Characterization of Kollár surfaces

Abstract

Kollár introduced in [Ko08] the surfaces $$(x_1^{a_1}{x}_2+x_2^{a_2}{x}_3+x_3^{a_3}{x}_4+x_4^{a_4}{x}_1=0)\subset \mathbb{P}(w_1,w_2,w_3,w_4)$$ where $w_i=W_i/w^{\ast }$, $W_i=a_{i+1}{a}_{i+2}{a}_{i+3}-a_{i+2}{a}_{i+3}+a_{i+3}-1$, and $w^{\ast }=$gcd$(W_1,\dots ,W_4)$. The aim was to give many interesting examples of $\mathbb{Q}$-homology projective planes. They occur when $w^{\ast }=1$. For that case, we prove that Kollár surfaces are Hwang-Keum [HK12] surfaces. For $w^{\ast }>1$, we construct a geometrically explicit birational map between Kollár surfaces and cyclic covers $z^{w^{\ast }}=l_1^{a_2{a}_3{a}_4}{l}_2^{-a_3{a}_4}{l}_3^{a_4}{l}_4^{-1}$, where $\{l_1,l_2,l_3,l_4\}$ are four general lines in ${\mathbb{P}}^2$. In addition, by using various properties on classical Dedekind sums, we prove that: (a) For any $w^{\ast }>1$, we have $p_g=0$ iff the Kollár surface is rational. This happens when $a_{i+1}\equiv 1$ or $a_i{a}_{i+1}\equiv -1($mod $w^{\ast })$ for some $i$. (b) For any $w^{\ast }>1$, we have $p_g=1$ iff the Kollár surface is birational to a K3 surface. We classify this situation. (c) For $w^{\ast }>>0$, we have that the smooth minimal model $S$ of a generic Kollár surface is of general type with $K_S^2/e(S)\to 1$.

References in corpus (1)

• Chern slopes of surfaces of general type in positive characteristic

Cited by in corpus (2)

• Open surfaces of small volume
• On the equality of Dedekind sums