Seminar
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| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
Singularities of ideals admitting a squarefree Gröbner degeneration
Let S be a polynomial ring over a field, I a homogeneous ideal, X the projective variety defined
by I, and < a monomial order on S. Assume that in_<(I) is squarefree. In 2018, with Conca we raised the
question whether, in this situation, X being smooth implies that S/I is Cohen-Macaulay with negative
a-invariant (equivalently, S/I is a F-rational singularity). In 2019, in a joint paper with Constantinescu
and De Negri, we gave a positive answer to the question in some cases, and we turned the question into
a conjecture. However, it is still widely open, even when X is a curve. In this case, rephrasing it the conjecture
says that, if X is a smooth projective curve admitting some embedding for which the defining ideal has a
squarefree initial ideal, then X must have genus 0. In this talk we will largely discuss this conjecture, giving
some evidence for it and explaining why it is difficult to show it in general. We will also discuss some recent
developments done in an ongoing work with Huang, Tarasova, and Witt.