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Register a new userOn minimal model theory for algebraic log surfaces
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Authors
Osamu Fujino
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We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.
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We establish the minimal model theory for $mathbb Q$-factorial log surfaces and log canonical surfaces in Fujikis class $mathcal C$.
Let $Gamma$ be a finite set, and $X i x$ a fixed klt germ. For any lc germ $(X i x,B:=sum_{i} b_iB_i)$ such that $b_iin Gamma$, Nakamuras conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor $E$ over $X i x$, such that $a(E,X,B)={rm{mld}}(X i x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamuras conjecture to the setting that $X i x$ is not necessarily fixed and $Gamma$ satisfies the DCC, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such $E$.
Let (S, BS) be the log-pair associated with a compactification of a given smooth quasi-projective surface V . Under the assumption that the boundary BS is irreducible, we propose an algorithm, in the spirit of the (log) Sarkisov program, to factorize any automorphism of V into a sequence of elementary links in the framework of the logarithmic Mori theory. The new noteworthy feature of our algorithm is that all the blow-ups and contractions involved in the process occur on the boundary.
In a previous work, we described the Minimal Model Program in the family of $Qbb$-Gorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical varieties. Here, we summarize the results of the previous work and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs $(X,D)$ when $X$ is a projective horospherical variety.
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.
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