Intersection Theory on Weighted Blowups of F-theory Vacua

TL;DR

This paper develops a formula for pushforward calculations in weighted blowups of algebraic varieties, with applications to F-theory compactifications and intersection theory on elliptic Calabi-Yau 4-folds.

Contribution

It generalizes previous results to arbitrary analytic functions of the exceptional divisor class in weighted blowups and applies this to compute intersection pairings in F-theory models.

Findings

Derived a general pushforward formula for weighted blowups.

Computed intersection numbers for F-theory related Calabi-Yau 4-folds.

Identified non-flat fibrations with 3-fold components in resolutions.

Abstract

Generalizing the results of 1211.6077 and 1703.00905, we prove a formula for the pushforward of an arbitrary analytic function of the exceptional divisor class of a weighted blowup of an algebraic variety centered at a smooth complete intersection with normal crossing. We check this formula extensively by computing the generating function of intersection numbers of a weighted blowup of the generic SU(5) Tate model over arbitrary smooth base, and comparing the answer to known results. Motivated by applications to four-dimensional F-theory flux compactifications, we use our formula to compute the intersection pairing on the vertical part of the middle cohomology of elliptic Calabi-Yau 4-folds resolving the generic F$_4$ and Sp(6) Tate models with non-minimal singularities. These resolutions lead to non-flat fibrations in which certain fibers contain 3-fold (divisor) components, whose physical interpretation in M/F-theory remains to be fully explored.