Penn Arts & Sciences Logo

Math-Physics Joint Seminar

Friday, March 30, 2018 - 3:00pm

Roberto Svaldi

Cambridge University, UK


University of Pennsylvania

DRL, 4N30

Note special time and day!

Given a pair (X, D), where X is a proper variety and D a 

divisor with mild singularities, it is natural to ask how to bound the 
number of components of D or rather the sum of their (positive) coefficients. 
In general such bound does not exist. But when -(K_X+D) is positive, i.e. 
ample (or nef), then a conjecture of Shokurov says this bound should 
coincide with the sum of the dimension of X and its Picard number. 
When the bound is achieved, Shokurov conjectured that then X would 
be a toric variety and D a choice of toric invariant divisor. We prove 
a strengthened version of this conjecture and show that it suffices 
that the number of components of D is close enough to said sum 
for X to be a toric variety and D to be very close to a toric invariant 
divisor. Moreover, if time permits I will explain how to use the same kind 
of techniques to prove rationality of certain varieties. 
This is joint work with M. Brown, J. McKernan, R. Zong.