# Del Pezzo surfaces of degree four and complete intersections of two quadrics

Let $$X = Q_1\cap Q_2$$ be a complete intersection of two smooth quadrics, over a field $$K$$, in $$\mathbb{P}^4$$ with homogeneous coordinates $$y_0,y_1,y_2,y_3,y_4$$.

Set $$Q_1 = \{F_1 = 0\}$$ and $$Q_2 = \{F_2 = 0\}$$ and assume that the monomial $$y_0^2$$ does not appear in $$F_1$$ so that $$Q_1$$ is rational and that the monimial $$y_0y_1$$ does not appear in $$F_2$$.

Under these hypotheses could we conclude anything about the unirationality over $$K$$ of $$X$$?

What if $$X$$ is a complete intersection of two quadrics in $$\mathbb{P}^n$$ with the same properties for $$n\geq 5$$?

## 1 Answer

No.

Let $$Q_1,Q_2$$ be arbitrary quadrics. Let $$a$$ be the coefficient of $$y_0^2$$ in $$F_1$$, $$b$$ the coefficient of $$y_0^2$$ in $$F_2$$, $$c$$ the coefficient of $$y_0 y_1$$ in $$F_1$$, $$d$$ the coefficient of $$y_0 y_1$$ in $$F_2$$.

Then the coefficient of $$y_0^2$$ in $$b F_1 - a F_2$$ and the coefficient of $$y_0 y_1$$ in $$d F_1 - c F_2$$ both vanish. If $$ad-bc \neq 0$$ (the generic case), then $$b F_1 - a F_2$$ and $$d F_1 - c F_2$$ is $$Q_1 \cap Q_2$$.

So any intersection of two quadrics that satisfies a mild genericity property can be written in this form, and thus the unirationality problem is as hard as for a general intersection of two quadrics.