# Maximally nodal degree 6 Fano threefolds

Let $$X$$ be a complete intersection of a quadric and a cubic in $$\mathbb{P}^5$$. In the smooth case it is a so-called Fano threefold of index one and degree six.

I would like to consider the case when $$X$$ has only ordinary double points as singularities. The question is: what's maximal number of ordinary double points that $$X$$ can have?

Here is a lower bound. The threefold $$X$$ defined by the following two equations $$x_0^2 + x_1^2 + x_2^2 - x_3^2 - x_4^2 - x_5^2 = (x_0 + x_1 + x_2 - x_3 - x_4 - x_5)^2$$ $$x_0^3 + x_1^3 + x_2^3 - x_3^3 - x_4^3 - x_5^3 = (x_0 + x_1 + x_2 - x_3 - x_4 - x_5)^3$$ has $$34$$ ordinary double points. (I found these by projecting a $$5$$-dimensional Segre cubic with $$35$$ ordinary double points from one of them.) The singularities are located at points $$[x_0, ..., x_5]$$ such that all $$x_i$$'s are $$0$$ or $$1$$, and the number of $$1$$'s among $$x_0, x_1, x_2$$ is either equal to or one more than the number of $$1$$'s among $$x_3, x_4, x_5$$.

Can there be more than $$34$$?

Another question: is $$X$$ necessarily a rational variety when the number of ordinary double points is large enough?

• @Pop: yes, thank you, I fixed this typo. – Evgeny Shinder Sep 3 '19 at 20:49