IPM Algebraic Geometry Seminar
Past Talks
Speaker: Roya Beheshti, Washington University (US)
Title: Asymptotic Enumerativity of Tevelev Degrees
Abstract: A Tevelev degree is a type of Gromov-Witten invariant where the
domain curve is fixed in the moduli. After reviewing the basic definitions and previously
known results, I will report on joint work with Lehmann, Lian, Riedl, Starr, and Tanimoto,
where we improve the Lian-Pandharipande bound on asymptotic enumerativity of Tevelev degrees of hypersurfaces and provide counterexamples to asymptotic enumerativity for certain other Fano varieties.
Date: December 11, 2024
Video link
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Speaker: Francesco Russo, University of Catania (Italy)
Title: On smooth rational complete intersections
Abstract:The known results about the rationality vs irrationality of smooth Fano complete
intersections $X^n\subset\mathbb P^{n+c}$ of dimension $n=3,4,5$ and fixed type $(d_1,\ldots, d_c)$ suggest
an uniform approach to treat several open cases: index one; index two; quartic fourfolds and fivefolds; etc.
From one hand one would like to decide the rationality/irrationality of every element in the numerous cases where the stable irrationality of
the very general element is known (e.g. quartic fourfolds and fivefolds, quintic fivefolds, etc); from
the other hand one hopes to put some further light on several longstanding
conjectures (e.g. the irrationality of the very general cubic fourfold). After an introduction
of the general problem and after recalling the state of the art, we shall present some of our recent results on these topics.
Date: November 27, 2024
Speaker: Pietro De Poi, University of Udine (Italy)
Title: The importance of being projected
Abstract: A set of points Z in $\mathbb{P}^3$
is an (a, b)-geproci set (for GEneral
PROjection is a Complete Intersection) if its projection from
a general point to a plane is a complete intersection of
curves of degrees a and b.
We will report on some results in order to pursue
classification of geproci sets. Specifically, we will show how
to classify (a, b)-geproci sets Z which consist of a points on
each of b skew lines, assuming the skew lines have two
transversals in common. We will show in this case that
b ≤ 6.
Moreover we will show that all geproci sets of this type and
with no points on the transversals are contained in the $F_4$
configuration. We conjecture that a similar result is true for
an arbitrary number a of points on each skew line,
replacing containment in $F_4$ by containment in a half grid
obtained by the so-called standard construction.
Date: November 13, 2024
Video link
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Speaker: Fatemeh Mohammadi, KU Leuven (Belgium)
Title: Computational tropical geometry and its applications
Abstract: Tropical geometry is a combinatorial counterpart of algebraic geometry, transforming
polynomials into piecewise linear functions and their solutions (varieties) into polyhedral fans. This
transformation is intricately linked to the concept of Gröbner bases, which provide a powerful tool in
computational algebra. Specifically, all possible Gröbner bases of an ideal are encoded within a polyhedral
fan, with the tropical variety appearing as a subfan. Despite its significance, the computational complexity of tropical varieties
often limits computations to small-scale instances.
In this talk, we introduce a geometric approach that enables the effective computation of various points within
tropical varieties. One application of this method is the computation of toric degenerations, which are important objects
in algebraic geometry. These degenerations can be modeled on polytopes, and there exists a dictionary between
their geometric properties and the combinatorial invariants of the corresponding polytopes.
This dictionary can be extended from toric varieties to arbitrary varieties through toric degenerations.
Date: October 16, 2024