Resolving nodes of a quintic CY 3-fold

Question

Let's consider the following quintic 3-fold $X$: \begin{equation} \{(x_i) \in \mathbb{P}^4 \ | \ x_1f(x)-x_2g(x)=0\} \end{equation} for generic homogeneous polynomials $f(x),g(x)$ of degree four. It is easy to se that $X$ has $16$ nodes at $\{x_1=x_2=f(x)=g(x)=0\}$. I now want a crepant resolution of $X$. There are four ways to blow-up the singularities, namely blow-up one of the four divisors: \begin{equation} \{x_1=x_2=0\},\ \{x_1=g(x)=0\},\ \{f(x)=x_2=0\},\ \{f(x)=g(x)=0\}. \end{equation} Is it true that any one of them gives a K\"{a}hler crepant resolution of $X$ and that the resulting Hodge numbers do not depend on the choice of resolutions?

Answer 1

I assume that you mean $x_1=x_2=0$ in your description of the divisors. I believe that you only really get two different resolutions, but yes, they are both projective varieties. Hodge numbers of birational complex Calabi-Yau varieties are always the same.

You can think of this as a determinantal variety for the matrix $A$ of the form $$ \Big(\begin{array}{cc}{\rm deg~1}& {\rm deg~1}\\ {\rm deg~4}& {\rm deg~4} \end{array}\Big) $$ in the variables $x$. The resolutions correspond to considering degenerate matrices together with either left or right kernel vectors.

This is also very much a toric example, in the sense that both resolutions are complete intersections of two hypersurfaces that correspond to "double-mirror" nef partitions, see for example the paper of my student Zhan Li "On the birationality of complete intersections associated to nef-partitions", arXiv:1310.2310.

Specifically, if I got my calculations right, one resolution is a complete intersection of degrees $(1,1)$ and $(1,4)$ in $\mathbb P^1\times \mathbb P^4$, and the other is a complete intersection in the $\mathbb P^1$ bundle $\mathbb P(\mathcal O(-1)+\mathcal O(-4)$) over $\mathbb P^4$ of the half-anticanonical divisor that corresponds to the tautological line bundle $\mathcal O(1)$ of this fibration. The singular variety in question is the projection to the $\mathbb P^4$ in both cases.