I'm trying to prove that the degree of an algebraic variety given by a polynomial parametrization of the form $$x_1=P_1(u_1,\dotsc,u_{m-g+1}), \dotsc, x_m=P_m(u_1,\dotsc, u_{m-g+1}),$$ where all $P_i$'s have degree $2$, is $2^{m-g}$.

I have an intuition on the answer "with the hands" that it should be this because of the dimension of this variety is (generically) $m-g+1$. Indeed, it seems that I have only $m-g$ degrees of freedom since when all the $u_i$'s but one are fixed, then the $x_i$'s are determined as roots of quadratic polynomials. This is basically restating that the degree is the number of solutions once the variety is cut by enough generic hyperplanes.

But it feels non rigorous. Also, for such varieties this looks like a known result, but I found nothing in the litterature apart from "such varieties are irreducible". I don't see any easy way to compute the Hilbert Polynomial from this generating system either.

Thanks in advances,

Phil It seems

  • 1
    $\begingroup$ degree typically refers to an embedding in projective space. Unless I am misunderstanding something, you haven't considered a particular embedding, correct? $\endgroup$ – John Martin May 9 '16 at 19:00

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