2
$\begingroup$

Consider a Del Pezzo surface X whose anticanonical divisor $-K$ has degree $2$. In this case $-K$ is ample but not very ample. But $-2K$ is very ample and I am interested in the corresponding embedding to $\mathbb{P}^6$ (which has degree $8$). More precisely, I am interested in the question of which irreducible threefolds contain the embedded $X$. Using Riemann-Roch one can show that there is a $7$-dimensional space of quadrics vanishing on $X$.

On the other hand, consider a threefold $Y$ in $\mathbb{P}^6$ of minimal degree, i.e., of degree $4$. It is known, that $Y$ is cut out by a $6$ dimensional space of quadrics. Thus if we add one more general quadric, we get a surface of degree $8$ which is cut out by a $7$-dimensional space of quadrics.

This coincidence of numbers motivates the following question:

Is the intersection of $Y$ with a general quadric a Del Pezzo surface?

Is every Del Pezzo surface $X\subset\mathbb{P}^6$ defined as above contained in a threefold of degree $4$?

$\endgroup$

1 Answer 1

2
$\begingroup$

If $Y \subset \mathbb{P}^6$ is the cone over a Veronese surface $S \subset \mathbb{P}^5$ and $$ X = Y \cap Q $$ is a smooth intersection with a quadric (in particular, $Q$ does not contain the vertex $P$ of the cone), then $X$ is a del Pezzo surface of degree 2 (its anticanonical projection to $\mathbb{P}^2$ is induced by the projection of the cone $Y$ to its base $S$ out of $P$).

Conversely, any del Pezzo surface of degree 2 can be obtained in this way.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .