Algebraic geometry6/27/2023 This conjecture is called CM minimization and a quantitative strengthening of the conjecture of separatedness of moduli spaces of K-stable varieties (K-moduli). Title: CM minimization and special K-stabilityĪbstract: Odaka proposed a conjecture predicting that the degrees of CM line bundles for families with fixed general fibers are strictly minimized if the special fibers are K-stable. Speaker: Masafumi Hattori (Kyoto University) Our results for (non-algebraic) schemes and rigid analytic spaces were previously only known in dimensions ≤ 3 and ≤ 2, respectively, and our results for formal schemes and Berkovich spaces are completely new. This not only recovers previous results for complex varieties, complex algebraic spaces, and complex analytic spaces, but also greatly extends the scope of the relative minimal model program with scaling to a broader class of geometric spaces, including formal schemes, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero. In this talk, I will discuss recent joint work with Shiji Lyu that establishes the relative minimal model program with scaling for excellent schemes, excellent algebraic spaces, and analytic spaces simultaneously in equal characteristic zero. Their result is now fundamental to our understanding of the birational classification of quasi-projective varieties and has numerous applications. Title: The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zeroĪbstract: In 2010, Birkar, Cascini, Hacon, and McKernan proved a relative version of the minimal model program for projective morphisms of complex quasi-projective varieties, called the relative minimal model program with scaling. Speaker: Takumi Murayama (Purdue University) This is a joint work with Quentin Gendron. Singularities of the differential give insights on the topological invariants of the fibers. For strata with several zeroes, isoresidual fibers are complex manifolds endowed with a matrix-valued meromorphic differential. The qualitative geometry of the latter translation surfaces is then classified by decorated trees, reducing the computation of the degree of the cover to a combinatorial problem. We give a formula to compute the degree of this cover and investigate its monodromy.The results are obtained using the dictionary between complex analysis and flat geometry of translation surfaces. For strata of 1-forms with only one zero, the isoresidual fibration is a cover of the space of configurations of residues ramified over an arrangement of complex hyperplanes. Fixing residues at the poles defined a fibration of any stratum to the vector space of configurations of residues. TITLE: Isoresidual fibration and resonance arrangementsĪBSTRACT: Meromorphic 1-forms on the Riemann sphere with prescribed orders of singularities form strata endowed with period coordinates. This is joint work with Brian Lawrence and Akshay Venkatesh. These are very useful when proving theorems about varieties of which we know almost nothing. In this talk, I’ll explain how to prove theorems about “sparsity” of rational points, a weaker notion which asks that the number of such with height less than B grows more slowly than any power of B it turns out that theorems of this kind can be proven for many varieties appearing as moduli spaces (though these still make up a very special subclass among the varieties we’d like to know about.) The two main ingredients are: 1) a classical trick allowing us to “trade” a single equation that’s hard to solve for many equations that are easier to solve 2) theorems of Heath-Brown type which provide bounds on points of bounded height on varieties which are uniform in the sense that they barely depend on what variety you’re studying. This might mean “there are only finitely many rational points” or “the rational points are contained in a proper closed subvariety.” Statements like these are extremely difficult to prove in any degree of generality, with Faltings’ finiteness theorem for rational points on high-genus curves a notable exception. Title: Sparsity of rational points on moduli spacesĪbstract: Varieties, if they are at all complicated, are expected to have very few rational points. Speaker: Jordan Ellenberg (University of Wisconsin)
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