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By definition, a(n abstract) complex projective variety $X$ has an expression $E$ (many actually) of the form $\bigcap_i V_i$, with each $V_i$ a hypersurface in a ${\Bbb P}^m$ ($m$ depending on $E$). Each $V_i$ has its degree, so one may associate to $E$ the multiset of these degrees. The family of all multisets that arise as $E$ varies manifestly consititutes an invariant, call it ${\cal H}_X$, of $X$.

How can one calculate ${\cal H}_X$ from familiar invariants of $X$ (e.g. cohomology)?

What closure properties do families ${\cal H}_X$ necessarily possess?

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  • $\begingroup$ A few quick comments: (a) This depends on the choice of embedding into projective space. So it's not an invariant of $X$ alone. (b) It is more natural to consider the ideal of polynomials vanishing along $X$. $\endgroup$
    – Donu Arapura
    Commented Aug 22, 2012 at 17:51
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    $\begingroup$ @Donu I don't understand. I intended my ${\cal H}_X$ to vary over all embeddings into projective space. $\endgroup$
    – David Feldman
    Commented Aug 22, 2012 at 17:56
  • $\begingroup$ OK, I misunderstood. $\endgroup$
    – Donu Arapura
    Commented Aug 22, 2012 at 17:57
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    $\begingroup$ This will not be so well-behaved for embeddings where $X$ is not a complete intersection. When it is, you can extract that multiset from the Hilbert polynomial of $X$. The set of Hilbert polynomials of line bundles of $X$ can be described using Hirzebruch-Riemann-Roch and the intersection theory on $X$. Is that what you're looking for? $\endgroup$
    – Will Sawin
    Commented Aug 22, 2012 at 20:36
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    $\begingroup$ @Will Yes, please. Posting I didn't know whether this was a mere exercise for an expert or an open problem. Evidently a little of both. And please say something more about the not a complete intersection story. $\endgroup$
    – David Feldman
    Commented Aug 22, 2012 at 22:38

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