The kernel of Veronese Embedding $\phi: k[w,x,y,z]\to k[a^3,a^2b,ab^2,b^3]$ is $(wz−xy, wy−x^2, xz−y^2)$.

Asked Jan 19 '18 at 22:03

Active Jan 19 '18 at 22:03

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I want to rigorously prove the kernel of Veronese Embedding $\phi: k[w,x,y,z]\to k[a^3,a^2b,ab^2,b^3]$, where $w \mapsto a^3$, $x \mapsto a^2b$, $y \mapsto ab^2$, $z \mapsto b^3$, is the ideal $I=(wz−xy, wy−x^2, xz−y^2)$.

Clearly $I\subset \operatorname{ker}\phi$. But I can't figure out the opposite containment. There must be some tricks that I am not familiar with.

asked Jan 19 '18 at 22:03

No One

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1

This can be shown using Gröbner bases. Does this post answer your question? Actually, this post seems to solve the exact question you are asking. – Viktor Vaughn Jan 19 '18 at 22:11
@Quasicoherent Thank you! But I am not familiar with Gröbner bases. A direct proof will be appreciated! – No One Jan 19 '18 at 22:16
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The best direct proof is: familiarize yourself with Groebner bases. It is really simple. The book by Cox et al. Ideals, Varieties and Algorithms is probably the best way to learn this. – Mariano Suárez-Álvarez Jan 19 '18 at 22:34

The kernel of Veronese Embedding $\phi: k[w,x,y,z]\to k[a^3,a^2b,ab^2,b^3]$ is $(wz−xy, wy−x^2, xz−y^2)$.

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