# Del Pezzos contained in threefolds of minimal degree

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 $$4$$?

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 $$P$$).