Rationality of the moduli spaces of plane curves of sufficiently large degree

On the rationality of the moduli space of instanton bundles on the projective 3-space

We prove the rationality and irreducibility of the moduli space of---what we call---the endomorphism-general instanton vector bundles of arbitrary rank on the projective space. In particular, we deduce the rationality of the moduli spaces of rank-two mathematical instantons. This problem was first studied by Hartshorne, Hirschowitz-Narasimhan in the late 1970s, and it has been reiterated within the framework of the ICM 2018.

Extensions of vector bundles and rationality of certain moduli spaces of stable bundles

In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over the projective plane P^2.

The rationality of the moduli space of two-pointed ineffective spin hyperelliptic curves

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 $g \geq 2$.

The rationality of the moduli space of one-pointed ineffective spin hyperelliptic curves via an almost del Pezzo threefold

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 the moduli spaces of 2-elementary K3 surfaces

K3 surfaces with non-symplectic involution are classified by open sets of seventy-five arithmetic quotients of type IV. We prove that those moduli spaces are rational except two classical cases.

Rationality and Brauer group of a moduli space of framed bundles

We prove that the moduli spaces of framed bundles over a smooth projective curve are rational. We compute the Brauer group of these moduli spaces to be zero under some assumption on the stability parameter.

On the Rationality of moduli spaces of vector bundles over chain-like curves

Let $C$ be a chain-like curve over $\mathbb{C}$. In this paper, we investigate the rationality of moduli spaces of $w$-semistable vector bundles on $C$ of arbitrary rank and fixed determinant by putting some restrictions on the Euler characteristics.

Rationality of the moduli spaces of Eisenstein K3 surfaces

K3 surfaces with non-symplectic symmetry of order 3 are classified by open sets of twenty-four complex ball quotients associated to Eisenstein lattices. We show that twenty-two of those moduli spaces are rational.

Moduli stacks of vector bundles on curves and the King-Schofield rationality proof

This mainly expository text translates into stack language the proof of King and Schofield for the rationality of moduli schemes of vector bundles on a curve in the coprime case. An appendix summarizes some basic properties of the relevant moduli stacks.

The rationality of the moduli space of curves of genus 3 after P. Katsylo

This article is a survey of P. Katsylo's proof that the moduli space of smooth projective complex curves of genus 3 is rational. We hope to make the argument more comprehensible and transparent by emphasizing the underlying geometry in the proof and its key structural features.

Rationality of Moduli space over reducible curve

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.

Rationality of moduli spaces of plane curves of small degree

We prove that the moduli space C(d) of plane curves of degree d (for projective equivalence) is rational except possibly if d= 6, 7, 8, 11, 12, 14, 15, 16, 18, 20, 23, 24, 26, 32, 48.

The rationality of certain moduli spaces of curves of genus 3

We show, for each algebraically closed field, the rationality of the following two moduli spaces: M(3,3) parametrizing pairs (C, η) where C has genus 3 and ηis a 3-torsion divisor class, respectively of M(3,<3>) parametrizing pairs (C, <η>) as above and where <η> is the cyclic subgroup of order 3 in Pic_0(C) generated by η.

On the rationality of the moduli of higher spin curves in low genus

The global geometry of the moduli spaces of higher spin curves and their birational classification is largely unknown for g >= 2 and r > 2. Using quite related geometric constructions, we almost complete the picture of the known results in genus g <= 4 showing the rationality of the moduli spaces of even and odd 4-spin curves of genus 3, of odd spin curves of genus 4 and of 3-spin curves of genus 4.

On the rationality of the moduli space of Lüroth quartics

We prove that the moduli space M_L of L"uroth quartics in P^2, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL_3(CC) is rational, as is the related moduli space of Bateman seven-tuples of points in P^2.

瞬子模空间的合理性

Rationality of instanton moduli

本文在Tikhomirov关于奇数荷瞬子模空间不可约性结果的基础上，证明了瞬子模空间的合理性，从而得到所有奇数荷及荷2、4的瞬子模空间均为有理的。

The rationality of the moduli space of genus four curves endowed with an order three subgroup of their Jacobian

Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and on a theorem of Dolgachev, the rationality of the moduli space R mentioned in the title is proved. Let M be the moduli space of 6 points in the plane, modulo the natural involution induced by double-six configurations on cubic surfaces. It is proved that R is birational to a tower of locally trivial projective bundles ending onto M. The rationality of R then…

Rationality of the Moduli Variety of Curves of Genus 3

We prove rationality of the moduli variety of curves of genus 3.

稳定映射的Kontsevich模空间的合理性及其伽罗瓦下降

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。

The Rationality Problem in Invariant Theory

This is a survey of the rationality problem in invariant theory. It also contains some new results, in particular in Chapter 2 on moduli spaces of plane curves with a theta-characteristic, and a detailed account of the relation of the Hesselink stratification of the Hilbert nullcone to the rationality problem, with an application to the rationality of the moduli space of plane curves of degree 34.

Rationality of moduli of vector bundles on curves

The moduli space M(r,d) of stable, rank r, degree d vector bundles on a smooth projective curve of genus g>1 is shown to be birational to M(h,0) x A, where h=hcf(r,d) and A is affine space of dimension (r^2-h^2)(g-1). The birational isomorphism is compatible with fixing determinants in M(r,d) and M(h,0) and we obtain as a corollary that the moduli space of bundles of rank r and fixed determinant of degree d is rational, when r and d are coprime. A key ingredient in the proof is the use of a naturally defined Brauer class for the function field of M(r,d).

The connectedness of the moduli space of maps to homogeneous spaces

We prove the connectedness of the moduli space of maps (of fixed genus and homology class) to the homogeneous space G/P by degeneration via the maximal torus action. In the genus 0 case, the irreducibility of the moduli of maps is a direct consequence of connectedness. An analysis of a related Bialynicki-Birula stratification of the map space yields a rationality result: the (coarse) moduli space of genus 0 maps to G/P is a rational variety. The rationality argument depends essentially upon rationality results for quotients of SL2 representations proven by Katsylo and Bogomolov.

Rationality and Poincare families for vector bundles with extra structure on a curve

Iterated Grassmannian bundles over moduli stacks of vector bundles on a curve are shown to be birational to an affine space times a moduli stack of degree 0 vector bundles, following the method of King and Schofield. Applications include the birational type of some Brill-Noether loci, of moduli schemes for vector bundles with parabolic structure or with level structure and for A. Schmitt's decorated vector bundles. A further consequence concerns the existence of Poincare families on finite coverings of the moduli schemes.

射影3空间上瞬子丛模空间的有理性与范畴性质

Rationality and categorical properties of the moduli of instanton bundles on the projective 3-space

本文证明了射影3空间上任意秩和荷的数学瞬子丛模空间的有理性与不可约性，并揭示其构成幺半范畴，基于Barth-Hulek monad构造与Hirzebruch曲面稳定丛模空间分析。

On the rationality of moduli spaces of pointed curves

Motivated by several recent results on the geometry of the moduli spaces $\bar{\Cal M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points, here we determine their birational structure for small values of $g$ and $n$ by exploiting suitable plane models of the general curve. More precisely, we show that ${\Cal M}_{g,n}$ is rational for $g=2$ and $1\le n\le 12$, $g=3$ and $1\le n\le 14$, $g=4$ and $1\le n\le 15$, $g=5$ and $1\le n\le 12$.

Geometry of genus 8 Nikulin surfaces and rationality of their moduli

Let S be a general complex Nikulin surface of genus 8, a geometric construction of S is given as follows. Consider a smooth 3-fold linear section T of the Grassmannian G(1,4) and the Hilbert scheme of rational normal sextic curves of T. In it consider the special family of sextics A which are also contained in the congruence of bisecant lines to a rational normal quartic curve of P^4. We show that S is biregular to a quadratic section of T containing a sextic A. In particular A admits a 1-dimensional family of bisecant lines contained in G(1,4) and 8 of them are in S. This explicit constructio…

Rationality of the universal K3 surface of genus 8

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 $k \leq n$.

Rationality of Moduli Spaces of Parabolic Bundles

The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities associated to the quasi-parabolic structure is equal to one. It follows that if rank and degree are coprime, the moduli space of vector bundles is stably rational, and the bound obtained on the level is strong enough to conclude rationality in many cases.

具有两个 $\frac12(1,1,1)$-点的亏格 $8$ Fano $3$-折的模空间：有理性和 K-稳定性

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。

Rationality of moduli spaces of stable bundles on curves over $\mathbb{R}$

Let $C$ be a smooth, projective, geometrically irreducible curve defined over $\mathbb{R}$ such that $C(\mathbb{R}) = \emptyset$. Let $r>0$ and $d$ be integers which are coprime. Let $L$ be a line bundle on $C$ which corresponds to an $\mathbb{R}$ point of ${\rm Pic}^d_{C/\mathbb{R}}$. Let $\mathcal{M}_{r,L}$ be the moduli space of stable bundles on the complexification of $C$ of rank $r$ and determinant $L$. We classify birational types of $\mathcal{M}_{r,L}$ over $\mathbb{R}$.

On two rationality conjectures for cubic fourfolds

Motivated by the question of rationality of cubic fourfolds, we show that a cubic X has an associated K3 surface in the sense of Hassett if and only if the variety F of lines on X is birational to a moduli space of sheaves on a K3 surface, but that having F birational to Hilb^2(K3) is more restrictive. We compare the loci in the moduli space of cubics where each condition is satisfied.

Rationality and Chow-Kuenneth decompositions for some moduli stacks of curves

In this paper, we show the existence of a Chow--Kuenneth decomposition for the moduli stack of stable curves of genus g with r marked points, for low values of g,r. We also look at the moduli space R of double covers of genus 3 curves, branched along 4 distinct points. We obtain a birational model of the moduli space R as a group quotient of a product of two Grassmanian varieties. This provides a Chow-Kuenneth decomposition over an open subset of R. The question of rationality of R is also discussed.

$\overline{\mathcal M}_{0,n}$ 形式的有理性

Rationality of forms of $\overline{\mathcal M}_{0,n}$

研究射影直线上点模空间在置换扭转下的等变几何与有理性，证明其有理性质。

Rationality of some tetragonal loci

We prove that the moduli space of tetragonal curves of genus g>6 is rational when g is congruent to 1, 2, 5, 6, 9, 10 modulo 12 and not equal to 9, 45.

On stable rationality of some conic bundles and moduli spaces of Prym curves

We prove that a very general hypersurface of bidegree (2, n) in P^2 x P^2 for n bigger than or equal to 2 is not stably rational, using Voisin's method of integral Chow-theoretic decompositions of the diagonal and their preservation under mild degenerations. At the same time, we also analyse possible ways to degenerate Prym curves, and the way how various loci inside the moduli space of stable Prym curves are nested. No deformation theory of stacks or sheaves of Azumaya algebras like in recent work of Hasset-Kresch-Tschinkel is used, rather we employ a more elementary and explicit approach via…

Stable rationality of Brauer-Severi surface bundles

For sufficiently ample linear systems on rational surfaces we show that a very general associated Brauer-Severi surface bundle is not stably rational.

Ulrich 丛的 Koszul 性质与 Del Pezzo 曲面上稳定丛模空间的有理性

Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces

本文研究 Del Pezzo 曲面上 Ulrich 丛的 Koszul 性质与迭代 syzygy 丛的斜率半稳定性，并证明无穷多个稳定丛模空间具有有理性，为 Costa 和 Miró-Roig 猜想提供新证据。

Series of rational moduli components of stable rank 2 vector bundles on $\mathbb{P}^3$

We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M(e,n)$ of rank 2 stable vector bundles with the first Chern class $e=0$ or -1 and all possible values of the second Chern class $n$ on the projective 3-space. The generalized null correlation bundles constituting open dense subsets of these components are defined as cohomology bundles of monads whose members are direct sums of line bundles of degrees depending on nonnegative integers $a,b,c$, where $b\ge a$ and $c>a+b$. We show that, in the wide range w…

阿贝尔簇模空间的有理性与算术

Rationality and arithmetic of the moduli of abelian varieties

本文研究主极化阿贝尔簇模空间的有理性，证明低维提升与单有理性质，并应用于算术，得出无条件结果。

无效 spin 亏格 4 thetanull 轨迹的有理性

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。

Motivic measures of moduli spaces of 1-dimensional sheaves on rational surfaces

We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by computing their motivic measures.

Peskine簇的有理性

Rationality of peskine varieties

本文研究了P^9中Peskine六重折叠的有理性，证明了特定除子情形下的有理性，并提出了一个上同调条件及涉及Debarre-Voisin hyperkähler四重折叠的猜想。

The universal K3 surface of genus 14 via cubic fourfolds

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.

Cobordism invariants of the moduli space of stable pairs

For a quasi-projective scheme M which carries a perfect obstruction theory, we construct the virtual cobordism class of M. If M is projective, we prove that the corresponding Chern numbers of the virtual cobordism class are given by integrals of the Chern classes of the virtual tangent bundle. Further, we study cobordism invariants of the moduli space of stable pairs introduced by Pandharipande-Thomas. Rationality of the partition function is conjectured together with a functional equation, which can be regarded as a generalization of the rationality and 1/q <-> q symmetry of the Calabi-Yau ca…

Dynamical Systems of Correspondences on the Projective Line I: Moduli Spaces and Multiplier Maps

We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We obtain the rationality of the moduli spaces. The rationality of the moduli space of degree $(d,e)$ correspondences is obtained from a representation-theoretic projection to the one for the usual dynamical systems of degree $d+e-1$. We also show that the multiplier maps for the fixed points and the multiplier index theorem (Woods Hole formula) are also reduced through the projection and obtain the reduced…

关于抛物联络模空间的周群

On the Chow group of 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。

Rank-3 stable bundles on rational ruled surfaces

In this paper, we compare the moduli spaces of rank-3 vector bundles stable with respect to different ample divisors over rational ruled surfaces. We also discuss the irreducibility, unirationality, and rationality of these moduli spaces.

判别式24的三次四重簇与有理性质

Cubic fourfolds of discriminant 24 and rationality

本文研究判别式24的三次四重簇，通过其扭曲曲面中Brauer类消失的条件，证明了有理性质，并构造了可数无穷多个新的有理例子。

On rational curves in n-space with given normal bundle

The stable rationality of components of the moduli space of (unparametrized) rational curves in projective $n$-space with fixed normal bundle is proved, provided these components dominate the moduli space of immersed rational curves in the plane.

关于曲面上稳定对模空间的虚拟欧拉示性数

On the Virtual Euler Characteristic of the Moduli Space of Stable Pairs on Surfaces

本文研究了环面上稳定对理论，确定了固定点轨迹上的虚拟切空间，提出计算虚拟欧拉示性数的程序并以射影平面为例说明，通过特例验证支持了有理性与对称性猜想。

Brauer group and birational type of moduli spaces of torsionfree sheaves on a nodal curve

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 $U'_Y(n,d)$.

Prym varieties and moduli of polarized Nikulin surfaces

We present a structure theorem for the moduli space R_7 of Prym curves of genus 7 as a projective bundle over the moduli space of 7-nodal rational curves. The existence of this parametrization implies the unirationality of R_7 and that of the moduli space of Nikulin surfaces of genus 7, as well as the rationality of the moduli space of Nikulin surfaces of genus 7 with a distinguished line. Using the results in genus 7, we then establish that R_8 is uniruled.

On the rationality problem for forms of moduli spaces of stable marked curves of positive genus

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…

The moduli space of (1,11)-polarized abelian surfaces is unirational

We prove that the moduli space A_{11}^{lev} of (1,11) polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface: a^2b+b^2c+c^2d+d^2e+e^2a=0 in P^4. Therefore, A_{11}^{lev} is unirational but not rational, and there are no Gamma_{11}-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of A_{9}^{lev}.