Triple planes with 𝑝_{𝑔}=𝑞=0

Faenzi, Daniele; Polizzi, Francesco; Vallès, Jean

doi:10.1090/tran/7276

Triple planes with $p_g=q=0$
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by Daniele Faenzi, Francesco Polizzi and Jean Vallès PDF

Trans. Amer. Math. Soc. 371 (2019), 589-639 Request permission

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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