The minimal volume of log surfaces of general type with positive geometric genus
The minimal volume of log surfaces of general type with positive geometric genus
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发表时间：
2017-06
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通讯作者：
Wenfei Liu
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作者：
Wenfei Liu
设X$是光滑的射影曲面，$\Delta$是X$上的法交曲线，使得$K_X+\Delta$大。我们证明了对$（X，\Delta）$的最小可能体积是$\frac{1}{143}$，如果它的（log）几何亏格是正的。在此基础上，我们建立了一个关于稳定对数曲面的Noether型不等式。在另一个方向上，我们证明了，如果$（X，\Delta）$的体积小于$\frac{1}{143}$，则$X$必须是有理曲面，并且$\Delta$的连通分支是光滑有理曲线的树。
Let $X$ be a smooth projective surface and $\Delta$ is a normal crossing curve on $X$ such that $K_X+\Delta$ is big. We show that the minimal possible volume of the pair $(X, \Delta)$ is $\frac{1}{143}$ if its (log) geometric genus is positive. Based on this, we establish a Noether type inequality for stable log surfaces, be they normal or non-normal. In the other direction, we show that, if the volume of $(X, \Delta)$ is less than $\frac{1}{143}$ then $X$ must be a rational surface and the connected components of $\Delta$ are trees of smooth rational curves.