No Arabic abstract
Let $mathcal Csubset(0,1]$ be a set satisfying the descending chain condition. We show that any accumulation point of volumes of log canonical surfaces $(X, B)$ with coefficients in $mathcal C$ can be realized as the volume of a log canonical surface with big and nef $K_X+B$ and coefficients in $overline{mathcal C}cup{1}$, with at least one coefficient in $Acc(mathcal C)cup{1}$. As a corollary, if $overline{mathcal C}subsetmathbb Q$ then all accumulation points of volumes are rational numbers, solving a conjecture of Blache. For the set of standard coefficients $mathcal C_2={1-frac{1}{n}mid ninmathbb N}cup{1}$ we prove that the minimal accumulation point is between $frac1{7^2cdot 42^2}$ and $frac1{42^2}$.
We characterize a $k$-th accumulation point of pseudo-effective thresholds of $n$-dimensional varieties as certain invariant associates to a numerically trivial pair of an $(n-k)$-dimensional variety. This characterization is applied towards Fujitas log spectrum conjecture for large $k$.
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.
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 $E$.
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.
In this work we show that the Weil-Petersson volume (which coincides with the CM degree) in the case of weighted points in the projective line is continuous when approaching the Calabi-Yau geometry from the Fano geometry. More specifically, the CM volume computed via localization converges to the geometric volume, computed by McMullen with different techniques, when the sum of the weights approaches the Calabi-Yau geometry.