Characterization of Kollár surfaces

Abstract

The aim was to give many interesting examples of Q-homology projective planes. They occur when w * = 1. For that case, we prove that Kollár surfaces are Hwang-Keum [HK12] surfaces. For w * > 1, we construct a geometrically explicit birational map between Kollár surfaces and cyclic covers z w * = l a 2 a 3 a 4

Open surfaces of small volume

“…In [HK12] Hwang and Keum construct, for any a 1 , a 2 , a 3 , a 4 ≥ 2, a surface T = T (a 1 , a 2 , a 3 , a 4 ) with ρ(T ) = 1 obtained by blowing up the 4-line configuration; it has two cyclic singularities corresponding to the chains [2 * (a 4 −1), a 3 , a 1 , 2 * (a 2 −1)] and [2 * (a 3 − 1), a 2 , a 4 , 2 * (a 1 − 1)]. In particular, these surfaces include all the surfaces S * (a 1 , a 2 , a 3 , a 4 ) with gcd(w 1 , w 3 ) = gcd(w 2 , w 4 ) = 1 by [UYn16].…”

“…For example, hypersurfaces S(a 1 , a 2 , a 3 , a 4 ) of the form x a1 1 x 2 + x a2 2 x 3 + x a3 3 x 4 + x a4 4 x 1 = 0 in weighted projective spaces P(w 1 , w 2 , w 3 , w 4 ) provide such examples under some mild conditions on the a i 's. These surfaces were studied in [OR77, Kou76,Kol08,HK12,UYn16]. The last three papers also study surfaces S * (a 1 , a 2 , a 3 , a 4 ) obtained by contracting two curves on such surfaces, as in the last section of [Kol08].…”

On the equality of Dedekind sums

“…Dedekind sums first appeared in the theory of modular forms; see [1]. But these sums have also interesting applications in a number of other fields, so in connection with class numbers, lattice point problems, topology, and algebraic geometry (see, for instance, [2,3,8,10,12,13]).…”