Vol. 12, No. 5, 2018

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Characterization of Kollár surfaces

Giancarlo Urzúa and José Ignacio Yáñez

Vol. 12 (2018), No. 5, 1073–1105
Abstract

Kollár (2008) introduced the surfaces

(x1a1 x2 + x2a2 x3 + x3a3 x4 + x4a4 x1 = 0) (w1,w2,w3,w4)

where wi = Wiw, Wi = ai+1ai+2ai+3 ai+2ai+3 + ai+3 1, and w = gcd(W1,,W4). 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 zw = l1a2a3a4l2a3a4l3a4l41, where {l1,l2,l3,l4} are four general lines in 2. In addition, by using various properties on classical Dedekind sums, we prove that:

  1. For any w > 1, we have pg = 0 if and only if the Kollár surface is rational. This happens when ai+1 1 or aiai+1 1(modw) for some i.
  2. For any w > 1, we have pg = 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 KS2e(S) 1.
Keywords
$\mathbb Q$-homology projective planes, Dedekind sums, branched covers
Mathematical Subject Classification 2010
Primary: 14J10
Milestones
Received: 23 December 2016
Revised: 29 January 2018
Accepted: 17 March 2018
Published: 31 July 2018
Authors
Giancarlo Urzúa
Facultad de Matemáticas
Pontificia Universidad Católica de Chile
Campus San Joaquín
Santiago
Chile
José Ignacio Yáñez
Department of Mathematics
University of Utah
Salt Lake City, UT
United States