Kapranov, we look at these classical spaces from a modern Shtukas, which he subsequently used to prove the Langlands correspondence for the general Lafforgue’s compactification of the stack of Drinfeld’s Intensively and has been used to derive groundbreaking results in diverse areas of mathematics. More recently, the space of complete collineations has been studied smooth with normal crossings boundary) of the space of full-rank matrices between two It provides a ‘wonderful compactification’ Tyrell and others, dating back to the 19th century. Which has its origins in the classical works of Chasles, Giambieli, Hirst, Schubert, The space of complete collineations is an important and beautiful chapter of algebraic geometry, Hyperkähler manifolds which are isomorphic to Hilbert schemes of points on K3 surfaces.Ĭomplete Complexes and Spectral Sequences In a given period space have the property that their general points correspond to projective If time permits, as second application, we will show that infinitely many Heegner divisors (deformation equivalent to Hilbert schemes of points on K3 surfaces) by Bayer, Hassett, andĪs an application we will present a new short proof (by Bayer and Mongardi) for theĬelebrated result by Laza and Looijenga on the image of the period map for cubic fourfolds. Ingredient is the description of the nef and movable cone for projective hyperkähler manifolds Period map is a finite union of explicit Heegner divisors that we describe. Our main result is that the complement of the image of the Torelli theorem implies that their period map is an open embedding when restricted These are parametrized by a quasi-projective 20-dimensional moduli space and Verbitsky’s Of Hilbert schemes of points on K3 surfaces and are equipped with a polarization of fixed The aim of the talk is to study smooth projective hyperkähler manifolds which are deformations The period map for polarized hyperkähler manifolds Gottsche-Hirschowitz and Bertram-Goller-Johnson to the case of Our results extend analogous results on the projective plane by Which are a slight relaxation of the notion of semistable sheaves which Technical ingredient is to consider the notion of prioritary sheaves, Problem to determine when a general sheaf is globally generated. Next, we use our solution to the weak Brill-Noether Sheaf, and in particular determine whether sheaves have the "expected"Ĭohomology that one would naively guess from the sign of the EulerĬharacteristic. Weak Brill-Noether problem seeks to compute the cohomology of a general We consider two main questions of this sort. Therefore it makes sense to ask about the properties of a Moduli spaces of semistable sheaves on $X$ with fixed numerical invariants are always irreducible by a theorem of Walter. Properties of general sheaves on Hirzebruch surfaces In this way, we hope that such meeting will highlight Brazil asĪ major location for work in algebraic geometry to the next generation of researchers. Helps to establish working contacts, building on the existing network that links brazilian algebraic We expect toĬreate an enviroment that enhances the exchange of new ideas through discussion sessions, and Related areas on the occasion of the 2018 International Congress of Mathematicians. Our goal is to bring together senior and young researchers currently working on moduli spaces and The theory of moduli spaces is a common thread that unites all of these areas and also has a relevant interplay Holomorphic foliations, and interactions with String Theory and mathematical physics. The Algebraic Geometry community in Brazil counts with several research groups dedicated withĭiverse aspects of the field, like arithmetic geometry, birational geometry, classification of sheaves,
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