained in $\Sigma$ can vary from $1$ up to, at least, $2^k-k-1$. So I do not see how an excess intersection theory computation will lead to so many different answers. Do you have some extra information about the hyperplane sections in your case of interest? $\endgroup$ – Jason Starr Aug 31 '14 at 15:31
  • $\begingroup$ No, I don't. And I know that for $k>3$, there are linear sections of codimension $k+1$ with more than $2^k-k-1$ points. Do you have any suggestion how I can tackle this problem? $\endgroup$ – user46071 Aug 31 '14 at 15:51
  • $\begingroup$ I think that intersection theory can help when I have a linear section of codimension $k$ that intersects $X$ in a curve $C$ and a finite set of points $\Gamma$. To get $|\Gamma|$ how much should I know about $C$? $\endgroup$ – user46071 Aug 31 '14 at 17:03

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