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On the Log Discrepancies in Toric Mori Contractions

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 Added by Valery Alexeev
 Publication date 2012
  fields
and research's language is English




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It was conjectured by McKernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive epsilon there is a positive delta, such that if X is epsilon-log terminal, then Y is delta-log terminal. We prove this conjecture in the toric case and discuss the dependence of delta on epsilon, which seems mysterious.



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302 - Shihoko Ishii 2019
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.
98 - Jingjun Han , Yujie Luo 2020
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 $E$.
93 - Jingjun Han , Jihao Liu , 2019
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67 - Sebastian Bozlee 2019
Let $C$ be a nodal curve, and let $E$ be a union of semistable subcurves of $C$. We consider the problem of contracting the connected components of $E$ to singularities in a way that preserves the genus of $C$ and makes sense in families, so that this contraction may induce maps between moduli spaces of curves. In order to do this, we introduce the notion of mesa curve, a nodal curve $C$ with a logarithmic structure and a piecewise linear function $overline{lambda}$ on the tropicalization of $C$. This piecewise linear function determines a subcurve $E$. We then construct a contraction of $E$ inside of $C$ for families of mesa curves. Resulting singularities include the elliptic Gorenstein singularities.
186 - Shihoko Ishii 2017
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