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Transactions of the American Mathematical Society

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

 
 

 

Triple planes with $ p_g=q=0$


Authors: Daniele Faenzi, Francesco Polizzi and Jean Vallès
Journal: Trans. Amer. Math. Soc. 371 (2019), 589-639
MSC (2010): Primary 14E20, 14J60
DOI: https://doi.org/10.1090/tran/7276
Published electronically: June 26, 2018
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Abstract: We show that general triple planes with genus and irregularity zero belong to at most 12 families, that we call surfaces of type I to XII, and we prove that the corresponding Tschirnhausen bundle is a direct sum of two line bundles in cases I, II, III, whereas it is a rank 2 Steiner bundle in the remaining cases.

We also provide existence results and explicit descriptions for surfaces of type I to VII, recovering all classical examples and discovering several new ones. In particular, triple planes of type VII provide counterexamples to a wrong claim made in 1942 by Bronowski.

Finally, in the last part of the paper we discuss some moduli problems related to our constructions.


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Additional Information

Daniele Faenzi
Affiliation: Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université de Bourgogne Franche-Comté, 9 Avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France
Address at time of publication: Université de Bourgogne, Institut de Mathématiques de Bourgogne, UMR CNRS 5584 UFR Sciences et Techniques, Bâtiment Mirande, 9 Avenue Alain Savary BP 47870, 21078 Dijon Cedex, France
Email: daniele.faenzi@u-bourgogne.fr

Francesco Polizzi
Affiliation: Dipartimento di Matematica, Università della Calabria, Cubo 30B, 87036 Arcavacata di Rende, Cosenza, Italy
Email: polizzi@mat.unical.it

Jean Vallès
Affiliation: Université de Pau et des Pays de l’Adour, Avenue de l’Université - BP 576, 64012 PAU Cedex, France
Email: jean.valles@univ-pau.fr

DOI: https://doi.org/10.1090/tran/7276
Keywords: Triple plane, Steiner bundle, adjunction theory
Received by editor(s): August 30, 2016
Received by editor(s) in revised form: April 29, 2017, and May 6, 2017
Published electronically: June 26, 2018
Additional Notes: The first and third authors were partially supported by ANR projects GEOLMI ANR-11-BS03-0011 and Interlow ANR-09-JCJC-0097-01.
The second author was partially supported by Progetto MIUR di Rilevante Interesse Nazionale Geometria delle Variet$à$ Algebriche e loro Spazi di Moduli, by the Gruppo di Ricerca Italo-Francese di Geometria Algebrica (GRIFGA) and by GNSAGA-INdAM.
Article copyright: © Copyright 2018 American Mathematical Society