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arXiv · 2208.07238

Multidegrees, prime ideals, and non-standard gradings

Alessio Caminata·Yairon Cid-Ruiz·Aldo Conca
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Abstract

We study several properties of multihomogeneous prime ideals. We show that the multigraded generic initial ideal of a prime has very special properties, for instance, its radical is Cohen-Macaulay. We develop a comprehensive study of multidegrees in arbitrary positive multigraded settings. In these environments, we extend the notion of Cartwright-Sturmfels ideals by means of a standardization technique. Furthermore, we recover or extend important results in the literature, for instance: we provide a multidegree version of Hartshorne's result stating the upper semicontinuity of arithmetic degree under flat degenerations, and we give an alternative proof of Brion's result regarding multiplicity-free varieties.

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Alessio Caminata, Yairon Cid-Ruiz, Aldo Conca. 2022-08-15. Multidegrees, prime ideals, and non-standard gradings. https://arxiv.org/abs/2208.07238

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