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Register a new userOn boundedness of divisors computing minimal log discrepancies for surfaces
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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 $E$.
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We prove the existence of $n$-complements for pairs with DCC coefficients and the ACC for minimal log discrepancies of exceptional singularities. In order to prove these results, we develop the theory of complements for real coefficients. We introduce $(n,Gamma_0)$-decomposable $mathbb{R}$-complements, and show its existence for pairs with DCC coefficients.
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.
This paper shows that Mustata-Nakamuras conjecture holds for pairs consisting of a smooth surface and a multiideal with a real exponent over the base field of positive characteristic. As corollaries, we obtain the ascending chain condition of the minimal log discrepancies and of the log canonical thresholds for those pairs. We also obtain finiteness of the set of the minimal log discrepancies of those pairs for a fixed real exponent.
In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy. We also show that this conjecture yields some basic properties of singularities; e.g. openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-Jacobian minimal log discrepancies, which are already known in characteristic 0 and open for positive characteristic case.We show some evidences of the conjecture: for example, for non-degenerate hypersurfaces of any dimension in arbitrary characteristic and 2-dimensional singularities in characteristic not 2. We aslo give a bound of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy.
In this short note, we extend the results of [Alexeev-Orlov, 2012] about Picard groups of Burniat surfaces with $K^2=6$ to the cases of $2le K^2le 5$. We also compute the semigroup of effective divisors on Burniat surfaces with $K^2=6$. Finally, we construct an exceptional collection on a nonnormal semistable degeneration of a 1-parameter family of Burniat surfaces with $K^2=6$.
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