Characterization of Kollár surfaces
| Autor: | José Ignacio Yáñez, Giancarlo Urzúa |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Algebra and Number Theory
Mathematics - Number Theory 14J10 010102 general mathematics Dedekind sum branched covers Type (model theory) Characterization (mathematics) Surface (topology) 01 natural sciences K3 surface Combinatorics Minimal model symbols.namesake Mathematics - Algebraic Geometry 0103 physical sciences symbols 010307 mathematical physics Projective plane 0101 mathematics $\mathbb Q$-homology projective planes Dedekind sums Mathematics |
| Zdroj: | Algebra Number Theory 12, no. 5 (2018), 1073-1105 |
| ISSN: | 1073-1105 |
| Popis: | 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 [math] , [math] , and [math] . The aim was to give many interesting examples of [math] -homology projective planes. They occur when [math] . For that case, we prove that Kollár surfaces are Hwang–Keum (2012) surfaces. For [math] , we construct a geometrically explicit birational map between Kollár surfaces and cyclic covers [math] , where [math] are four general lines in [math] . In addition, by using various properties on classical Dedekind sums, we prove that: ¶ 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 . 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. 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 . |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|