Han-Bom Moon, David Swinarski
We study new effective curve classes on the moduli space of stable pointed rational curves given by the fixed loci of subgroups of the permutation group action. We compute their numerical classes and provide a strategy for writing them as effective linear combinations of F-curves, using Losev-Manin spaces and toric degeneration of curve classes.
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In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac}(X)) \otimes Q$ contains the totally real cubic number field $Q(ζ_ 7 + \overlineζ_7)$. We construct explicit two-dimensional families defined over $Q(s; t)$ whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when X is hyperelliptic was studied by the authors Hoffman and Wang in a previous work. We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.
In this work, we introduce the moduli stack $\widetilde{\mathcal{M}}_{g,n}^r$ of $n$-pointed, $A_r$-stable curves of genus $g$ and use it to compute the Chow ring of $\overline{\mathcal{M}}_3$. As a byproduct, we also compute the Chow ring of $\widetilde{\mathcal{M}}_3^7$. All the Chow rings are assumed to be with coefficients in $\mathbb{Z}[1/6]$.
We show that the pseudoeffective cone of divisors $\overline{\text{Eff}}^1(\overline{\mathcal{M}}_{g,n})$ for $g\geq 2$ and $n\geq 2$ is not polyhedral by showing that the class of the fibre of the morphism forgetting one point forms an extremal ray of the dual nef cone of curves $\overline{\text{Nef}}_1(\overline{\mathcal{M}}_{g,n})$ and the cone at this ray is not polyhedral.
We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space $\overline{\mathcal M}_{g,n}$ of stable genus $g$ curves with $n$ ordered marked points. In particular, we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in $\overline{\mathcal M}_{3,1}$ and the locus of hyperelliptic curves in $\overline{\mathcal M}_4$ are extremal cycles. In addition, we exhibit infinitely many extremal…
We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $\overline{\mathcal{M}}_{g,n}$ is not pseudo-effective in some range, implying that $\overline{\mathcal{M}}_{12,6},\overline{\mathcal{M}}_{12,7},\overline{\mathcal{M}}_{13,4}$ and $\overline{\mathcal{M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $\overline{\mathcal{M}}_{12,8}$ and $\overline{\mathcal{M}}_{16}$. We also show that the moduli of $(4g+5)$-pointed hyperelliptic curves $\mathcal{H}_{g,4g+5}$ is uniruled. Together with a re…
In this article we will construct a universal moduli space of stable parabolic vector bundles over the moduli space of marked Deligne-Mumford stable curves $\overline{M}_{_{g, n}}$. The objects that appear over the boundary of $\overline{M}_{_{g, n}}$ i.e., over singular curves will remain vector bundles. The total space and the fibers over $\overline{M}_{_{g, n}}$ will have good singularities.
We revisit the work of Chang and Ran on bounding the slopes of $\overline{\mathscr{M}_{15}}$ and $\overline{\mathscr{M}_{16}}$, correct one of the formulas used at the conclusion of the argument, and recompute the lower bounds on the slopes, yielding $s(\overline{\mathscr{M}_{15}})>6.5$ but not for $\overline{\mathscr{M}_{16}}$. Our contribution only involves plugging in formulas.
Stable cuspidal curves and the integral Chow ring of $\overline{\mathscr{M}}_{2,1}$
本研究引入n标记至多尖点稳定曲线的模栈$\tilde{\text{mod}}_{g,n}$，以此确定了$\bar{\text{mod}}_{2,1}$的整周环，并附带得到$\bar{\text{mod}}_{1,2}$的整周环结果。
Extremal effective curves and non-semiample line bundles on $\overline{\rm{M}}_{g,n}$
本文通过构造新的nef分割和可处理的归纳结构，证明了$\overline{\mathrm{M}}_{g,n}$上有效曲线闭锥的非半 ample 部分，并给出了首个非可缩极值射线的例子。
Pascal's Theorem gives a synthetic geometric condition for six points $a,\ldots,f$ in $\mathbb{P}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap\overline{de}$, $\overline{af}\cap\overline{dc}$, $\overline{ef}\cap\overline{bc}$ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for $d+4$ points in $\mathbb{P}^d$ to lie on a degree $d$ rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of $d+4$ ordered p…
For $n\geq 2$, let $K=\overline{\mathbb{Q}}(\mathbb{P}^n)=\overline{\mathbb{Q}}(T_1, \ldots, T_n)$. Let $E/K$ be the elliptic curve defined by a minimal Weiestrass equation $y^2=x^3+Ax+B$, with $A,B \in \overline{\mathbb{Q}}[T_1, \ldots, T_n]$. There's a canonical height $\hat{h}_{E}$ on $E(K)$ induced by the divisor $(O)$, where $O$ is the zero element of $E(K)$. On the other hand, for each smooth hypersurface $Γ$ in $\mathbb{P}^n$ such that the reduction mod $Γ$ of $E$, $E_Γ / \overline{\mathbb{Q}}(Γ)$ is an elliptic curve with the zero element $O_Γ$, there is also a canonical height $\hat{h…
The open subvariety $\overline{M}_g^{\leq k}$ of $\overline{M}_g$ parametrizes stable curves of genus $g$ having at most $k$ rational components. By the work of Looijenga, one expects that the cohomological excess of $\overline{M}_g^{\leq k}$ is at most $g-1+k$. In this paper we show that when $k=0$, the conjectured upper bound is sharp by showing that there is a constructible sheaf on $\overline{H}_g^{\leq k}$ (the hyperelliptic locus) which has non-vanishing cohomology in degree $3g-2$.
Let $\overline{\mathcal{M}}_{g,A[n]}$ be the Hassett moduli stack of weighted stable curves, and let $\overline{M}_{g,A[n]}$ be its coarse moduli space. These are compactifications of $\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, obtained by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \leq 1$ to the markings; they are defined over $\mathbb{Z}$, and therefore over any field. We study the first order infinitesimal deformations of $\overline{\mathcal{M}}_{g,A[n]}$ and $\overline{M}_{g,A[n]}$. In particular, we show that $\overline{M}_{0,A[n]}$ is rigid over any field, if $g\ge…
Let $C \subset \mathbb{P}^2$ be a plane curve of degree at least three. A point $P$ in projective plane is said to be Galois if the function field extension induced by the projection $π_P: C \dashrightarrow \mathbb P^1$ from $P$ is Galois. Further we say that a Galois point is extendable if any birational transformation induced by the Galois group can be extended to a linear transformation of the projective plane. This article is the second part of [2], where we showed that the Galois group at an extendable Galois point $P$ has a natural action on the dual curve $C^* \subset \mathbb{P}^{2*}$ w…
Given a smooth plane curve $\overline{C}$ of genus $g\geq 3$ over an algebraically closed field $\overline{k}$, a field $L\subseteq\overline{k}$ is said to be a \emph{plane model-field of definition for $\overline{C}$} if $L$ is a field of definition for $\overline{C}$, i.e. $\exists$ a smooth curve $C'$ defined over $L$ where $C'\times_L\overline{k}\cong \overline{C}$, and such that $C'$ is $L$-isomorphic to a non-singular plane model $F(X,Y,Z)=0$ in $\mathbb{P}^2_{L}$. {In this short note, we construct a smooth plane curve $\overline{C}$ over $\overline{\mathbb{Q}}$, such that the field of m…
Belyi pair for the orientation cover of $ \overline{{\mathcal M}_{0,5}^{\mathbb R}}$
本文针对具有5个标记点的零亏格实曲线模空间Deligne-Mumford紧化的定向覆盖，计算其亏格4儿童画所对应的Belyi对，并发现该曲线为著名的Bring曲线。
We give an explicit description of $\overline{\mathcal{M}}_{1,2}$ as a weighted blow-up of a weighted projective stack. We use this description to compute the Brauer group of $\overline{\mathcal{M}}_{1,2;S}$ over any base scheme $S$ where 6 is invertible, and the integral Chow rings of $\overline{\mathcal{M}}_{1,2}$ and $\mathcal{M}_{1,2}$.
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.
Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $α_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right] $ in the Jacobian of $C$. In this work we extend $α_{L}$ to a morphism $\overlineα_{L}:C\rightarrow\overline{J}_{E}^{P}$ taking values on Esteves' compactified Jacobian for any given polarization $E$. The maps $\overlineα_{L}$ are limits of Abel maps of smooth curves of the type $α_{L}$.
Basepoint free cycles on the moduli space $\overline{M}_{0,n}$ of stable n-pointed rational curves, defined using Gromov-Witten invariants of smooth projective homogeneous spaces X are studied. Intersection formulas to find classes are given, with explicit examples for X a projective space, and X a smooth projective quadric hypersurface. When X is projective space, divisors are shown equivalent to conformal blocks divisors for type A at level one, giving maps from $\overline{M}_{0,n}$ to birational models constructed as GIT quotients, parametrizing configurations of weighted points supported o…
For a curve $C$ of genus $6$ or $8$ and a torsion bundle $η$ of order $\ell$ we study the vanishing of the space of global sections of the twist $E_C \otimes η$ of the rank two Mukai bundle $E_C$ of $C$. The bundle $E_C$ was used in a well-known construction of Mukai which exhibits general canonical curves of low genus as sections of Grassmannians in the Plücker embedding. Globalizing the vanishing condition, we obtain divisors on the moduli spaces $\overline{\mathcal{R}}_{6,\ell}$ and $\overline{\mathcal{R}}_{8,\ell}$ of pairs $[C, η]$. First we characterize these divisors by different condit…
We prove that the Chow quotient parametrizing configurations of n points in $\mathbb{P}^d$ which generically lie on a rational normal curve is isomorphic to $\overline{M}_{0,n}$, generalizing the well-known $d = 1$ result of Kapranov. In particular, $\overline{M}_{0,n}$ admits birational morphisms to all the corresponding geometric invariant theory (GIT) quotients. For symmetric linearizations the polarization on each GIT quotient pulls back to a divisor that spans the same extremal ray in the symmetric nef cone of $\overline{M}_{0,n}$ as a conformal blocks line bundle. A symmetry in conformal…
The Kodaira classification of the moduli space of pointed curves in genus $3$
本文通过证明 $n \geq 15$ 时带标记点曲线模空间 $\overline{\mathcal{M}}_{3,n}$ 为一般型，完成了亏格 $3$ 模空间的 Kodaira 分类，并分析了其奇点与典范类的性质。
Some singular curves in Mukai's model of $\overline{M}_7$
本文研究 Mukai 的亏格 7 曲线 GIT 模型边界，借助 Macaulay2、Magma 和 Sage 分析线性子空间产生的三类奇异曲线，并用不变多项式证明其 Spin(10)-半稳定性。
The fifteenth cohomology group of moduli spaces of stable curves
本研究针对稳定曲线模空间的第十五上同调群问题，利用半重言扩张新框架，证明了特定情形下的相关结论，确定了对应向量空间的关系与显式基，并给出了若干新小对的结果。
It is known that the moduli space $\overline{\mathcal{H}}_{g,n}$ of genus $g$ stable hyperelliptic curves with $n$ marked points is uniruled for $n \leq 4g+5$. In this paper we consider the complementary case. We calculate the canonical divisor of $\overline{\mathcal{H}}_{g,n}$ and show that it is effective for $n=4g+6$ and big for $n\leq 4g+7$. This leads us to conjecture that $\overline{\mathcal{H}}_{g,n}$ has non-negative Kodaira dimension for $n = 4g+6$ and is of general type for $n \geq 4g+7$.
In this series of papers, we investigate a new anabelian phenomenon of curves over algebraically closed fields of positive characteristic. Let $\overline M_{g, n}$ be the moduli space of curves of type $(g, n)$ over $\overline {\mathbb{F}}_p$. We introduce a topological space $\overline Π_{g, n}$ which can be determined group-theoretically from admissible fundamental groups of pointed stable curves of type $(g, n)$. By introducing a certain equivalence relation $\sim_{fe}$ on the underlying topological space $|\overline M_{g, n}|$ of $\overline M_{g, n}$, we obtain a topological space $\overli…
We study the variety of rank $\leq k$ quadrics containing a general projective curve and show that it has the expected dimension in the range $g-d+r\leq 1$. By considering the loci where this expectation is not true, we construct new divisor classes in $\overline{\mathcal{M}}_{g,n}$. We use one of these classes to show that $\overline{\mathcal{M}}_{15,9}$ is of general type.
Log algebraic hyperbolicity of $\overline{M}_{0,n}$
本文证明了带边界 Δ 的稳定 n 点有理曲线模空间 $\bar{M}_{0,n}$ 具有代数双曲性。
In a recent paper Chiodo and Farkas described the singular locus and the locus of non-canonical singularities of the moduli space of level curves. In this work we generalize their results to the moduli space $\overline{\mathcal R}_{g,G}$ of curves with a $G$-cover for any finite group $G$. We show that non-canonical singularities are of two types: $T$-curves, that is singularities lifted from the moduli space $\overline{\mathcal M}_g$ of stable curves, and $J$-curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case $G=S_3$, th…
The stack $\overline{\mathcal{M}}_{g,n}$ of stable curves and its coarse moduli space $\overline{M}_{g,n}$ are defined over $\mathbb{Z}$, and therefore over any field. Over an algebraically closed field of characteristic zero, Hacking showed that $\overline{\mathcal{M}}_{g,n}$ is rigid (a conjecture of Kapranov). Bruno and Mella for $g=0$, and the second author for $g\geq 1$ showed that its automorphism group is the symmetric group $S_n$, permuting marked points unless $(g,n)\in\{(0,4),(1,1),(1,2)\}$. The methods used in the papers above do not extend to positive characteristic. We show that i…
Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal M}_{1,n}$
通过Hassett模空间墙穿越，将曲线模空间上同调计算归结为最小权重情形，并给出亏格1的显式多项式公式，实现高效计算。
Effective cone of a Grassmann bundle over a curve defined over $\overline{\mathbb F}_p$
本文证明了定义在有限域代数闭包上的光滑射影曲线上的Grassmann丛的有效锥与伪有效锥重合，推广了Moriwaki在秩为1或余秩为1情形下的已有结果。
In this paper we introduce the stack of polarized twisted conics and we use it to give a new point of view on $\overline{\mathcal{M}}_2$. In particular, we present a new and independent approach to the computation of the integral Chow ring of $\overline{\mathcal{M}}_2$, previously determined by Eric Larson.
A Murphy's law for supports of forgetful maps between moduli spaces of stable pointed curves
本文研究稳定带标曲线模空间间忘却映射的分解定理，证明当m趋于无穷时，$\bar{\textit{M}}_{g,n}$的每个层闭包为分解定理的直和项支集，证明用到Hassett的模空间及跨壁论证。
We prove that the moduli space of $n$-pointed stable maps $\overline{M}_{0,n}(\mathbb{P}^1,1)$ is a Mori dream space whenever the moduli space $\overline{M}_{0,n+3}$ of $(n+3)$-pointed rational curves is. We also show that $\overline{M}_{0,n}(\mathbb{P}^1,1)$ is a log Fano variety for $n \le 5$.
A faithful action of Gal($\overline{\mathbb{Q}}/\mathbb{Q}$) on Zariski multiplets
本文研究了 Gal($\overline{\mathbb{Q}}/\mathbb{Q}$) 在 Zariski 多重组上的忠实作用，构造了新的算术 Zariski 对和多重组，并探讨了伽罗华表示与奇异性平面曲线几何的相互作用。
The automorphism group of $\overline{M}_{g,n}$
本文证明在 $2g-2+n\geq 3$ 时，稳定曲线模叠及其粗模空间的自同构群同构于对称群 $S_n$，并计算了其余情形。
We perform a variation of geometric invariant theory stability analysis for 2nd Hilbert points of bi-log-canonically embedded pointed curves of genus two. As a result, we give a GIT construction of the last three non-trivial log canonical models of the moduli space of pointed genus two curves, and obtain a VGIT presentation of the second flip in its Hassett-Keel program.
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 $\overline{M}_{2,2}({\mathcal Z})$.
In this paper we study fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov's description we are able to prove that the only such morphisms are forgetful maps. This allow us to show that the Automorphism group of $\overline{M}_{0,n}$ is the permutation group on $n$ elements as soon as $n\geq 5$.
The spaces of point configurations on the projective line up to the action of $\mathrm{SL}(2,\mathbb K)$ and its maximal torus are canonically compactified by the Grothdieck-Knudsen and Losev-Manin moduli spaces $\overline M_{0,n}$ and $\overline L_n$ respectively. We examine the configuration space up to the action of the maximal unipotent group $\mathbb G_a\subseteq \mathrm{SL}(2,\mathbb K)$ and define an analogous compactification. For this we first assign a canonical quotient to the action of a unipotent group on a projective variety. Moreover, we show that similar to $\overline M_{0,n}$ a…
We identify a set of initial rational contractions of fiber type on $\overline{M}_{0,6}$. Our proof uses a new algorithm we develop for verifying descriptions of the cone of effective divisors on varieties without elementary rational contractions of fiber type. On Mori dream spaces, our algorithm identifies faces of the cone of nef curves associated to initial rational contractions of fiber type.
Congruences on tropical rational function semifields and tropical curves
本文定义了热带有理函数半域，通过具体映射证明热带曲线可作为同余簇实现，并揭示了商半域在决定同构类中发挥类似坐标环的作用。
In the paper we consider the Jacobian locus $\overline{J_g}$ and the Prym locus $\overline{P_{g+1}}$, in the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$, for $g\geq 7$, and we study the extrinsic geometry of $\overline{J_g}\subset \overline{P_{g+1}}$, under the inclusion provided by the theory of generalized Prym varieties as introduced by Beauville. More precisely, we study certain geodesic curves with respect to the Siegel metric of $A_g$, starting at a Jacobian variety $[JC]\in A_g$ of a curve $[C]\in M_g$ and with direction $ζ\in T_{[JC]}J_g$. We prove th…
K-stability of Thaddeus' moduli of stable bundle pairs on genus two curves
本文研究了Thaddeus稳定束对模空间的K-稳定性，证明其与四次del Pezzo三次型中的线的GIT模空间同构，并建立了与稳定向量包模空间的自然映射。
We prove a general formula for the intersection form of two arbitrary monomials in boundary divisors. Furthermore we present a tree basis of the cohomology of $\overline {M}_{0,n}$. With the help of the intersection form we determine the Gram matrix for this basis and give a formula for its inverse. This enables us to calculate the tensor product of the higher order multiplications arising in quantum cohomology and formal Frobenius manifolds. In the context of quantum cohomology this establishes the explicit Künneth formula.
We prove that the moduli space of double covers ramified at two points $\mathcal{R}_{g,2}$ is uniruled for $3\leq g\leq 6$ and of general type for $g\geq 16$. Furthermore, we consider Prym-canonical divisorial strata in the moduli space $\overline{\mathcal{C}^n\mathcal{R}}_g$ parametrizing $n$-pointed Prym curves, and we compute their classes in $\mathrm{Pic}_\mathbb{Q}(\overline{\mathcal{C}^n\mathcal{R}}_g)$.
Cup products on curves over finite fields
该研究针对有限域上的光滑射影几何不可约曲线，计算了两类平展上同调群元素的上积，明确了代数闭包下上积与Weil配对的关联，以及基域上因取值群不同带来的微妙差异。
We study the moduli space ${V}_4\mathcal{M}_{g}$ of Klein four covers of genus $g$ curves and its natural compactification. This requires the construction of a related space which has a choice of basis for the Klein four group. This space has the interesting property that the two components intersect along a component of the boundary. Further, we carry out a detailed analysis of the boundary, determining components, degrees of the components over their images in $\overline{\mathcal{M}_g}$, and computing the canonical divisor of $\overline{{V}_4\mathcal{M}_{g}}$.
Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles
本文通过构造大量非对偶的超椭圆曲线双有理映射到阿贝尔曲面，并利用这些曲线在零环的Chow群中找到许多有理等价，从而推进了Beilinson猜想关于零环的证明。
Building on the work of Goto, Nishida and Watanabe on symbolic Rees algebras of monomial primes, we prove that the moduli space of stable rational curves with n punctures is not a Mori Dream Space for n>133. This answers the question of Hu and Keel.
The Kodaira dimensions of $\overline{\mathcal{M}}_{22}$ and $\overline{\mathcal{M}}_{23}$
本文通过计算与秩 6 线性系相关的小斜率虚拟除子类，并引入新的热带方法证明其有效性，从而证明了亏格 22 和 23 的曲线模空间是一般型的。
We construct moduli spaces of representations of quivers over arbitrary schemes and show how moduli spaces of pointed curves of genus zero like the Grothendieck-Knudsen moduli spaces $\overline{M}_{0,n}$ and the Losev-Manin moduli spaces $\overline{L}_n$ can be interpreted as inverse limits of moduli spaces of representations of certain bipartite quivers. We also investigate the case of more general Hassett moduli spaces $\overline{M}_{0,a}$ of weighted pointed stable curves of genus zero.
We define an effective divisor of the moduli space of stable curves $\overline{M_g}$, which is denoted $\overline{S^{2}W}$. Writing the class of $\overline{S^{2}W}$ in the Picard group of the moduli functor Pic$_{\text{fun}}(\overline{M_{g}})\otimes \mathbb{Q}$ in terms of the so-called Harer basis $λ,δ_0,\ldots,δ_{[g/2]}$, we prove that the relations among the coefficients of $δ_1,\ldots,δ_{[g/2]}$ are the same relations on coefficients as the Brill-Noether divisors. We present a result on effective divisors of $\overline{M_g}$ which could be useful to get the same relations on coefficients f…
We prove that the moduli space $\overline{\mathcal{R}}_{15}$ of Prym curves of genus $15$ is of general type. To this end we exhibit a virtual divisor $\overline{\mathcal{D}}_{15}$ on $\overline{\mathcal{R}}_{15}$ as the degeneracy locus of a globalized multiplication map of sections of line bundles. We then proceed to show that this locus is indeed of codimension one and calculate its class. Using this class, we can conclude that $K_{\overline{\mathcal{R}}_{15}}$ is big. This complements a 2010 result of Farkas and Ludwig: now the spaces $\overline{\mathcal{R}}_g$ are known to be of general t…
In this work, we compute the effective cone of the space of $n$ pointed genus 0 rational curves, $\bar{M}_{0,n}$ for odd $n$. We will, in fact, use the equivalence of $\bar{M}_{0,n}$ and the space of (semi) stable configurations of $n$ weighted points on $\mathbb{CP}^{1}$ with special weight $(1,..., 1)$ up to symmetric group action.
We consider the cones of curves and divisors on the moduli space of stable pointed rational curves,M_n, and on the quotient by the symmetric group, Q_n, which is a moduli space of pairs. We find generators for contractible extremal rays of the cone of curves NE_1(M_n), and for the cone of divisors NE^1(Q_n). This second cone turns out to be simplicial. We give complete descriptions of NE_1(M_n) and NE_1(Q_n) for small n (< 8 in the first case, < 11 in the second). We also have results of independent interest on when curves in a divisor generate the cone of curves of the ambient variety.
Smoothness results for the schemes of special divisors on general k-gonal curves
本文研究一般k-边曲线C上的特殊除子方案W^r_d(C)的光滑性，通过Brill-Noether退化方案的细化，证明其在特定开子集上光滑，并探讨invertible sheaves的Petri映射性质。
Liaison theory and the birational geometry of the Hilbert scheme of curves in $\mathbb{P}^{3}$
本研究运用联络理论，探究$\mathbb{P}^{3}$中曲线希尔伯特概形的双有理几何，证明算术Cohen-Macaulay曲线存在唯一分量，验证特定Rao模秩为1的子簇含可约除子，且除子分量线性无关并生成有效锥的极射线。
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