Vol. 12, No. 5, 2018

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Characterization of Kollár surfaces

Giancarlo Urzúa and José Ignacio Yáñez

Vol. 12 (2018), No. 5, 1073–1105

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.

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$\mathbb Q$-homology projective planes, Dedekind sums, branched covers
Mathematical Subject Classification 2010
Primary: 14J10
Received: 23 December 2016
Revised: 29 January 2018
Accepted: 17 March 2018
Published: 31 July 2018
Giancarlo Urzúa
Facultad de Matemáticas
Pontificia Universidad Católica de Chile
Campus San Joaquín
José Ignacio Yáñez
Department of Mathematics
University of Utah
Salt Lake City, UT
United States