Maksym Fedorchuk, David Ishii Smyth
We introduce a new technique for proving positivity of certain divisor classes on $\bar{M}_{0,n}$ and its weighted variants. Our methods give an unconditional description of the spaces of symmetric weighted pointed rational curves as log canonical models of $\bar{M}_{0,n}$.
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We study relations between the property of being log abundant for lc pairs and the termination of log MMP with scaling. We prove that any log MMP with scaling of an ample divisor starting with a projective dlt pair contains only finitely many log abundant dlt pairs. We also discuss log MMP for slc pairs.
We show that minimal models of $\mathbb{Q}$-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a $\mathbb{Q}$-factorial NQC log canonical generalised pair $ (X,B+M) $ we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that $K_X+B+M$ is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.
We study the log canonical models of the moduli space MBar_{0,n} of pointed stable genus zero curves with respect to the standard log canonical divisors K+aD, where D denotes the boundary. In particular we show that, as a formal consequence of a conjecture by Fulton regarding the ample cone of MBar_{0,n}, these log canonical models are equal to certain of Hassett's weighted pointed curve spaces.
Remarks on the existence of minimal models of log canonical generalized pairs
本文研究了NQC log canonical广义对的极小模型存在性，证明了在较弱意义下存在极小模型或非pseudo-effective时带缩放的MMP可终止，并在边界含ample divisor等附加条件下证明了pseudo-effective广义对的极小模型存在性。
By using classical invariant theory, we reduce the $S_{n}$-invariant F-conjecture to a feasibility problem in polyhedral geometry. We show by computer that for $n \le 19$, every integral $S_{n}$-invariant F-nef divisor on the moduli space of genus zero stable pointed curves is semi-ample, over arbitrary characteristic. Furthermore, for $n \le 16$, we show that for every integral $S_{n}$-invariant nef (resp. ample) divisor $D$ on the moduli space, $2D$ is base-point-free (resp. very ample). As applications, we obtain the nef cone of the moduli space of stable curves without marked points, and t…
Let X_n := \bar M_{0,n}, the moduli space of n-pointed stable genus zero curves, and let X_{n,m} be the quotient of X_n by the action of the symmetric group S_{n-m} on the last n-m marked points. The cones of effective divisors of X_{n,m}, m = 0,1,2, are calculated. Using this, upper bounds for the cones Mov(X_{n,m}) generated by divisors with moving linear systems are calculated, m = 0,1, along with the induced bounds on the cones of ample divisors of \bar M_g and \bar M_{g,1}. As an application, the cone of effective divisors of \bar M_{2,1} is analyzed in detail.
Using Kollár's semipositivity results, we produce a number of nef and ample tautological divisors on Hassett's spaces of weighted stable pointed curves. As an application, we prove that Hassett's spaces are log canonical models of Deligne-Mumford spaces of stable pointed curves.
The moduli space $\M_{g,n}$ of $n-$pointed stable curves of genus $g$ is stratified by the topological type of the curves being parametrized: the closure of the locus of curves with $k$ nodes has codimension $k$. The one dimensional components of this stratification are smooth rational curves (whose numerical equivalence classes are) called F-curves. The F-conjecture asserts that a divisor on $\M_{g,n}$ is ample if and only if it positively intersects the $F-$curves. In this paper the F-conjecture on $\M_{g,n}$ is reduced to showing that certain divisors in $\M_{0,N}$ for $N \leq g+n$ are equi…
Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture
本文研究了极小族的有限性问题，证明了在固定点曲线和极小极种变种下，局部平凡族的同构类有限，并在假设条件成立时进一步证明了投影族的有限性。
Pointed Evaluation Fibers of Rational Curves on del Pezzo Manifolds
本文研究 del Pezzo 流形上 Kontsevich 空间的带点赋值映射，证明了单点赋值映射的一般纤维几何不可约，并在超丰情形下证明了两点赋值映射的一般纤维几何不可约。
Let A be an integral Banach ring, and X/A be a projective scheme of finite type, endowed with a semi-ample line bundle L. We define a class PSH(X,L) of plurisubharmonic metrics on L on the Berkovich analytification X^an and prove various basic properties thereof. We focus in particular on the case where A is a hybrid ring of complex power series and X/A is a smooth variety, so that X^an is the hybrid space associated to a degeneration X of complex varieties over the punctured disk. We then prove that when L is ample, any plurisubharmonic metric on L with logarithmic growth at zero admits a can…
In this paper we study the ample cone of the moduli space $\mgn$ of stable $n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on $\mgn$ is ample iff it has positive intersection with all 1-dimensional strata (the components of the locus of curves with at least $3g+n-2$ nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for $g=0$. More pr…
We introduce a smooth projective variety $T_{d,n}$ which compactifies the space of configurations of $n$ distinct points on affine $d$-space modulo translation and homothety. The points in the boundary correspond to $n$-pointed stable rooted trees of $d$-dimensional projective spaces, which for $d = 1$, are $(n+1)$-pointed stable rational curves. In particular, $T_{1,n}$ is isomorphic to $\bar{M}_{0,n+1}$, the moduli space of such curves. The variety $T_{d,n}$ shares many properties with $\bar{M}_{0,n}$. For example, as we prove, the boundary is a smooth normal crossings divisor whose componen…
Let X be a smooth projective variety over the complex numbers, and let D be an ample divisor in X. For which spaces Y is the restriction map r: Hom(X, Y) -> Hom(D, Y) an isomorphism? Using positive characteristic methods, we give a fairly exhaustive answer to this question. An example application of our techniques is: if dim(X) > 2, Y is smooth, the cotangent bundle of Y is nef, and dim(Y) < dim(D), the restriction map r is an isomorphism. Taking Y to be the classifying space of a finite group BG, the moduli space of pointed curves M_{g,n}, the moduli space of principally polarized Abelian var…
Complete Subvarieties of $\rm{M}_{g,n}$ and a Lifting Problem
本文通过引入并证明提升猜想，将完全子簇的存在性转化为模空间线丛性质，从而在正特征下构造出 $\rm{M}_g$ 中维数为 $d$ 的完全子簇及 $\rm{M}_{g,n}$ 中的完全曲面，揭示了特征 $0$ 与特征 $p$ 的几何差异。
Let $(X,ω)$ be a closed symplectic manifold. A loop $ϕ: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $π: P_ϕ \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of $π$, Lalonde, McDuff and Polterovich proved that if $ϕ$ lifts to the Hamiltonian group $\mathrm{Ham}(X,ω)$, then the rational cohomology of $P_ϕ$ splits additively. We prove, with the same assumptions, that the $\mathbb{E}$-generalised cohomology of $P_ϕ$ splits additively for any complex-oriented cohomology theory $\mathbb{E}$, in particular the integral cohomology splits. …
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