AI 中文总结
本文研究给定基本群的一般型复射影曲面的陈斜率，证明其在[1/2,3]稠密，在[1/2,2]稠密的情况首次验证了相关猜想，所用工具为乘积构造的丰富性判据。
AI 中文摘要
设G为非奇异复射影曲面的拓扑基本群。Troncoso与Urzúa证明，满足基本群π₁(S)≃G的一般型极小曲面S的陈斜率c₁²/c₂在区间[1,3]上稠密，且区间[1/3,1)的情况留待研究。本文证明该斜率在[1/2,3]上稠密，此区间若扩大则会与Mendes Lopes和Pardini的定理或Reid猜想矛盾；还证明这类曲面中满足K_S丰富的斜率在[1/2,2]上稠密，首次验证了Troncoso与Urzúa关于丰富典范类的猜想。所用工具为针对其乘积构造的精确丰富性判据，该判据特别表明，无论定义截面如何，他们构造的曲面永远不会具有丰富典范类。
英文摘要
Let $G$ be the topological fundamental group of a nonsingular complex projective surface. Troncoso and Urzúa proved that the Chern slopes $c_1^2/c_2$ of minimal surfaces of general type $S$ with $π_1(S)\simeq G$ are dense in $[1,3]$, and left $[1/3,1)$ open. We prove that they are dense in $[1/2,3]$, an interval that cannot be enlarged without contradicting either a theorem of Mendes Lopes and Pardini or Reid's conjecture. We prove more: the slopes of such surfaces with $K_S$ ample are dense in $[1/2,2]$, the first case in which the conjecture of Troncoso and Urzúa on ample canonical classes is established. The tool is an exact ampleness criterion for their product construction, which shows in particular that their own surfaces never have ample canonical class, whatever the defining sections.
