The moduli b-divisor of an lc-trivial fibration

Okounkov bodies associated to abundant divisors and Iitaka fibrations

The aim of this paper is to study the Okounkov bodies associated to abundant divisors. As a main result, we prove that the valuative Okounkov bodies of an abundant divisor encode all the numerical properties. We apply this result to recover the asymptotic base loci of an abundant divisor from the valuative Okounkov bodies. We also give a criterion of when the valuative and limiting Okounkov bodies of an abundant divisor coincide by comparing their Euclidean volumes. To obtain these results, we prove some variants of Fujita's approximations for Okounkov bodies using Iitaka fibrations.

Bounding non-rationality of divisors on 3-fold Fano fibrations

In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations $(X,B)\to Z$ under mild conditions. We show that if $D$ is a component of $B$ with coefficient $\ge t>0$ which is contracted to a point on $Z$, then $D$ is birational to $\mathbb{P}^1\times C$ where $C$ is a smooth projective curve with gonality bounded depending only on $t$. Moreover, if $t>\frac{1}{2}$, then genus of $C$ is bounded depending only on $t$.

Rationality of moduli of elliptic fibrations with fixed monodromy

We prove rationality results for moduli spaces of elliptic K3 surfaces and elliptic rational surfaces with fixed monodromy groups.

On Deformations and Moduli of Genus 2 Fibrations

In this paper, we investigate the moduli of surfaces of general type admitting genus 2 fibrations with irregularity q = g_b + 1, where g_b >= 2 is the genus of the base. We prove that smooth fibrations are parametrized by a unique component in the moduli space. The same result applies to nonsmooth fibrations with special values of g_b. In the general case, we give a bound on the dimension of the corresponding connected components.

具有 nef 反典范除子的射影簇的自同构群与齐次纤维化

Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors

本文研究具有 nef 反典范除子的代数簇的 Albanese 态射与极大有理链连通纤维化的齐次结构，构造自然群同态诱导 Aut^0 的 Chevalley 分解（相差等源），并类比 Nishi-Matsumura 定理，用现代语言重证之。

On The Moduli of Surfaces Admitting Genus Two Fibrations Over Elliptic Curves

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes ${\cal H}(1,X(d),n)$ of morphisms of degree $n$ from elliptic curves to the modular curve $X(d)$, $d\geq 3$. Ultimately, we show that the moduli spaces in…

关于抛物纤维化的奇异纤维

On singular fibers of parabolic fibrations

本文描述了模除子数值平凡且判别除子为零的抛物纤维化的奇异纤维结构。

曲线上对数卡拉比-丘纤维化的射影模空间

Projective moduli of log Calabi--Yau fibrations over curves

研究曲线上对数卡拉比 - 丘纤维化，引入新稳定性条件，证明该条件能产生有射影粗模空间的恰当层，其边界可参数化曲线上的平坦纤维化。

Compact moduli for certain Kodaira fibrations

We describe explicitly the possible degenerations of a class of double Kodaira fibrations in the moduli space of stable surfaces. Using deformation theory we also show that under some assumptions we get a connected component of the moduli space of stable surfaces.

曲线中的极小次数纤维化与除子的非有理度渐近

Minimal degree fibrations in curves and the asymptotic degree of irrationality of divisors

本文研究复射影簇中高次数超曲面的非有理度，证明相关映射经由有理纤维化分解，并引入极小纤维化次数给出紧界，推出不平衡高次完全交的非有理度渐近为次数乘积。

Hitchin fibrations on moduli of irregular Higgs bundles and motivic wall-crossing

In this paper we give a complete description of the Hitchin fibration on all 2-dimensional moduli spaces of rank 2 irregular Higgs bundles with two poles on the projective line. We describe the dependence of the singular fibers of the fibration on the eigenvalues of the Higgs fields, and describe the corresponding motivic wall-crossing phenomenon in the parameter space of parabolic weights.

Hitchin Fibrations on Two-Dimensional Moduli Spaces of Irregular Higgs Bundles with One Singular Fiber

We analyze and completely describe the four cases when the Hitchin fibration on a $2$-dimensional moduli space of irregular Higgs bundles over $\mathbb{C}P^{1}$ has a single singular fiber. The case when the fiber at infinity is of type $I_0^*$ is further analyzed, and we give constructions of all the possible configurations of singular curves inelliptic fibrations having this type of singular fiber at infinity.

Moduli of Weierstrass fibrations with marked section

We study the the moduli space of KSBA stable pairs $(X,sS+\sum a_i F_i)$, consisting of a Weierstrass fibration $X$, its section $S$, and some fibers $F_i$. We find a compactification which is a DM stack, and we describe the objects on the boundary. We show that the fibration in the definition of Weierstrass fibration extends to the boundary, and it is equidimensional when $s \ll 1$. We prove that there are wall-crossing morphisms when the weights $s$ and $a_i$ change. When $s=1$, this recovers the work of La Nave (arXiv:math/0205035); and a special case of the work of Ascher-Bejleri (arXiv:17…

Lagrangian fibration in duality on moduli space of rank two logarithmic connections over the projective line

We study the moduli space of logarithmic connections of rank 2 on the Riemann sphere minus n points with fixed spectral data. There are two natural Lagrangian maps: one towards apparent singularities of the associated fuchsian scalar equation, and another one towards moduli of parabolic bundles. We show that these are transversal and dual to each other. In case n=5, we recover the beautiful geometry of Del Pezzo surfaces of degree 4.

具有数值平凡canonical除数的不规则三维流形

Irregular threefolds with numerically trivial canonical divisor

本研究对正特征下具有数值平凡canonical除数的不规则三维流形进行分类，揭示其双纤维结构及显式描述。

关于正特征中反-canonical 分割子在Iitaka猜想中的研究

On the Iitaka conjecture for anticanonical divisors in positive characteristic

本文研究正特征下纤维化中的Iitaka型不等式，证明在三重或曲线目标时，若一般纤维正则且诱导对强F-正则，则成立。同时给出特征2和3下非正常纤维的反例。

Extremal effective divisors on the moduli space of n-pointed genus one curves

For every $n\geq 3$, we exhibit infinitely many extremal effective divisors on the moduli space of genus one curves with $n$ marked points.

Relations among divisors on the moduli space of curves with marked points

We prove relations among the classes of certain divisors on the moduli spaces of curves with marked points, generalizing the Brill-Noether Ray Theorem of Eisenbud and Harris.

Descendent bounds for effective divisors on the moduli space of curves

The slope of the moduli space of genus g curves is bounded from below by 60/(g+4) via a descendent calculation.

The moduli space of twisted canonical divisors

The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in the moduli space of Deligne-Mumford stable pointed curves which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors. In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus g curves are o…

Enumerative aspect of symplectic log Calabi-Yau divisors and almost toric fibrations

In this paper we are interested in the isotopy classes of symplectic log Calabi-Yau divisors in a fixed symplectic rational surface. We give several equivalent definitions and prove the stability, finiteness and rigidity results. Motivated by the problem of counting toric actions, we obtain a general counting formula of symplectic log Calabi-Yau divisors in a restrictive region of $c_1$-nef cone. A detailed count in the case of 2- and 3-point blow-ups of complex projective space for all symplectic forms is also given. In our framework the complexity of the combinatorics of analyzing Delzant po…

Birational models of ${\mathcal M}_{2,2}$ arising as moduli of curves with nonspecial divisors

We study birational projective models of ${\mathcal M}_{2,2}$ obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of ${\mathcal Z}$-stable curves $\overline{\mathcal M}_{2,2}({\mathcal Z})$ defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space $\overline{M}_{2,2}({\mathcal Z})$.

Koszul divisors on moduli spaces of curves

Given a moduli problem with a nice coarse moduli space, which is the most intrinsic and natural divisor on this moduli space? We describe a general method of constructing effective divisors on a large class of moduli spaces using the syzygies of the parametrized objects. In this paper we treat the case of the moduli space of curves and we compute the slope of a doubly infinite sequence of Koszul divisors on M_g. We present a single formula which apart from giving myriads of new counterexamples to the Harris-Morrison Slope Conjecture it also encodes virtually all interesting divisor class calcu…

Log Iitaka conjecture for abundant log canonical fibrations

We prove that the log Iitaka conjecture holds for log canonical fibrations when log canonical divisor of a sufficiently general fiber is abundant.

One remark on the b-semiampleness of the moduli part

In this short note we reduce the b-semiampleness conjecture for lc-trivial fibrations to the b-semiampleness conjecture for klt-trivial fibrations.

Divisors on the moduli space of curves from divisorial conditions on hypersurfaces

In this note, we extend work of Farkas and Rimányi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of possible values of $g\in \{1,\ldots,N\}$ has density $Ω(\log(N)…

Fundamental groups of moduli stacks of smooth Weierstrass fibrations

We give finite presentations for the fundamental group of moduli stacks of smooth Weierstrass curves over complex projective space P^n which extend the classical result for elliptic curves to positive dimensional base. We thus get natural generalisations of SL_2(Z) and pave the way to understanding the fundamental group of moduli stacks of elliptic surfaces in general. Our approach exploits the natural involution on Weierstrass curves and the identification of its fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over P^n. The fundamental group of…

Stable pair compactifications of the moduli space of degree one del pezzo surfaces via elliptic fibrations

A degree one del Pezzo surface is the blowup of P^2 at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces.

The Moduli Space of Abelian Varieties and the Singularities of the Theta Divisor

In this paper we give a lower bound for the codimension of the Andreotti-Mayer loci in the moduli space of principally polarized complex abelian varieties. We also present a conjecture on this codimension.

Variety of power sums and divisors in the moduli space of cubic fourfolds

We show that a cubic fourfold F that is apolar to a Veronese surface has the property that its variety of power sums VSP(F,10) is singular along a K3 surface of genus 20. We prove that these cubics form a divisor in the moduli space of cubic fourfolds and that this divisor is not a Noether-Lefschetz divisor. We use this result to prove that there is no nontrivial Hodge correspondence between a very general cubic and its VSP.

Weierstrass纤维模堆的算术几何

Arithmetic geometry of the moduli stack of Weierstrass fibrations over $\mathbb{P}^1$

本文研究了无参数化射影线上Weierstrass纤维的模堆，构造了其显式形式，并确定了其在Grothendieck环中的动机。

Heegner divisors in the moduli space of genus three curves

S. Kondō used periods of $K3$ surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by: hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational "splitting curves".

Weierstrass 纤维化模空间的整系数 Chow 环

The integral Chow rings of moduli of Weierstrass fibrations

本文计算了射影直线上极小 Weierstrass 纤维化模空间的整系数 Chow 环，利用商栈表示和等变交叠理论确定了各 Chow 环的完整生成元与关系，并对 $N=1$ 和 $N=2$ 的情况给出了更明确的计算。

Extremal divisors on moduli spaces of rational curves with marked points

We study effective divisors on $\overline{M}_{0,n}$, focusing on hypertree divisors introduced by Castravet and Tevelev and the proper transforms of divisors on $\overline{M}_{1,n-2}$ introduced by Chen and Coskun. Results include a database of hypertree divisor classes and closed formulas for Chen--Coskun divisor classes. We relate these two types of divisors, and from this construct extremal divisors on $\overline{M}_{0,n}$ for $n \geq 7$ that furnish counterexamples to the conjectural description of the effective cone of $\overline{M}_{0,n}$ given by Castravet and Tevelev.

Divisors in the moduli spaces of curves

In this mostly expository paper we review several known results about the cohomology of moduli spaces of smooth and stable curves, focusing in particular on low degree cohomology. We also give a new proof of Harer's theorem describing the second cohomology group of the moduli space of smooth n-pointed curves of given genus

The wobbly divisors of the moduli space of rank-$2$ vector bundles

Let $X$ be a smooth projective complex curve of genus $g \geq 2$ and let $\M_X(2,Λ)$ be the moduli space of semi-stable rank-$2$ vector bundles over $X$ with fixed determinant $Λ$. We show that the wobbly locus, i.e., the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field is a union of divisors $\Ww_k \subset \M_X(2,Λ)$. We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors $\Ww_k$ in the Picard group of $\M_X(2,Λ)$.

Effective divisors on moduli spaces of curves and abelian varieties

This paper is an expository survey of results about the effective divisors on moduli spaces, with a focus on what is known about the effective cones of moduli spaces of stable curves and of principally polarized abelian varieties. This version incorporates clarifications and suggestions of Ana-Maria Castravet and the referee, both of whom we thank. To appear in the proceedings of the conference, "Joe at 60:A Celebration of Algebraic Geometry" edited by Brendan Hassett, James McKernan, Jason Starr and Ravi Vakil.

The class of a Hurwitz divisor on the moduli of curves of even genus

We calculate the cycle class of the Hurwitz divisor $D_2$ on the moduli space of stable curves of genus $g=2k$ given by the degree $k+1$ covers of the projective line with simple ramification points, two of which lie in the same fibre. We also study some aspects of the geometry of the natural map of the Hurwitz space of admissible covers of degree $k+1$ and genus $2k$ to the moduli space of stable curves of genus $2k$.

Stability and Fourier-Mukai Transforms on Higher Dimensional Elliptic Fibrations

We consider elliptic fibrations with arbitrary base dimensions, and generalise previous work by the second author. In particular, we check universal closedness for the moduli of semistable objects with respect to a polynomial stability that reduces to PT-stability on threefolds. We also show openness of this polynomial stability. On the other hand, we write down criteria under which certain 2-term polynomial semistable complexes are mapped to torsion-free semistable sheaves under a Fourier-Mukai transform. As an application, we construct an open immersion from a moduli of complexes to a moduli…

有限生成的诺特-利夫谢茨除子与立方四重面模空间的斜率

Finite generation of Noether-Lefschetz divisors and the slope of the moduli space of cubic fourfolds

本文研究了立方四重面模空间和K3表面的除子结构，通过引入斜率量和显式界限，揭示了有效锥的性质，并给出了K3模空间有理Picard群的生成方式。

On the restriction of the moduli part to a reduced divisor

Let $f \colon (X,Δ) \to Y$ be a fibration such that $K_X + Δ$ is torsion along the fibres of $f$. Assume that $Y$ has dimension 2, or that $Y$ has dimension 3 and the fibres have dimension at most 3. Then the restriction of the moduli part to its augmented base locus is semiample.

Ample divisors on moduli spaces of weighted pointed rational curves, with applications to log MMP for $\bar{M}_{0,n}$

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 $\bar{M}_{0,n}$.

非Fano型纤维上的Stein度

Stein degree on non-Fano type fibrations

研究非Fano型log Calabi-Yau纤维上垂直除子的Stein度无界性。

通过10节点平面六次曲线研究立方四维流形的非特殊除子

A nonspecial divisor in the moduli space of cubic fourfolds via 10-nodal plane sextics

本文通过研究10节点平面六次曲线，发现立方四维流形模空间中存在一个非诺特-勒夫施茨类型的不可约除子。

Homogeneous fibrations on log Calabi-Yau varieties

We prove a structure theorem for the Albanese maps of varieties with Q-linearly trivial log canonical divisors. Our start point is the action of a nonlinear algebraic group on a projective variety.

Moduli of curves with nonspecial divisors and relative moduli of $A_\infty$-structures

In this paper for each $n\ge g\ge 0$ we consider the moduli stack $\widetilde{\mathcal U}^{ns}_{g,n}$ of curves $(C,p_1,\ldots,p_n,v_1,\ldots,v_n)$ of arithmetic genus $g$ with $n$ smooth marked points $p_i$ and nonzero tangent vectors $v_i$ at them, such that the divisor $p_1+\ldots+p_n$ is nonspecial (has no $h^1$) and ample. With some mild restrictions on the characteristic we show that it is a scheme, affine over the Grassmannian $G(n-g,n)$. We also construct an isomorphism of $\widetilde{\mathcal U}^{ns}_{g,n}$ with a certain relative moduli of $A_\infty$-structures (up to an equivalence)…

Moduli of cubic surfaces and their anticanonical divisors

We consider the moduli space of log smooth pairs formed by a cubic surface and an anticanonical divisor. We describe all compactifications of this moduli space which are constructed using Geometric Invariant Theory and the anticanonical polarization. The construction depends on a weight on the divisor. For smaller weights the stable pairs consist of mildly singular surfaces and very singular divisors. Conversely, a larger weight allows more singular surfaces, but it restricts the singularities on the divisor. The one-dimensional space of stability conditions decomposes in a wall-chamber struct…

Divisors on the moduli spaces of stable maps to flag varieties and reconstruction

We determine generators for the codimension 1 Chow group of the moduli spaces of genus zero stable maps to flag varieties G/P. In the case of SL flags, we find all relations between our generators, showing that they essentially come from $\bar M_{0,n}$. In addition, we analyze the codimension 2 classes on the moduli spaces of stable maps to Grassmannians and prove a new codimension 2 relation. This will lead to a partial reconstruction theorem for the Grassmannian of 2 planes.

The tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the jacobian

We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on $Pic^{g-1}C$ which are linearly equivalent to $2Θ$. The embedded tangent space at a semi-stable non-stable bundle $ξ\oplusξ^{-1}$, where $ξ$ is a degree zero line bundle, is shown to consist of those divisors in $|2Θ|$ which contain $Sing(Θ_ξ)$ where $Θ_ξ$ is the translate of $Θ$ by $ξ$. We also obtain geometrical results on the structure of this tangent space.

Log canonical models for the moduli space of curves: First divisorial contraction

In this paper, we initiate our investigation of log canonical models for the moduli space of curves with the boundary divisor $\a \d$ as we decrease $\a$ from 1 to 0. We prove that for the first critical value $\a = 9/11$, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that $\a = 7/10$ is the next critical value, i.e., the log canonical model stays the same in the interval $(7/10, 9/11]$. In the appendix, we develop a theory of log canonical models of stacks that explains how these can be expressed in …

On some Fano manifolds admitting a rational fibration

Let X be a smooth, complex Fano variety. For every prime divisor D in X, we set c(D):=dim ker(r:H^2(X,R)->H^2(D,R)), where r is the natural restriction map, and we define an invariant of X as c_X:=max{c(D)|D is a prime divisor in X}. In a previous paper we showed that c_X<9, and that if c_X>2, then either X is a product, or X has a flat fibration in Del Pezzo surfaces. In this paper we study the case c_X=2. We show that up to a birational modification given by a sequence of flips, X has a conic bundle structure, or an equidimensional fibration in Del Pezzo surfaces. We also show a weaker prope…

Families of divisors

We establish a new moduli theory for divisors, that interpolates between the Hilbert scheme and the Cayley-Chow variety. This completes the last step in the construction of a good moduli theory for stable pairs $(X,Δ)$.

A boundary divisor in the moduli space of stable quintic surfaces

We give a bound on which singularities may appear on Kollár--Shepherd-Barron--Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with $K^2=χ=5$) whose unique non Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space $\overline{\mathcal{M}}_{5,5}$ corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of de…

Rational maps between moduli spaces of curves and Gieseker-Petri divisors

We perform an intersection theoretic study of the rational map between two different moduli spaces of stable curves which associates to a curve its corresponding Brill-Noether locus (in the case this locus has virtual dimension 1). We then use these results to describe the cone of moving divisors on M_g. Several other applications to moduli spaces of Prym varieties are presented. In a different direction, we prove that the locus in M_g of curves failing to satisfy the Gieseker-Petri theorem is supported in codimension 1 for every possible type of linear series.