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 $n\geq 5$?



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


