In Lemma 6.5, p 306 of their paper, Clemens and Griffiths proved that for a Cubic Hypersurface V in P^n with only one ordinary double Point x_0, lines in V passing x_0 can be realised as a type (2,3) Complete Intersection in P^(n-1). In p 311, they say that lines in V passing a general point is a type (2,3) Complete intersection in P^(n-2). Does anyone see how this follows? Thanks in advance
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4$\begingroup$ They are both special cases of the following: If the local expansion of $f(x_1,x_2,\cdots,x_n)$ at the origin is $f_1+f_2+\cdots f_d$, where $f_i$ has degree $i$, then the space of lines through the origin in $f=0$ is an intersection of the hypersurfaces corresponding to $f_1,f_2,\cdots, f_d$ in $\mathbb{P}^{d-1}.$ This follows from a short computation. $\endgroup$– dhyCommented Nov 8, 2018 at 20:58
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$\begingroup$ Okay, I think I now see how to prove your claim( It's some local calculation). The intersection needn't be complete or smooth and can have unexpected dimension. Please ignore my earlier comment $\endgroup$– user69183Commented Nov 13, 2018 at 18:38
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