Algebra & Number Theory

Characterization of Kollár surfaces

Giancarlo Urzúa and José Ignacio Yáñez

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Abstract

Kollár (2008) introduced the surfaces

( x 1 a 1 x 2 + x 2 a 2 x 3 + x 3 a 3 x 4 + x 4 a 4 x 1 = 0 ) P ( w 1 , w 2 , w 3 , w 4 )

where w i = W i w , W i = a i + 1 a i + 2 a i + 3 a i + 2 a i + 3 + a i + 3 1 , and w = gcd ( W 1 , , W 4 ) . The aim was to give many interesting examples of -homology projective planes. They occur when w = 1 . For that case, we prove that Kollár surfaces are Hwang–Keum (2012) surfaces. For w > 1 , we construct a geometrically explicit birational map between Kollár surfaces and cyclic covers z w = l 1 a 2 a 3 a 4 l 2 a 3 a 4 l 3 a 4 l 4 1 , where { l 1 , l 2 , l 3 , l 4 } are four general lines in P 2 . In addition, by using various properties on classical Dedekind sums, we prove that:

  1. For any w > 1 , we have p g = 0 if and only if the Kollár surface is rational. This happens when a i + 1 1 or a i a i + 1 1 ( mod w ) for some i .
  2. For any w > 1 , we have p g = 1 if and only if the Kollár surface is birational to a K3 surface. We classify this situation.
  3. For w 0 , we have that the smooth minimal model S of a generic Kollár surface is of general type with K S 2 e ( S ) 1 .

Article information

Source
Algebra Number Theory, Volume 12, Number 5 (2018), 1073-1105.

Dates
Received: 23 December 2016
Revised: 29 January 2018
Accepted: 17 March 2018
First available in Project Euclid: 14 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1534212097

Digital Object Identifier
doi:10.2140/ant.2018.12.1073

Mathematical Reviews number (MathSciNet)
MR3840871

Zentralblatt MATH identifier
06921170

Subjects
Primary: 14J10: Families, moduli, classification: algebraic theory

Keywords
$\mathbb Q$-homology projective planes Dedekind sums branched covers

Citation

Urzúa, Giancarlo; Yáñez, José Ignacio. Characterization of Kollár surfaces. Algebra Number Theory 12 (2018), no. 5, 1073--1105. doi:10.2140/ant.2018.12.1073. https://projecteuclid.org/euclid.ant/1534212097


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