Projective toric varieties of codimension 2 with maximal Castelnuovo--Mumford regularity

Preston Cranford; Alan Peng; Vijay Srinivasan

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arXiv · 2106.12667

Projective toric varieties of codimension 2 with maximal Castelnuovo--Mumford regularity

Preston Cranford·Alan Peng·Vijay Srinivasan

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Abstract

The Eisenbud--Goto conjecture states that $\operatorname{reg} X\le\operatorname{deg} X -\operatorname{codim} X+1$ for a nondegenerate irreducible projective variety $X$ over an algebraically closed field. While this conjecture is known to be false in general, it has been proven in several special cases, including when $X$ is a projective toric variety of codimension $2$. We classify the projective toric varieties of codimension $2$ having maximal regularity, that is, for which equality holds in the Eisenbud--Goto bound. We also give combinatorial characterizations of the arithmetically Cohen--Macaulay toric varieties of maximal regularity in characteristic $0$.

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Preston Cranford, Alan Peng, Vijay Srinivasan. 2021-06-23. Projective toric varieties of codimension 2 with maximal Castelnuovo--Mumford regularity. https://arxiv.org/abs/2106.12667

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