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2016 | OriginalPaper | Chapter

Classical Moduli Spaces and Rationality

Author : Alessandro Verra

Published in: Rationality Problems in Algebraic Geometry

Publisher: Springer International Publishing

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Abstract

Moduli spaces and rational parametrizations of algebraic varieties have common roots. A rich album of moduli of special varieties was indeed collected by classical algebraic geometers and their (uni)rationality was studied. These were the origins for the study of a wider series of moduli spaces one could define as classical. These moduli spaces are parametrize several type of varieties which are often interacting: curves, abelian varieties, K3 surfaces. The course will focus on rational parametrizations of classical moduli spaces, building on concrete constructions and examples.

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previous chapter Derived Categories View on Rationality Problems

next chapter Unirationality of Moduli Spaces of Special Cubic Fourfolds and K3 Surfaces

A first counterexample to this outstanding problem has been finally produced by B. Hassett, A. Pirutka and Y. Tschinkel in 2016. See: Stable rationality of quadric surfaces bundles over surfaces, arXiv1603.09262

We recall that \(kod(X):= \mbox{ min}\{\dim f_{m}(X^{{\prime}}),\ m \geq 1\}\). Here X ^′ is a complete, smooth birational model of X. Moreover f _m is the map defined by the linear system of pluricanonical divisors \(P_{m}:= \mathbf{P}H^{0}(det(\Omega _{X^{{\prime}}}^{1})^{\otimes m})\). If P _m is empty for each m ≥ 1 one puts kod(X): = −∞.

Unless differently stated, we assume g ≥ 2 to simplify the exposition.

To simplify the notation we identify Pic(P ¹) to \(\mathbb{Z}\) via the degree map.

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Title: Classical Moduli Spaces and Rationality
Author: Alessandro Verra
Publisher: Springer International Publishing
Book: Rationality Problems in Algebraic Geometry
Print ISBN: 978-3-319-46208-0

Electronic ISBN: 978-3-319-46209-7

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-3-319-46209-7_4

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