# Asymptotic syzygies of normal crossing varieties

## TL;DR

This paper investigates the asymptotic syzygies of normal crossing varieties, revealing their behavior aligns with smooth components unless intersection cohomology introduces obstructions, which are explicitly computed.

## Contribution

It provides a formula for asymptotic syzygies of normal crossing varieties considering intersection cohomology, and applies these results to degenerations of smooth projective varieties.

## Key findings

- Asymptotic syzygies match those of smooth components in absence of obstructions.
- Cohomology of the associated simplicial complex determines syzygy behavior.
- Results extend to degenerations of smooth projective varieties.

## Abstract

Asymptotic syzygies of a normal crossing variety follow the same vanishing behavior as one of its smooth components, unless there is a cohomological obstruction arising from how the smooth components intersect each other. In that case, we compute the asymptotic syzygies in terms of the cohomology of the simplicial complex associated to the normal crossing variety. We combine our results on normal crossing varieties with knowledge of degenerations of certain smooth projective varieties to obtain some results on asymptotic syzygies of those smooth projective varieties.