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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The generalized Mukai conjecture for symmetric varieties
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by Giuliano Gagliardi and Johannes Hofscheier PDF
Trans. Amer. Math. Soc. 369 (2017), 2615-2649

Abstract:

We associate to any complete spherical variety $X$ a certain nonnegative rational number $\wp ({X})$, which we conjecture to satisfy the inequality $\wp ({X}) \le \dim X - \mathrm {rank} X$ with equality holding if and only if $X$ is isomorphic to a toric variety. We show that, for spherical varieties, our conjecture implies the generalized Mukai conjecture on the pseudo-index of smooth Fano varieties due to Bonavero, Casagrande, Debarre, and Druel. We also deduce from our conjecture a smoothness criterion for spherical varieties. It follows from the work of Pasquier that our conjecture holds for horospherical varieties. We are able to prove our conjecture for symmetric varieties.
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Additional Information
  • Giuliano Gagliardi
  • Affiliation: Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
  • Address at time of publication: Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
  • MR Author ID: 1040639
  • Email: gagliardi@math.uni-hannover.de
  • Johannes Hofscheier
  • Affiliation: Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
  • Address at time of publication: Institut für Algebra und Geometrie, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
  • Email: johannes.hofscheier@ovgu.de
  • Received by editor(s): December 30, 2014
  • Received by editor(s) in revised form: April 15, 2015
  • Published electronically: May 2, 2016
  • © Copyright 2016 by the authors
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 2615-2649
  • MSC (2010): Primary 14M27; Secondary 14J45, 14L30, 52B20
  • DOI: https://doi.org/10.1090/tran/6738
  • MathSciNet review: 3592522