Set theoretic equation for Veronese varieties

Question

Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety of dimension $n$.

The polynomials generating the ideal of $V_{n,\, d}$ are well-known: they are minors of a suitable matrix.

Question. Is it possible to cut-out $V_{n, \, d}$ set-theoretically with fewer equations?

For instance, we need three quadrics to generate the ideal of the twisted cubic $V_{1,3}\subset\mathbb{P}^3$, but $V_{1,3}$ is the set-theoretic intersection of a quadric and a cubic.

I am particularly interested in the case $V_{2,\,3}\subset\mathbb{P}^9$.

Answer 1

This is an attempt to address the special case of $V_{2, 3} \subseteq \mathbb{P}^9$. Denote the homogeneous coordinates on $\mathbb{P}^9$ by $z_{ij}$, $i + j \leq 3$, so that the embedding $\mathbb{A}^2 \hookrightarrow V_{2, 3} \subseteq \mathbb{P}^9$ is given by $z_{ij} = x^iy^j$. Then I believe the following $7$ equations cut $V_{2, 3}$ as a set theoretic complete intersection on $\mathbb{P}^9$: $$z_{20}z_{00} = z_{10}^2 \qquad z_{30}^2z_{00} = z_{20}^3 \qquad z_{02}z_{00} = z_{01}^2 \qquad z_{03}^2z_{00} = z_{02}^3$$ $$z_{12}z_{30} = z_{21}^2 \qquad z_{03}^2z_{30} = z_{12}^3 \qquad z_{11}^3 = z_{30}z_{03}z_{00} $$ (The first six equations are precisely the cubics and quadrics giving as set theoretic intersections the three twisted cubics coming from the "coordinate lines" on $\mathbb{P}^2$, and the last equation takes care of the remaining coordinate. An inductive procedure like this might work in general.)