Algebraic Geometry in String Theory Seminar

Speaker: Iacopo Brivio (Harvard CMSA)

Title: On (semi)stable reduction and KSBA moduli in positive characteristic

Abstract: The moduli space M_g of genus g stable curves is perhaps the most studied of all algebraic varieties. Its higher-dimensional generalization is the moduli functor M_{n,v} of n-dimension stable varieties of volume v. It was proven only recently, and thanks to the joint effort of many over many years, that such functors are represented by projective algebraic spaces when working over the complex numbers. In this talk I will give some examples showing that the same moduli functors in positive characteristic are not even proper and, more in general, that the MMP fails to be functorial even in very nice families. In the second part I am going to explore some possible generalizations of the notion of stable variety that could be used as a replacement in positive characteristic.