0
$\begingroup$

I found this statement, and I can't get a proof:

Given $X\subset \mathbb{P}^{n}$ an irreducible projective set of dimension $d$ and degree $e$, and given $H_{1},...,H_{d}$ hypersurfaces with degrees $e_{1},...,e_{d}$, respectively, such that $X\cap H_{1}\cap...\cap H_{d}$ is a finite set of points, then the number of points is precisely $ee_{1}...e_{d}$, counted with multiplicity.

Nothing is assumed about the regularity of the sequence of the polynomials defining the hypersurfaces.

Could anyone give me a hint about how can it be proven?

Thanks a lot in advance.

Improve this question
$\endgroup$
4
  • 3
    $\begingroup$ See for example Fulton's Intersection Theory, Prop. 8.4. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 11, 2020 at 23:36
  • 1
    $\begingroup$ If you want to prove it yourself, I think you should be able to do it using Hilbert polynomials (use additivity in short exact sequences). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 12, 2020 at 0:19
  • 1
    $\begingroup$ See Harris Algebraic Geometry (a first course), Theorem 18.4 (perhaps less intimidating than Fulton). $\endgroup$
    – abx
    Commented Jun 12, 2020 at 4:59
  • $\begingroup$ Thanks! My first approach was using Hilbert polynomial, but I’m not sure how to get rid of the undesirable embedded components that may appear when treating with the sum of the ideals, just to get nice exact sequences that give you the degree of the intersection (counting multiplicity) $\endgroup$
    – Carnby
    Commented Jun 14, 2020 at 22:53

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .