The Moduli Space of Curves (Progress in Mathematics)

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In particular, we show that the closure of the twisted cubic curves in the Hilbert scheme is the flip of over the Chow variety.

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Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume Department of Mathematics, Harvard University. Correspondence to be sent to: dchen math. Oxford Academic.

Google Scholar. Cite Citation. Permissions Icon Permissions. It is conjectured that the CM line bundle yields a polarization on the conjectured moduli space of K-polystable Q-Fano varieties. This boils down to showing semi-positivity and positivity statements about the CM-line bundle for families with K-semi-stable and K-polystable Q-Fano fibers, respectively. I present a joint work with Giulio Codogni where we prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for very general fibers.

Our statements work in the most general singular situation klt singularities , and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. I also present applications to fibered Fano varieties.

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I will present an approach towards answering this question. This is joint work with Jenia Tevelev. There is by now an extensive theory of rational Chow rings of moduli spaces of smooth curves. The integral version of these Chow rings is not as well understood. I will survey what is known. In the last part of the talk I will discuss the Chow ring of the stack of stable curves of genus 2, which has been recently calculated by Eric Larson. I will present a different approach to the calculation, which offers an interesting point of view on stack of stable curves of genus 2. This part is joint work with Andrea Di Lorenzo.

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I will discuss Kulikov and stable degenerations of K3 surfaces and and describe explicit, geometric compactifications of their moduli spaces in several interesting cases. Based on joint work with Philip Engel and Alan Thompson.

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I will discuss joint work with Daniel Bragg on the geometry of supersingular K3 surfaces and their moduli. In particular, I will discuss a proof that for very general supersingular K3 surfaces, no non-Jacobian elliptic structure can carry a purely inseparable multisection. This appears to invalidate the published proof of Artin's unirationality conjecture.

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We consider rationality questions for threefolds over non-closed fields that become rational over an algebraic closure, like smooth complete intersections of two quadrics. Brody hyperbolic projective varieties over the complex numbers are extremely rich in properties. For instance, such varieties have only finitely many automorphisms, and every surjective endomorphism is actually an automorphism of finite order. In this talk, we will see how to establish some of these properties, and thereby verify many of the predictions made by the conjectures of Green-Griffiths and Lang. On the other hand, the introduction and study of a number of tropical moduli spaces of curves along with its realization as skeletons of their classical compactified counterparts allows for a deeper understanding of combinatorial aspects of moduli spaces and in particular of their compactifications.

This is joint work with Massimo Bagnarol and Fabio Perroni.

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We compute the motive in question, which depends on the curve, the integers n and rank E, but not the specific bundle. If time allows, I will outline applications by Bagnarol and Perroni which motivated this work. The techniques are from tropical geometry and graph complexes. Joint work with Faber, Galatius, Payne. I will first review the relationship between the classical Bessel equation and the Kloosterman sum. Then I will discuss the generalizations of this story for arbitrary reductive groups using ideals from the geometric Langlands program, based on the works by Frenkel-Gross, Heinloth-Ngo-Yun, myself, and the recent joint work in progress with Daxin Xu.

Progress in Mathematics 129

In recent years, understanding moduli spaces of relatively simple geometric objects has been a crucial ingredient in many advances in number theory in a subfield now called "arithmetic statistics. Using explicit descriptions of certain moduli spaces of genus one curves with extra structure, sometimes rationally and sometimes integrally, we show that the second moment and the average of the number of integral points on elliptic curves over Q is bounded.

This is joint work with Levent Alpoge.

Progress in Mathematics 129

The problem discussed in this talk is to compactify the moduli space of pairs C,L with C a smooth projective curve of genus g and L a line bundle of degree d on C. Such a compactification was first found by Caporaso a quarter century ago, but there is an obstruction to the existence of a universal family over it. Show less.