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Classical Moduli Spaces and Rationality

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Rationality Problems in Algebraic Geometry

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2172))

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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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Notes

  1. 1.

    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

  2. 2.

    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): = −∞.

  3. 3.

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

  4. 4.

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

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Acknowledgements

I wish to thank CIME and the scientific coordinators of the School ‘Rationality problems’ for the excellent and pleasant realization of this event. Let me also thank the referee for a patient reading and several useful suggestions. Supported by MIUR PRIN-2010 project ‘Geometria delle varietá algebriche e dei loro spazi di moduli’ and by INdAM-GNSAGA.

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  1. Dipartimento di Matematica, Universitá Roma Tre, Largo San Leonardo Murialdo, 1-00146, Roma, Italy

    Alessandro Verra

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  1. Dipartimento di Matematica, Università di Pisa Dipartimento di Matematica, Pisa, Italy

    Rita Pardini

  2. Dipartimento di matematica, University of Pavia, Pavia, Italy

    Gian Pietro Pirola

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Verra, A. (2016). Classical Moduli Spaces and Rationality. In: Pardini, R., Pirola, G. (eds) Rationality Problems in Algebraic Geometry. Lecture Notes in Mathematics(), vol 2172. Springer, Cham. https://doi.org/10.1007/978-3-319-46209-7_4

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