On the rationality of moduli spaces of pointed curves
Gianfranco Casnati, Claudio Fontanari
备注：
Final version accepted for publication on the Journal of the London Mathematical Society (Section 6 of the previous version has been removed since the proof of Claim 6.3 contained a gap)
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 $1\le n\le 12$.
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 $g\geq 2$.
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 $g \geq 2$.
We prove that the moduli spaces of K3 surfaces with non-symplectic involution are rational for four deformation types. With the previous results, this establishes the rationality of those moduli spaces except two classical cases.
We show that each of the irreducible components of moduli of rank 2 torsion-free sheaves with odd Euler characteristic over a reducible nodal curve is rational.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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…
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.
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).
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.
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.
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…
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 $k \leq n$.
We run Mori's program for the moduli space of pointed stable rational curves with divisor $K +\sum a_{i}ψ_{i}$. We prove that, without assuming the F-conjecture, the birational model for the pair is the Hassett's moduli space of weighted pointed stable rational curves, without any modification of weight coefficients.
The aim of this paper is to study some modular contractions of the moduli space of stable pointed curves. These new moduli spaces, which are modular compactifications of the moduli space of smooth pointed curves, are related with the minimal model program for the moduli space of stable pointed curves and have been introduced in a previous work of the authors. We interpret them as log canonical models of adjoints divisors and we then describe the Shokurov decomposition of a region of boundary divisors on the moduli space of stable pointed curves.
The notion of $m/Γ$-pointed stable curves is introduced. It should be viewed as a generalization of the notion of m-pointed stable curves of a given genus, where the labels of the marked points are only determined up to the action of a group of permutations $Γ$. The classical moduli spaces and moduli stacks are generalized to this wider setting. Finally, an explicit construction of the new moduli stack of $m/Γ$-pointed stable curves as a quotient stack is given.
The aim of the present paper is to study the (abstract) Picard group and the Picard group scheme of the moduli stack of stable pointed curves over an arbitrary scheme. As a byproduct, we compute the Picard groups of the moduli stack of stable or smooth pointed curves over a field of characteristic different from two.
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.
This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given.
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 $\mathbb{R}$.
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.
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.
The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation and we study their geometric properties. As a particular case, we recover the first few Hassett-Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.
The object of this note is the moduli spaces of cubic fourfolds (resp., Gushel-Mukai fourfolds) which contain some special rational surfaces. Under some hypotheses on the families of such surfaces, we develop a general method to show the unirationality of the moduli spaces of the $n$-pointed such fourfolds. We apply this to some codimension 1 loci of cubic fourfolds (resp., Gushel-Mukai fourfolds) appeared in the literature recently.
We study the log canonical models of the moduli space MBar_{0,n} of pointed stable genus zero curves with respect to the standard log canonical divisors K+aD, where D denotes the boundary. In particular we show that, as a formal consequence of a conjecture by Fulton regarding the ample cone of MBar_{0,n}, these log canonical models are equal to certain of Hassett's weighted pointed curve spaces.
We construct the Mumford-Knudsen space of n pointed stable rational curves by a sequence of explicit blow-ups from the GIT quotient (P^1)^n//SL(2) with respect to the symmetric linearization O(1,...,1). The intermediate blown-up spaces turn out to be the moduli spaces of weighted pointed stable curves for suitable ranges of weights. As an application, we provide a new unconditional proof of M. Simpson's Theorem about the log canonical models of the Mumford-Knudsen space. We also give a basis of the Picard group of the moduli spaces of weighted pointed stable curves.
We determine the action of the product of symmetric groups on the cohomology of certain moduli of weighted pointed rational curves. The moduli spaces that we study are of stable rational curves with m+n marked points where the first m marked points are distinct from all the others where as the last n may coincide among themselves. We give a recipe for calculating the equivariant Poincaré polynomials and list them for small m and n.
This is the authors doctoral thesis written at the Humboldt-University Berlin. It contains material from the three separate papers: "On the Kodaira dimension of the moduli space of nodal curves", "On quotients of $\overline{\mathcal{M}}_{g,n}$ by certain subgroups of $S_n$" and "The moduli space of hyperelliptic curves with marked points".
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.
We prove that the moduli space ${\Cal M}_{g,n}$ of smooth curves of genus $g$ with $n$ marked points is rational for $g=6$ and $1 \le n \le 8$, and it is unirational for $g=8$ and $1 \le n \le 11$, $g=10$ and $1 \le n \le 3$, $g=12$ and $n = 1$.
We prove that, assuming the F-conjecture, the log canonical model of the pair $(\bar{M}_{0,n}, \sum a_i ψ_i)$ is the Hassett's moduli space of weighted pointed stable rational curves without any modification of weight coefficients. For the boundary weight cases, we prove that the birational model is the GIT quotient of the product of the projective lines. This is a generalization of Simpson's theorem for symmetric weight cases.
A weighted pointed curve consists of a nodal curve and a sequence of marked smooth points, each assigned a number between zero and one. A subset of the marked points may coincide if the sum of the corresponding weights is no greater than one. We construct moduli spaces for these objects using methods of the log minimal model program, and describe the induced birational morphisms between moduli spaces as the weights are varied. In the genus zero case, we explain the connection to Geometric Invariant Theory quotients of points in the projective line, and to compactifications of moduli spaces stu…
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 $\bar{M}_{0,n}$.
The dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the market point is computed for three families of symmetric semigroups of multiplicity six. We also collect the dimensions of such moduli spaces for all semigroups of genus not greater than seven. A question related to an improvement of a Deligne--Pinkham's bound is also formulated, suggesting that the positive graded part of the first module of the cotangent complex associated to a semigroup algebra is a missing invariant. The answer for this question is positive for all these moduli that we k…
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…
