*9th Meeting, October 14, 2017, BICMR, PKU*

Kevin Tucker (UIC)

*Bertini theorems for F-signature*

In characteristic zero, it is well known that multiplier ideals and log terminal singularities satisfy Bertini-type theorems for hyperplane sections. In characteristic p > 0, the analogous statements for test ideals are more complicated: while F-regular singularities satisfy Bertini, the test ideal does not. In this talk, I will describe joint work with Karl Schwede and Javier Carvajal-Rojas showing that the F-signature – a numerical invariant of singularities that detects F-regularity – satisfies the relevant Bertini statements for hyperplane sections. In particular, one can view this as a generalization of the corresponding results for F-regularity.

Roberto Svaldi (Cambridge)

*Log birational boundedness of Calabi-Yau pairs*

I will discuss new results towards the birational boundedness of Calabi-Yau pairs. Recent work in the minimal model program suggests that pairs with trivial log canonical class should satisfy some boundedness properties.

I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are indeed log birationally bounded. This implies birational boundedness of elliptically fibered Calabi-Yau manifolds with a section, in dimension up to 5. This is joint work with Gabriele Di Cerbo.

Xuanyu Pan (AMSS, CAS)

*Cycles on Fano manifolds*

In this talk, I will give a survey about cycles on Fano manifolds including a recent joint work with Cristian Minoccheri. The philosophy is that the positivity of the tangent bundle implies the simplicity of the cycle relations. At the end of this talk, I will talk about some speculation on 1-cycles on Fano manifolds.

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