# Local inequalities for $cA_k$ singularities

## TL;DR

This paper extends local intersection inequalities to weighted blowups and applies them to establish birational rigidity of certain Fano 3-folds with specific singularities, advancing understanding of their birational properties.

## Contribution

It generalizes an intersection inequality to weighted blowups and derives a new inequality for $cA_k$ singularities, leading to results on birational rigidity.

## Key findings

- Established the $4n^2/(k+1)$-inequality for $cA_k$ singularities.
- Proved birational rigidity for families of Fano 3-folds with $cA_k$ singularities.
- Extended intersection-theoretic inequalities to weighted blowups.

## Abstract

We generalize an intersection-theoretic local inequality of Fulton-Lazarsfeld to weighted blowups. As a consequence, we obtain the $4n^2/(k+1)$-inequality for isolated $cA_k$ singularities, an analogue of the $4 n^2$-inequality for smooth points. We use this to prove birational rigidity of many families of Fano 3-fold weighted complete intersections with terminal quotient singularities and isolated $cA_k$ singularities, including sextic double solids with $cA_1$ and ordinary $cA_2$ points.