Projections in toric degenerations and standard monomials

Abstract: I will report on joint work in progress with Takuya Murata. We study toric degenerations, i.e. flat morphism of a normal variety to the affine line whose generic fibre is isomorphic to a fixed projective variety and whose special fibre is a projective toric variety. Although such a flat morphism may be given abstractly (i.e. without an embedding, for example a toric scheme over the affine line) using valuations and Gröbner theory we may restrict our attention to the case where our family comes endowed with an embedding. I will illustrate an example of an elliptic curve where a toric degeneration admits a projection from the generic fibre (the elliptic curve) to the special fibre (the toric curve). We want to understand which kind of (embedded) toric degenerations admit such a projection. The notion of standard monomials in Gröbner theory proves to be a useful tool in constructing projections in arbitrary toric degenerations.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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