Complements of hypersurfaces in projective spaces

TL;DR

This paper investigates whether the isomorphism of complements of hypersurfaces in projective spaces implies the hypersurfaces themselves are isomorphic, providing new counterexamples and positive results across various degrees and dimensions.

Contribution

It offers counterexamples for the complement problem in higher dimensions and degrees, and establishes positive results for quadrics and certain cases, also addressing a question in the Grothendieck ring.

Findings

Counterexamples for all n, d ≥ 3 with (n, d) ≠ (3, 3)

Positive results for degree 2 hypersurfaces

A new criterion for rational normal projective surfaces in the Grothendieck ring

Abstract

We study the complement problem in projective spaces $\mathbb{P}^n$ over any algebraically closed field: If $H, H' \subseteq \mathbb{P}^n$ are irreducible hypersurfaces of degree $d$ such that the complements $\mathbb{P}^n \setminus H$, $\mathbb{P}^n \setminus H'$ are isomorphic, are the hypersurfaces $H$, $H'$ isomorphic? For $n = 2$, the answer is positive if $d\leq 7$ and there are counterexamples when $d = 8$. In contrast we provide counterexamples for all $n, d \geq 3$ with $(n, d) \neq (3, 3)$. Moreover, we show that the complement problem has an affirmative answer for $d = 2$ and give partial results in case $(n, d) = (3, 3)$. In the course of the exposition, we prove that rational normal projective surfaces admitting a desingularisation by trees of smooth rational curves are piecewise isomorphic if and only if they coincide in the Grothendieck ring, answering affirmatively a question posed by Larsen and Lunts for such surfaces.