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Let $H_1,H_2\cdots,H_{d-1}$ be hypersurfaces in $\mathbb{P}^d$, if the intersection $B:=H_1\cap H_2\cap \cdots \cap H_{d-1}$ is $1$-dimensional then it is called a complete intersection curve.

What are some sufficient conditions on $H_1,H_2,\cdots,H_{d-1}$ that will ensure that $B$ is $1$-dimensional and so a complete intersection curve?

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  • $\begingroup$ If the Jacobian matrix is everywhere of the correct rank, you get a smooth complete intersection. $\endgroup$
    – Felipe Voloch
    Commented Oct 10, 2014 at 1:32
  • $\begingroup$ Thanks, is there is a sufficient condition like the hypersurfaces do not share an irreducible component? Or even in the case when all the hypersurfaces are irreducible, can we say something? $\endgroup$
    – Sivakanth Gopi
    Commented Oct 10, 2014 at 2:00
  • $\begingroup$ Of course not. Take a bunch of hyperplanes passing through a fixed $(d-2)$-plane. $\endgroup$
    – Alex Degtyarev
    Commented Oct 10, 2014 at 6:43
  • $\begingroup$ It's kind of lame, but if the hypersurfaces are general. I can't even make good sense of how one would state transversality. Contemplate how one would distinguish the ideal (f,g) from the ideal (hf,hg). For me this says your question is very hard. $\endgroup$
    – meh
    Commented Oct 10, 2014 at 13:38

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