3
$\begingroup$

Let $V \subset \mathbb{P}^n$ be a projective variety. The homogeneous vanishing ideal is generated by the forms $f_1, \ldots , f_r$. Now consider the morphism

$\phi: \mathbb{P}^n \to \mathbb{P}^n, (x_0 : \ldots : x_n) \mapsto (x_0^2: \ldots : x_n^2).$

Is there a way to find generators of the vanishing ideal of the image $\phi(V$) under this morphism, i.e. an easier or more conceptional way than computung the image via Groebner bases?

$\endgroup$
2
  • $\begingroup$ Here is a method that works for hypersurfaces. I think one can make a version of it for other varieties but it is computationally intractable. You use Galois theory. Take $2^{n+1}$ copies of $f$, in each of which you multiply a different subset of $(x_0,...,x_n)$ by $-1$. Multiply these all together. This function will have only even powers of $x_i$ in it, so divide all those powers by two. Take the radical if necessary. $\endgroup$
    – Will Sawin
    Commented Jun 7, 2012 at 19:37
  • $\begingroup$ Okay, thank you. Considering symmetry of the $f_1, \ldots , f_r$ could lead to an acceptable result. $\endgroup$
    – Döni
    Commented Jun 10, 2012 at 15:12

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.