Daniele Di Tullio
The aim of the present paper is to prove the rationality of the universal family of polarized $ K3 $ surfaces of degree 14. This is achieved by identifying it with the moduli space of cubic fourfolds plus the data of a quartic scroll. The last moduli space is finally proved to be rational since it has a natural structure of $\mathbb P^n$-bundle over a $ k $-stably rational variety with $k \leq n$.
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The rationality of the moduli spaces of trigonal curves of odd genus
证明了奇亏格$g>4$三角曲线的模空间是有理的。
Irreducibility and stable rationality of the loci of Weierstrass points on curves of genus at most six
证明了对亏格g不大于6的任意数值半群H，亏格g的曲线上的Weierstrass点的轨迹是不可约的，且对除可能两个半群外的所有半群，该轨迹是稳定有理的.
On the rationality of the moduli of higher spin curves in low genus
高自旋曲线的模空间的整体几何和它们的双有理分类对于g >= 2和r > 2是未知的。利用相当相关的尺规作图法，我们几乎完成了亏格g <= 4的已知结果的图象，表明亏格3的偶、奇4-旋曲线、亏格4的奇旋曲线和亏格4的3-旋曲线的模空间的合理性。
The rationality of the moduli space of curves of genus 3 after P. Katsylo
本文综述了Katsylo关于亏格为3的光滑射影复曲线的模空间是有理的证明。我们希望通过强调证明中的潜在几何及其关键结构特征，使论证更易于理解和透明。
The rationality of certain moduli spaces of curves of genus 3
对于每个代数闭域，我们证明了以下两个模空间的合理性：M（3，3）个参数化对（C，η），其中C的亏格为3，η是M（3，）<3>个参数化对（C，<η>）的3-扭除子类，其中<η>是由η生成的Pic_0（C）中的3阶循环子群.
The rationality of ineffective spin genus-4 thetanull loci
本文证明了由亏格4且thetanull消失的曲线C与无效thetacharacteristic θ构成的对[C,θ]所给出的除子是有理簇，并得出Prym模空间中的类似除子也是有理的。
Rationality of the Moduli Variety of Curves of Genus 3
证明了亏格为3的曲线模簇的合理性。
The rationality of the moduli space of genus four curves endowed with an order three subgroup of their Jacobian
参考版本出现在密歇根数学杂志。上一版本最后一节中的一个错误已得到纠正。新标题准确地描述了所获得的主要结果。本文利用三次曲面的几何学和Dolgachev的一个定理，证明了文中所提的模空间R的合理性。设M是平面上6点的模空间，模是三次曲面上的双六构形诱导的自然对合。证明了R对于一个终止于M上的局部平凡投射丛塔是双有理的。R的有理性则由多尔加乔夫定理得出，M是有理的。
Geometry of genus 8 Nikulin surfaces and rationality of their moduli
设S是亏格为8的一般复Nikulin曲面，给出了S的一个尺规作图法.考虑Grassmannian G（1，4）的光滑三重线性截面T和T的有理正规六次曲线的Hilbert格式。其中考虑了特殊的六次曲线族A，它也包含在有理正规四次曲线P^4的双斜线的同余中。我们证明了S对于T的一个包含六次曲线A的二次截面是双正则的。特别地，A允许包含在G（1，4）中的一个1维双割线族，其中8条在S中.然后利用这种显式构造证明亏格为8的Nikulin曲面的模空间是有理的。
We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except for the case of genus $9$, where we assume $n \ge 5$. Furthermore, we prove that Mukai varieties of genus $g \in \{7,8,9,10\}$ and dimension $n \ge 5$ contain cylinders if they have a $\mathsf{k}$-point. Finally, we prove that the embedding $X \hook…
On the rationality of certain Fano threefolds
本文研究具典范奇点的 Gorenstein Fano 三维簇有理性，证明三角、含平面、高亏格及含普通直线等四类情形在相应阈值下均为有理簇。
The moduli space of genus $8$ Fano $3$-folds with two $\frac12(1,1,1)$-points: rationality and K-stability
研究具有两个 $\frac12(1,1,1)$ 型终端商奇点的亏格 $8$ Fano $3$-折的模空间，证明其有理性和一般成员的 K-稳定性。
The rationality of the moduli spaces of trigonal curves
证明了当亏格被4整除时三角曲线的模空间是有理的。
Using the isomorphism between the moduli space of polarized K3 surfaces of genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we establish the rationality of the universal K3 surface of genus 14. Precisely, we show that the universal K3 surface of genus 14 is a projective bundle over a certain moduli space of nodal scrolls in P^5, whose rationality we prove using a degenerate version of Mukai's structure theorem for curves of genus 8.
Rationality of some tetragonal loci
证明了亏格g>6的四元曲线的模空间是有理的，当g与1，2，5，6，9，10模12全等且不等于9，45时.
The connectedness of the moduli space of maps to homogeneous spaces
本文证明了（固定亏格和同调类的）映射的模空间通过极大环面作用退化到齐性空间G/P的连通性。在亏格为0的情况下，映射模的不可约性是连通性的直接结果。对映射空间的一个相关的Bialynicki-Birula分层的分析得到了一个合理性结果：亏格0映射到G/P的（粗）模空间是一个有理簇.合理性论证基本上取决于由Katsylo和Bogomolov证明的SL 2表示的合理性结果。
The rationality of quaternionic Darmon points over genus fields of real quadratic fields
p-adic环面上的Darmon点和Q上Shimura曲线的Jacobians在以前的联合工作中与Rotger一起引入，作为Darmon的Stark-Heegner点的推广。本文研究了Darmon点到椭圆曲线的投影在实二次域K上的扩张的代数性。更精确地说，我们证明了椭圆曲线上的Darmon点的线性组合在K的亏格域上是有理的。这将Bertolini和Darmon得到的关于Stark-Heegner点的合理性的主要定理推广到任意四元数的情形，同时也证明了与Rotger和M.格林伯格在他关于斯塔克-希格纳观点的文章中。鉴于这一结果，四元数达蒙点代表了第一个实例的系统供应点的斯塔克-希格纳型以外的达蒙的原始的明确的合理性结果是已知的。
By the geometry of the 3-fold quadric we show that the coarse moduli space of genus g ineffective spin hyperelliptic curves with two marked points is a rational variety for every $g \geq 2$.
Using the geometry of an almost del Pezzo threefold, we show that the moduli space of genus $g$ one-pointed ineffective spin hyperelliptic curves is rational for every $g\geq 2$.
Rationality of Kontsevich moduli of stable maps, and their Galois descents
本文研究域k上几何有理簇X的低次数零亏格稳定映射的Kontsevich模空间的k-合理性，定义其扭曲伽罗瓦下降，针对不同类型的X展开分析，复数域齐次簇情形对应Kim-Pandharipande的已知结论。
We use the universal generation of algebraic cycles to relate (stable) rationality to the integral Hodge conjecture. We show that the Chow group of 1-cycles on a cubic hypersurface is universally generated by lines. Applications are mainly in cubic hypersurfaces of low dimensions. For example, we show that if a generic cubic fourfold is stably rational then the Beauville--Bogomolov form on its variety of lines, viewed as an integral Hodge class on the self product of its variety of lines, is algebraic. In dimension $3$ and $5$, we relate stable rationality with the geometry of the associated i…
Rationality and Parametrizations of Algebraic Curves under Specializations
本研究针对含参数的有理代数曲线特殊化问题，通过划分参数仿射空间明确亏格变化与曲线可约性的参数区域，在亏格零情形下优化参数化描述并给出应用实例，完善了代数曲线特殊化的理论刻画。
Rationality and Chow-Kuenneth decompositions for some moduli stacks of curves
本文证明了当g，r的值较小时，具有r个标记点的亏格g的稳定曲线的模栈的Chow-Kuenneth分解的存在性。我们也看看模空间R的双覆盖的亏格3曲线，沿4个不同的点分支。我们得到了模空间R作为两个Grassmanian簇的乘积的群商的双有理模型。这提供了R的开子集上的Chow-Kuenneth分解。本文还讨论了R的合理性问题。
Universal torsors over quartic del Pezzo surfaces and stable rationality
针对特征零域上的光滑四次 del Pezzo 曲面，证明其通用挠子若有 k-点则为 k-有理，进而得到有理数域上所有维数大于 2 的稳定有理光滑三次超曲面例子。
Let $M_{g, n}$ (respectively, $\overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $\overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n \geqslant 0$ if $g \geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ wit…
On the rationality of moduli spaces of pointed curves
受最近几个关于含$n$标记点的亏格$g$稳定曲线的模空间$\bar{\Cal M}_{g,n}$的几何结果的启发，本文利用一般曲线的适当平面模型，确定了$g$和$n$的小值的双有理结构.更准确地说，我们证明了${\Cal M}_{g,n}$对于$g=2$和$1\le n\le 12$，$g=3$和$1\le n\le 14$，$g=4$和$1\le n\le 15$，$g=5$和$1\le n\le 12$是合理的。
Rationality of moduli of vector bundles on curves
证明了亏格g>1的光滑射影曲线上稳定的秩r、度d的向量丛的模空间M（r，d）对M（h，0）x A是双有理的，其中h=hcf（r，d），A是维数为（r^2-h^2）（g-1）的仿射空间.双有理同构与M（r，d）和M（h，0）中的不动行列式相容，并作为推论得到当r和d互质时，秩为r的丛与d次不动行列式的模空间是有理的.在证明中的一个关键因素是使用一个自然定义的布劳尔类的功能领域的M（r，d）。
About the equivalence of divisor classes on hyperelliptic curves and a quotient of linear forms by an algebraic group action
对于亏格为$g$的超椭圆曲线，给出了一个在$g+1$次一般位置上的因子的多项式方程.有一个行动，从代数群的代表性的因子多项式固定的因子类。这种结构将因子类的合理性问题简化为多项式的合理性问题，更容易控制。
Rationality of Moduli Spaces of Parabolic Bundles
证明了亏格大于1的光滑曲线上行列式固定的抛物丛的模空间是有理的，只要与拟抛物结构相关的重数之一等于1.证明了若秩与度互质，则向量丛的模空间是稳定有理的，并且在该水平上得到的界强到足以在许多情况下得出有理的结论。
Prym varieties and moduli of polarized Nikulin surfaces
本文给出亏格为7的Prym曲线模空间R7作为7结点有理曲线模空间上的射影丛的一个结构定理。这一参数化的存在性暗示了R_7和亏格为7的Nikulin曲面的模空间的单合理性，以及亏格为7的Nikulin曲面的模空间的有区别线的合理性.利用亏格7中的结果，我们建立了R_8是单圆的.
Universal relations on stable map spaces in genus zero
本文对稳定映射的模空间之间的映射引入了一种因子分解，这种分解忽略了一个标记点。这导致了研究的普遍关系的上同调稳定的映射空间亏格为零。
On the rationality of $SU(r,d)$
设SU（r，d）是亏格为$g\ge 2$的光滑射影曲线上的秩r，度d向量丛的模空间.如果（r，d）=1且d整除r+1，则SU是有理数。此外，如果$0<δ< r$和$δ$的所有素因子整除r，并且如果d整除$r-δ$，则SU是有理数。证明是一个变化的结果纽斯特德和修改由于巴利科，然后博登和横河。
On universal Severi varieties of low genus K3 surfaces
我们证明了不可约的通用Severi品种parametrizing不可约，减少，节超平面部分的原始K3曲面的亏格g，与3 \le g \le 11，g \neq 10。
Universal cohomological expressions for singularities in families of genus 0 stable maps
我们认为家庭的曲线到曲线的地图，没有奇异性，除了那些属0稳定的地图，并满足一个多角性条件，在每个奇点。我们提供了一个通用的表达式的上同调类庞加莱对偶的轨迹的任何给定的奇点。我们的表达式适用于满足上述性质的任何曲线到曲线映射族。
Universal equations for higher genus Gromov-Witten invariants from Hodge integrals
本文通过研究曲线模空间上Hodge丛的陈特征与陈类，建立了靶流形高亏格Gromov-Witten不变量的新普适方程，并由此导出稳定曲线模空间上的新前推关系。
Rational Chow ring of the universal moduli space of semistable rank two bundles over genus two curves
本文确定亏格 2 曲线上秩 2 半稳定丛泛模空间的有理 Chow 环，证明其由重言类生成，并顺带得到相关泛 Jacobian 的 Chow 环。
Primitively polarized genus $g$ Nikulin surfaces $(S,M,H)$ are of two types, that we call standard and non-standard depending on whether the lattice embedding $\mathbb{Z}[H] \oplus_{\perp} \mathbf{N} \subset \rm{Pic}(S)$ is primitive. Here $H$ is the genus $g$ polarization and $\mathbf{N}$ is the Nikulin lattice. We concentrate on the non-standard case, which only occurs in odd genus. In particular, we study the birational geometry of the moduli space of non-standard Nikulin surfaces of genus $g$ and prove its rationality for $g=7,11$ and the existence of a rational double cover of it when $g=…
Cycles in the universal moduli stack of bundles of rank two over genus two curves
本文提出并证明了双曲曲线上的 universal moduli 堆叠的 Chow 环猜想，确定了生成元和关系，并计算了 genus 二曲线的 universal Jacobians 乘积的 Chow 环。
We classify all positive integers n and r such that (stably) non-rational complex r-fold quadric bundles over rational n-folds exist. We show in particular that for any n and r, a wide class of smooth r-fold quadric bundles over projective n-space are not stably rational if r lies in the interval from $2^{n-1}-1$ to $2^{n}-2$. In our proofs we introduce a generalization of the specialization method of Voisin and Colliot-Thélène--Pirutka which avoids universally $CH_0$-trivial resolutions of singularities.
We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2,3) in P^2 x P^3 is not stably rational. Via projections onto the two factors, X is a cubic surface bundle over P^2 and a conic bundle over P^3, and we analyze the stable rationality problem from both these points of view. This provides another example of a smooth family of rationally connected fourfolds with rational and nonrational fibers. Finally, we in…
Let $\mathfrak p$ be any point in the moduli space of genus-two curves $\mathcal M_2$ and $K$ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal C_{α, β}$ defined over $K(α, β)$, corresponding to $\mathfrak p$, where $α$ and $β$ satisfy a quadratic $α^2+ b β^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{α, β}$ is defined over the field of moduli $K$ if and only if the quadratic has a $K$-rational point $(α, β)$. We discover some interesting symmetries of the Weierstrass equation of $\mathcal C_{α, β}$. …
For each universal genus-$g$ polarization $μ$ of degree $d$, we construct a universal tropical Jacobian $J_{μ,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several properties of the space $J_{μ,g}^{trop}$. In particular, we prove that the natural compactification of $J_{μ,g}^{trop}$ is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus-$g$ curves.
We bound the dimensions of the graded pieces of the tautological ring of the universal curve from below for genus up to 27 and from above for genus up to 9. As a consequence we obtain the precise structure of the tautological ring of the universal curve for genus up to 9. In particular, we see that it is Gorenstein for these genera.
The singularities and birational geometry of the universal compactified Jacobian
本文证明了当亏格至少为4时，泛紧化雅可比矩阵的奇点是正则的。作为一个推论，我们确定的Kodaira尺寸和Iitaka纤维化的普遍紧雅可比矩阵的每一个度和属。我们还确定了双有理自同构群的每度，如果亏格至少是12。这推广了G. Farkas和A. Verra和G.比尼角Fontanari和第三作者。
A remark on the Moduli of parabolic connections
本文研究亏格至少为3的紧Riemann曲面上有理一般权重抛物联络的模空间，确定底层抛物丛稳定情形的周群，并讨论其有理性与有理连通性。
We construct a moduli space for Riemann surfaces that is universal in the sense that it represents compact Riemann surfaces of any finite genus. This moduli space is stratifed according to genus, and it carries a metric and a measure that induce a Riemannian metric and a finite volume measure on each stratum. Applications to the Plateau-Douglas problem for minimal surfaces of varying genus and to the partition function of Bosonic string theory are outlined. The construction starts with a universal moduli space of Abelian varieties. This space carries a structure of an infinite dimensional loca…
The Universal Severi Variety of Rational Curves on K3 Surfaces
研究了K3曲面上的有理曲线的泛Severi簇，它在固定亏格的不同K3曲面上的固定类中参数化不可约有理曲线.利用稳定映射的变形理论研究了该空间在本原类情形下的截断不可约性，得到了部分结果。在附录中我们给出了K3曲面上高亏格曲线的万有Severi簇的不可约性的一个简短证明。
We compute the Brauer group of the universal moduli stack of vector bundles on (possibly marked) smooth curves of genus at least three over the complex numbers. As consequence, we obtain an explicit description of the Brauer group of the smooth locus of the associated moduli space of semistable vector bundles, when the genus is at least four.
We introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus 1 and prove combinatorial classification results for fine compactified Jacobians in the case of a single nodal curve and in the case of the universal family over the moduli space of stable pointed curves. We show that if the fine compactified Jacobian of a nodal curve of genus 1 can be extended to a smoothing of the curve, then it can be described as the moduli space of stable sheaves with respect to some polarisation. In the universal case we construct new examples of ge…
The Chow ring of $\mathcal{S}_5^-$ is tautological
本文证明了亏格5的奇自旋曲线模空间$\mathcal{S}_5^-$的Chow环是重言的，通过分析典范亏格5曲线和全切超平面的几何性质，并证明了$\mathcal{M}_{5,4}$中支配$\mathcal{S}_5^-$的微分层的理性。
Universal polynomials for tropical refined invariants in genus 0
本文证明了亏格0热带精细不变量中小编余度系数存在仅依赖于亏格和余度的通用多项式，并给出显式公式，证明基于地板图操作。
We carry out a complete birational classification of the universal theta divisor Th_g over the moduli space of curves of genus g, and show that Th_g enjoys good rationality properties for g<12, and is a variety of general type for g\geq 12. The key ingredient is an intersection-theoretic study of the universal antiramification locus of the Gauss map. We also present a complete classification of the universal symmetric product of degree g-2 over M_g.
Using the connection discovered by Hassett between the Noether-Lefschetz moduli space of special cubic fourfolds of discriminant 42 and the moduli space F_{22} of polarized K3 surfaces of genus 22, we show that the universal K3 surface over F_{22} is unirational.
Towards large genus asymtotics of intersection numbers on moduli spaces of curves
我们显式地计算了$n$ -尖复代数曲线的模空间的Weil-Petersson体积的大亏格渐近中的发散因子。模一个普遍的乘法常数，我们证明了一个完整的渐近展开的Weil-Petersson体积的逆幂的属的系数是多项式在$n$。这是通过分析更一般的重言式类的相交数的各种递归来完成的，重言式类的大亏格渐近行为也被广泛研究。
Let $X$ be a del Pezzo surface of degree $5$ defined over a field $F$. A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree $5$ is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree $5$ over a field $F$ are precisely the twisted $F$-forms of the moduli space $\overline{M_{0, 5}}$ of stable curves of genus $0$ with $5$ marked points. Suppose $n \geq 5$ is an integer, and $F$ is an infinite field of characteristic $\neq 2$. It is easy to see that every twisted $F$-form of $\overline{M_{0, n}}$ is unira…
We will prove that given a genus-2 fibration $f: X \rightarrow C$ on a smooth projective surface $X$ such that $b_1(X)=b_1(C)+2$, the fundamental group of $X$ is almost isomorphic to $π_1(C) \times π_1(E)$, where $E$ is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces $X$ with genus-2 fibration $X\rightarrow C$ such that $b_1(X)>b_1(C)$.
We construct the universal Mumford curve of given genus as a family of Mumford curves over the deformation space of degenerate curves in the category of arithmetic formal geometry. Furthermore, we give explicit formulas of abelian differentials and their periods of the universal Mumford curve.
On O'Grady's generalized Franchetta conjecture for genus 11 K3 surfaces
本文通过 Mukai 程序和 tautological 生成结果，证实了 genus 11 K3 表面的广义 Franchetta 猜想。
Toric genera of homogeneous spaces and their fibrations
本文的目的是进一步研究紧致齐性空间的泛环面亏格及其齐性纤维化。考虑具有正欧拉示性数的齐性空间。在极大环面$T^k$的规范作用下，这类空间具有许多稳定的等变复结构。由于这种情况下的环面作用只有孤立的不动点，因此可以有效地应用泛环面亏格的局部化公式。在此基础上，我们证明了在齐性空间上，仅用表示论就可以得到与Hirzebruch亏格的刚性和乘法性有关的著名拓扑结果.在这方面的齐性$SU$ -空间，我们证明了著名的结果刚性的Krichever属。我们还证明了对于一大类稳定的复齐性空间，任何由奇幂级数给出的$T^k$ -等变Hirzebruch亏格为零.与问题的乘法性，我们提供了稳定的复杂的$T^k$ -纤维化的泛环面属是扭曲乘法的建设。我们证明了它总是扭曲乘法几乎复杂的齐次纤维化，并描述这些纤维化，它是乘法。因此，这种纤维化的刚性和乘法之间的强关系的等变Hirzebruch属建立。还考虑了基不允许任何稳定复结构的纤维化的泛环面亏格。这里我们计算泛环面亏格的主要例子是四元数射影空间上的齐次纤维化。
(Uni)rational parametrizations of $\mathcal R_{g,2}$, $\mathcal R_{g,4}$ and $\mathcal R_{g,6}$ in low genera
本文研究了亏格为 $g$ 的光滑曲线在偶数个点处分歧的双重覆盖模空间 $R_{g,2n}$，并证明了低亏格下 $R_{g,2}$、$R_{g,4}$ 和 $R_{g,6}$ 的（单）有理性。
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