Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings

TL;DR

This paper demonstrates that Lyubeznik numbers of cones over projective schemes can vary depending on the embedding choice, providing a negative answer to a longstanding question in characteristic zero.

Contribution

It constructs examples showing Lyubeznik numbers depend on projective embeddings, contrasting with known results in positive characteristic and addressing an open question.

Findings

Lyubeznik numbers depend on projective embeddings in characteristic zero

Constructed examples of schemes with variable Lyubeznik numbers

Contrasts with positive characteristic case where Frobenius is used

Abstract

We construct complex projective schemes with Lyubeznik numbers of their cones depending on the choices of projective embeddings. This answers a question of G. Lyubeznik in the characteristic 0 case. It contrasts with a theorem of W. Zhang in the positive characteristic case where the Frobenius endomorphism is used. Reducibility of schemes is essential in our argument. B. Wang recently constructed examples of irreducible projective schemes (which are not normal) from our examples of reducible ones. So the question is still open in the normal singular case.