Log Geometry and the Joy of Desingularisation.

Navid Nabijou (Glasgow)

Monday 25th February 16:00-17:00 Maths 311B

Abstract

A toric variety is an algebraic variety with a sufficiently “nice” action by an algebraic torus; these objects are arguably the most well-understood in all of algebraic geometry, because they can be studied entirely in terms of the combinatorics of an associated piecewise-linear “fan”.

Because they possess large symmetry groups, toric varieties are vanishingly rare in nature. On the other hand, many geometries look “locally” like toric varieties, even if they have no global toric structure. The idea of log geometry, roughly, is to encode this local information in the data of something called a “log structure”. With this at hand, various standard constructions in toric geometry (for instance: toric blowups) can be extended to the logarithmic setting. This has myriad applications, one of the most exciting of which is to the study of moduli spaces. In this talk I will discuss this circle of ideas, with an emphasis on explicit computations.
 
Time permitting, I will discuss joint work with L. Battistella and D. Ranganathan, in which we use log geometry to produce a desingularisation of the moduli space of relative stable maps in genus one, and apply this to derive recursion formulae for Gromov-Witten invariants.

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