Dung Nguyen
In this paper we compute the number of rational curves with one node passing through a given number of points, lines and tangent to a given number of planes in $\mathbb{P}^3$.
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A Simple Proof that Rational Curves on K3 are Nodal
给出了一般K3曲面的本原类中的每一条有理曲线都是结点曲线的简单证明。
Non-rational nodal quartic threefolds
结点四次3重数的$\mathbb{Q}$ -因式性暗示了它的非理性。我们证明了一个结点不超过8个的四次三次结点是$\mathbb{Q}$ -阶乘的，并且证明了一个结点为9个的四次三次结点不是$\mathbb{Q}$ -阶乘的当且仅当它包含一个平面.然而，在$\mathbb{P}^4$中存在非有理非$\mathbb{Q}$ -阶乘结点四次3-折叠.特别是，我们证明了非理性的一般非$\mathbb{Q}$阶乘节点四次3倍，其中包含一个平面或一个光滑的del Pezzo表面的程度为4。
On the Chow ring of the stack of rational nodal curves
本文的目的是计算亏格为0且至多3个结点的结点曲线构成的栈的有理Chow环：它是一个具有10个生成元和11个关系的Q-代数。
Tautological Classes of the Stack of Rational Nodal Curves
这是第二次在一系列的三篇论文中，我们探讨了合理的周环的堆栈组成的节点曲线的属。在这里，我们定义基本类：阶层类和芒福德类。
The Stack of Rational Nodal Curves
在这一系列的三篇论文中，我们开始研究由亏格为0的节点曲线组成的堆栈的有理Chow环，特别是我们完全确定了由至多3个节点的曲线组成的子堆栈的有理Chow环。在这第一篇论文中，我们构造了堆栈的合理节点曲线和分层的节点，并表明，从通用曲线的堆栈的映射是不可表示的类别的计划。
Enumeration of non-nodal real plane rational curves
该研究探讨计数带非结点奇点的实有理平面曲线的不变量，证明唯一的此类不变量对应实有理三尖点四次曲线，且任意四对复共轭点的一般构型均存在一对这类曲线。
Hirota Varieties and Rational Nodal Curves
本文研究了由有理节点曲线导出的Hirota簇，定义了其主分支，证明其为不可约分支等价于求解弱Schottky问题，并利用计算工具解决了直到亏格9的该问题。
We prove that the quartic threefolds defined by $$ \sum_{i=0}^{5}x_i=\sum_{i=0}^{5}x_i^4-t\left(\sum_{i=0}^{5}x_i^2\right)^2=0 $$ in $\mathbb{P}^5$ are rational for $t=\frac{1}{6}$ and $t=\frac{7}{10}$.
The integral Chow ring of the stack of at most 1-nodal rational curves
给出了6维不可约表示中至多有1个结点的有理曲线与开集GL（3）的商的叠加.然后，我们使用等变相交理论来计算该堆栈的积分Chow环。
We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational, and nodal quartic double solids with at least eleven singular points are rational.
In this article we obtain the rigid isotopy classification of generic rational curves of degre $5$ in $\mathbb{R}\mathbb{P}^{2}$. In order to study the rigid isotopy classes of nodal rational curves of degree $5$ in $\mathbb{R}\mathbb{P}^{2}$, we associate to every real rational nodal quintic curve with a marked real nodal point a nodal trigonal curve in the Hirzebruch surface $Σ_3$ and the corresponding nodal real dessin on $\mathbb{C}\mathbb{P}^{1}/(z\mapsto\bar{z})$. The dessins are real versions, proposed by S. Orevkov, of Grothendieck's dessins d'enfants. The dessins are graphs embedded i…
We consider singular holomorphic foliations on compact complex surfaces with invariant rational nodal curve of positive self-intersection. Then, under some assumptions, we list all possible foliations.
Enumeration of One-Nodal Rational Curves in Projective Spaces
给出了一个计算复射影空间中通过适当约束集合的单节点有理曲线个数的公式。我们结合的方法和结果从三个不同的文件。
Nodal rational curves on Enriques surfaces of base change type
本文提出一种纯几何无格构造方法，证明了基变换型Enriques曲面上双截面通常是节点的，并展示了此类曲面承认形成秩8子群的节点有理曲线集。
We study the existence of components with the expected number of moduli of the Hilbert scheme of integral nodal curves $C \subset \mathbb {P}^r$ with prescribed degree, arithmetic genus and number of singular points.
Let $M(2,\textbf{\underline{w}},χ)$ be the moduli space of rank $2$ torsion-free sheaves over a reducible nodal curve with each component having utmost two nodal singularities. We show that in each component of $M(2,\textbf{\underline{w}},χ)$, the closure of rank $2$ vector bundles with fixed determinant is rational.
On factoriality of nodal threefolds
我们证明了$\mathbb{Q}$ -具有最多${\frac{(n-1)^{2}}{4}}$节点的$n$度$\mathbb{P}^{4}$中节点超曲面的因式性，以及$\mathbb{Q}$ -在具有最多${\frac{(2r-1)r}{3}}$节点的$2r$度节点表面上分支的$\mathbb{P}^{3}$的双覆盖的因式性。
We prove an elementary method to classify, up to rigid isotopy, all nodal degree 4 real rational curves in $\mathbb{RP}^2$ that have only real double points. We show how to associate a chord diagram to a nodal real degree 4 planar rational curve and prove that such curves are defined up to rigid isotopy by their chord diagrams.
Nodal Enriques surfaces are Reye congruences
本文证明了任何包含光滑有理曲线的经典恩里克斯曲面均可实现为雷耶线汇，揭示了这两类代数几何对象之间的对应关系。
On nodal deformations of singular surfaces in $\mathbb P^3$
本文研究了 $\mathbb P^3$ 中奇异曲面的节点变形，探讨了孤立奇点和沿线奇点的变形问题。
Cohomology of Moduli Stacks of Principal $\Bbb C^*$-Bundles Over Nodal Algebraic Curves
研究结点曲线上主$\Bbb C^*$-丛模栈，用陈类确定其有理上同调代数。
Singularities of Rational Curves on K3 surfaces
证明了一般K3曲面的任一亏格的本原类中的每一条有理曲线都是结点曲线。
We generalize Werner's defect formula for nodal hypersurfaces in $\mathbb P^{4}$ to the case of a nodal complete intersection threefold.
The surfaces considered are real, rational and have a unique smooth real $(-2)$-curve. Their canonical class $K$ is strictly negative on any other irreducible curve in the surface and $K^2>0$. For surfaces satisfying these assumptions, we suggest a certain signed count of real rational curves that belong to a given divisor class and are simply tangent to the $(-2)$-curve at each intersection point. We prove that this count provides a number which depends neither on the point constraints nor on deformation of the surface preserving the real structure and the $(-2)$-curve.
Let $(X,L)$ be a general primitively polarized K3 surface with $c_1(L)^2 = 2g-2$ for some integer $g \geq 2$. The Severi variety $V^{L,δ} \subset |L|$ is defined to be the locus of reduced and irreducible curves in $|L|$ with exactly $δ$ nodes and no other singularities. When $δ=g$, any curve $C \in V^{L,g}$ is a rational curve; in fact, Chen \cite{Chen02} has shown that all rational curves in $|L|$ are nodal, and the number of such rational curves is given by the Yau-Zaslow formula \cite{YZ96}. In this paper, we consider the next case where $δ= g-1$ and the Severi variety $V^{L,g-1}$ parametr…
Suppose that $C\subset\mathbb P^2$ is a general enough nodal plane curve of degree $>2$, $ν\colon \hat C\to C$ is its normalization, and $π\colon \hat C\to\mathbb P^1$ is a finite morphism simply ramified over the same set of points as a projection $\mathrm{pr}_p\circ ν\colon\hat C \to\mathbb P^1$, where $p\in\mathbb P^2\setminus C$ (if $\mathrm{deg}\, C=3$, one should assume in addition that $\degπ\ne4$). We prove that the morphism $π$ is equivalent to such a projection if and only if it extends to a finite morphism $X\to(\mathbb P^2)^*$ ramified over $C^*$, where $X$ is a smooth surface. As …
Rational Curves on K3 Surfaces
证明了一般K3曲面上的线性系组存在有理曲线，且超平面类中的所有有理曲线都是小亏格的一般K3曲面上的节点.
Geometry of families of nodal curves on rational surfaces
设P^2_r是在r个类属点处展开的射影平面。用E_0、E_1、.表示，E_r分别是P^2上一般直线的严格变换和P^2_r上爆破点的例外因子。我们考虑所有不可约曲线C的簇V，|dE_0-d_1E_1-...- d_rE_r|以k个结点为唯一奇点，给出了其光滑性、不可约性和非空性的渐近近最优充分条件.此外，我们推广了可约曲线族的光滑性和不可约性条件。当r<10时，我们给出了V中节曲线存在性的完整答案.
We present a rigid isotopy classification of irreducible sextic curves in $\mathbb{RP}^2$ which have non-real ordinary double points as their only singularities. Our approach uses periods of K3 surfaces and V. Nikulin's classification of involutions with condition on unimodular lattices. The classification obtained generalizes Nikulin's rigid isotopy classification of non-singular sextics in $\mathbb{RP}^2$.
On nodal sextic fivefold
证明了6次$\mathbb{P}^{6}$中至多具有孤立的普通二重点的超曲面的双有理超刚性和非有理性.
The Chow Ring of the Stack of Rational Curves with at most 3 Nodes
本文给出了亏格为0且至多3个结点的结点曲线构成的叠的有理Chow环：它是一个具有10个生成元和11个关系的Q-代数。
Let U^{'s}_{L}(n,d) be the moduli space of stable vector bundles of rank $n$ and fixed determinant L of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree $d$ will be denoted by U'_Y(n,d). We calculate the Brauer groups of U^{'s}_{L}(n,d)$. We study the question of rationality of $U^{'s}_{L}(n,d)$ and $U'_Y(n,d)$.
We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one.
An enriched count of nodal orbits in an invariant pencil of conics
本文研究了有限群线性作用下的圆锥曲线不变丛中结点轨道的等变富化计数问题，证明了除三个例外群外所有有限群的结点轨道加权和公式均成立，并给出例外反例。
The Wahl map of the normalization of nodal curves on Hirzebruch surfaces
本文研究Hirzebruch曲面上节点曲线的正则化Wahl映射，证明其corank与曲面的canonical bundle相关，并探讨了节点曲线嵌入的限制条件。
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 $α_{L}$.
In this paper we consider the Brill-Noether locus $W_{\underline d}(C)$ of line bundles of multidegree $\underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $g\geq2$. We give an explicit description of the irreducible components of $W_{\underline d}(C)$ for a semistable multidegre $\underline d$. As a consequence we show that, if two semistable multidegrees of total degre $g-1$ on a curve with no rational components differ by a twister, then the respective Brill-Noether loci have isomorphic components.
We suggest an invariant way to enumerate nodal and nodal-cuspidal real deformations of real plane curve singularities. The key idea is to assign Welschinger signs to the counted deformations. Our invariants can be viewed as a local version of Welschinger invariants enumerating real plane rational curves.
We determine the scrollar invariants of the normalization $C$ of a nodal curve $Γ$ of type $(k,a)$ on a smooth quadric $\mathbb{P}^1 \times \mathbb{P}^1$ associated to the $g^1_k$ defined by the pencil of lines of type $(0,1)$ in case all nodes are contained in at most $k-1$ lines of type $(1,0)$. This result is very much related to results obtained by E. Ballico, but in this paper the proof follows directly from an easy lemma. Also a result of E. Ballico on the existence of curves with prescribed scrollar invariant is a consequence of that lemma making the arguments much shorter.
In this paper, we show the full flag Hilbert scheme of points of nodal curves has only complete intersection singularities. We study the Hilbert scheme of points of the cusp curve and show the punctual Hilbert scheme is isomorphic to \mathbb{P}^1. Furthermore Hilbert scheme has only one singularty along the punctual one.
Irreducibility of Some Quot Schemes on Nodal Curves
本文证明了对于算术亏格 $g \geqslant 2$ 的整射影节点曲线 $C$，当次数 $d$ 足够大时，向量丛 $E$ 的 Quot scheme $\textrm{Quot}_{C/\mathbb C}(E,k,d)$ 是不可约的。
Maximally nodal sextic surfaces and linear determinantal representations
研究极多节点六次曲面的节点结构，证明其存在35个对称半偶节点，从而得到线性行列式表示，且半二次sheaf为乌里奇sheaf。
In this paper we show that a nodal complete intersection threefold $X$ in $\mathbb{P}^{3+c}$ with defect, but without induced defect, has at least $\sum_{i\leq j} (d_i-1)(d_j-1)$ nodes, provided either $c=2$ or $d_c>\sum_{i=1}^{c-1} d_i$ holds.
We determine the rational Chow ring of the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$ when $n \leq 2g+6$. We also show that the Chow ring of the partial compactification $\mathcal{I}_{g,n}$, parametrizing $n$-pointed irreducible nodal hyperelliptic curves, is generated by tautological divisors. Along the way, we improve Casnati's result that $\mathcal{H}_{g,n}$ is rational for $n \leq 2g+8$ to show $\mathcal{H}_{g,n}$ is rational for $n \leq 3g+5$.
We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to certain polyhedral cones which we describe explicitly. Not all homology classes in the cone correspond to rational curves, but all classes sufficiently far from the walls of the cone do. The proof for $X_5$ is explicit, the proof for $X_6$ was carried out using computers.
Nodal deformations of hypersurfaces with an ordinary $m$-fold point
本文研究具有普通 $m$ 重点的节点超曲面的形变，利用谱半连续性给出节点数上界，并构造多项式下界，改进了 Ciliberto-Galati 的结果。
Weil's Theorem for Logarithmic Connections on Irreducible Nodal Curves
本研究将André Weil经典定理推广至不可约射影结点曲线，证明其上不可分解向量丛或无挠凝聚层存在关于对偶化层的全纯对数联络等价于次数为0，还构造了有理结点三次曲线上的显式单参数平坦联络族。
We show that the compactification of the moduli space of $n-$nodal curves of genus g, i.e. $\mathcal{N}_{g,n}:= \mathcal{M}_{g,2n} /G$, with $G:=(\mathbb{Z}_2)^n \rtimes S_n$, is of general type for $g \geq 24$, for all $n \in \mathbb{N}$. While this is a fairly easy result, it requires completely different techniques to extend it to low genus $5 \leq g \leq 23$. Here we need that the number of nodes varies in a band $n_{\mathrm{min}}(g) \leq n \leq n_{\mathrm{max}}(g)$, where $n_{\mathrm{max}}(g)$ is the largest integer smaller than (or in some cases equal to) $\frac{7}{2}(g-1)-3$. The lower …
Chow ring of the stack of plane nodal curves
本文计算固定次数平面结点曲线模空间栈的有理周环并用重言类刻画，同时把 Vial 关于 Brauer–Severi 簇周群的结果推广至 $G$-等变情形。
Nodal surfaces in $\mathbb{P}^3$ and coding theory
本文证明 $\mathbb{P}^3$ 中具有 65 个节点的六次曲面（如 Barth 曲面）关联的二元线性码唯一，并给出可能达到节点数上界的七次曲面的候选码。
Let six points $1, ...6$ lie in general position in the real projective plane and consider the pencil of nodal cubics based at these points, with node at one of them, say 1. This pencil has five reducible cubics. We call combinatorial cubic a topological type (cubic, points), and combinatorial pencil the cyclic sequence of five combinatorial reducible cubics. Up to the action of the symmetric group $S_5$ on $\{2, ...6\}$, there are seven possible combinatorial pencils with node at 1. Consider now the set of six pencils obtained, making the node to be $1, ...6$. Up to the action of S_6 on ${1, …
Cosmic String, Harvey-Moore Conjecture and Family Seiberg-Witten Theory
本文研究了某些卡-丘K3纤维化的浸入节线的虚数计数问题。利用宇宙弦的概念，我们验证了弦理论（Harvey-Moore）中浸入节曲线生成函数的模性猜想。基于代数曲面上节线虚数的定义理论（适用于一束格点极化K3），给出了Calabi-Yau K3纤维化的浸入有理曲线虚数（所谓的Gopakumar-Vafa数）的数学定义，并与弦理论的预言相符合.
We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.
Recursive Formulas for the Characteristic Numbers of Rational Plane Curves
我们推导出递归方程的有理节面曲线的特征数与最多一个尖点，受点的条件，切线条件和标志条件，开发技术类似于量子上同调的模空间的稳定升降机。
We determine the maximum number of $\mathbb{F}_q$-rational points that a nonsingular threefold of degree $d$ in a projective space of dimension $4$ defined over $\mathbb{F}_q$ may contain. This settles a conjecture by Homma and Kim concerning the maximum number of points on a hypersurface in a projective space of even dimension in this particular case.
Limits of nodal surfaces and applications
本文研究平坦族中δ-结点曲面趋于双重正规交叉中心纤维时的极限结构，证明极限曲面由两光滑曲面沿δ-结点曲线相交组成，并利用形变理论给出Severi簇正则不可约分支的存在性结果。
Enumeration of rational curves in a moving family of $\mathbb{P}^2$
本文针对 $\mathbb{P}^3$ 中位于移动 $\mathbb{P}^2$ 内并穿过指定点、线的有理曲线，给出满足 $r+2s=3d+2$ 的数目递推公式，并验证其支持 Laarakker 结点曲线多项式的枚举性猜想。
Nodal curves on surfaces of general type
本文研究了曲面$S$上具有光滑一般成员的自由线性系统$|C|$的不可约曲线族，使得$|K_S|$是充分的。在假设$C$在数值上等价于$pK_S$，$p\ge 3$有理数的情况下，给出了一个$δ$的上界，保证了$|C|$中余维$δ$族是光滑的（加上另一个同类型的稍有不同的结果）.我们给出的例子表明，界限是尖锐的。证明了节曲线线性正规性的有关结果。
Marked nodal curves with vector fields
本文研究带对数向量场的标记结点曲线，构造类似 Knudsen 稳定化的族操作及其逆，并用于证明 Losev–Manin 紧化的同平凡退化及群作用的整体相容性。
BPS States, String Duality, and Nodal Curves on K3
通过将亏格$g$的超对称圈与K3上具有$g$双点的有理曲线的个数联系起来，描述了K3上II型弦的BPS态的计数.这种曲线的生成函数是玻色弦的左移配分函数。
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