Mark Gross, Sorin Popescu
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}.
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The new result is the unirationality of the moduli space of curves of genus 14. The method applies to lower genus.
Unirationality of instanton moduli space for small charges
本文证明了射影3-空间上数学瞬子模空间在电荷≤7时的单有理性质，解决了小电荷情形下该模空间的单有理问题。
We illustrate the use of the computer algebra system Macaulay2 for simplifications of classical unirationality proofs. We explicitly treat the moduli spaces of curves of genus g=10, 12 and 14.
We show that the coarse moduli space $\cR_5$ of étale double covers of curves of genus~5 over the complex numbers is unirational. We give two slightly different arguments, one purely geometric and the other more computational.
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.
We prove that the moduli spaces of K3 surfaces with non-symplectic involutions are unirational. As a by-product we describe configuration spaces of 4<d<9 points in the projective plane as arithmetic quotients of type IV.
We show that $\mathcal{M}_{g,n}$, the moduli space of smooth curves of genus $g$ together with $n$ marked points, is unirational for $g=12$ and $2 \leq n\leq 4$ and for $g=13$ and $1 \leq n \leq 3$, by constructing suitable dominant families of projective curves in $\mathbb{P}^1 \times \mathbb{P}^2$ and $\mathbb{P}^3$ respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genus $g$ together with $n$ unordered points, establishing their unirationality for $g=11, n=7$ and $g=12, n =5,6$.
We provide methods to construct explicit examples of $K3$ surfaces. This leads to unirational constructions of Noether--Lefschetz divisors inside the moduli space of $K3$ surfaces of genus $g$. We implement Mukai's unirationality construction of the moduli spaces of $K3$ surfaces of genus $g\in\{6,\dots, 10, 12\}$, and we also present a new constructive proof of the unirationality of the moduli space of $K3$ surfaces of genus $11$. Furthermore, we show the existence of three unirational hypersurfaces in any moduli space of $K3$ surfaces of genus $g$.
We provide explicit descriptions of the generic members of Hassett's divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and for $d=44$. In doing so, we prove that $\mathcal C_d$ is unirational for $18\leq d\leq 38,d=44$. As a corollary, we prove that the moduli space $\mathcal N_{d}$ of polarized K3 surfaces of degree $d$ is unirational for $d=14,26,38$. The case $d=26$ is entirely new, while the other two cases have been previously proven by Mukai.
In this note we look at the moduli space $\cR_{3,2}$ of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to {\mathcal A}_4$ to Siegel space. We show that there is a birational model of $\cR_{3,2}$ as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of $\cR_{3,2}$ and hence a new proof for the unirationality of ${\mathcal A}_4$.
We show that the moduli space of $U\oplus \langle -2k \rangle$-polarized K3 surfaces is unirational for $k \le 50$ and $k \notin \{11,35,42,48\}$, and for other several values of $k$ up to $k=97$. Our proof is based on a systematic study of the projective models of elliptic K3 surfaces in $\mathbb{P}^n$ for $3\le n \le 5$ containing either the union of two rational curves or the union of a rational and an elliptic curve intersecting at one point.
Unirationality of the universal moduli space of semistable bundles over smooth curves
本文通过构造从有理簇上的射影丛到模空间的显式支配有理态射，证明了秩、次数任意且亏格在2到9之间的半稳定向量丛万有模空间 $U_{r,d,g}$ 的单有理性，并将结论推广到 $U_{r,\mathcal{L},g}$。
The unirationality of $S_9^-$ and moduli spaces of pointed spin curves
本文研究了 genus 9 的奇旋曲线模空间的合理性，并展示了通过局部平凡射影包将其与标定点的奇稳定旋曲线模空间联系起来，从而得到最高 genus 的合理性结果，同时探讨了标定点旋曲线模空间的 Kodaira 维数。
Irreducible unirational and uniruled components of moduli spaces of polarized Enriques surfaces
本文利用极化为各向同性类有效和的分解刻画极化 Enriques 曲面模空间分支，并证明存在无穷多个单有理或单直纹分支，涵盖亏格与 ϕ-不变量任意大的情形。
We show that the moduli space of SU_X(r,L) of rank r bundles of fixed determinant L on a smooth projective curve X is separably unirational.
We combine the idea of Chang and Ran [Invent. Math. 76 (1984), 41-54] of using monads of vector bundles on the projective 3-space to prove the unirationality of the moduli spaces of curves of low genus with our classification of globally generated vector bundles with small first Chern class $c_1$ on the projective 3-space to get an alternative argument for the unirationality of the moduli spaces of curves of degree at most 13 (based on the general framework of Chang and Ran).
We prove that the moduli space A_3(1,1,4) of polarized abelian threefolds with polarization of type (1,1,4) is unirational. By a result of Birkenhake and Lange this implies the unirationality of the isomorphic moduli space A_3(1,4,4). The result is based on the study the Hurwitz space H_{4,n}(Y) of quadruple coverings of an elliptic curve Y simply branched in n points. We prove the unirationality of its codimension one subvariety H^{0}_{4,A}(Y) which parametrizes quadruple coverings π:X --> Y with Tschirnhausen modules isomorphic to A^{-1}, where A\in Pic^{n/2}Y, and for which π^*:J(Y)--> J(X)…
We study the Picard groups of moduli spaces in positive characteristics and we give a "$p$-adic" proof that the Picard group of moduli of vector bundles of fixed determinant is isomorphic to the group of integers. Along the way we prove that the local fundamental group scheme of a normal unirational projective variety is trivial. This is reminiscent of results of Serre and Nygaard who studied the fundamental groups of smooth, projective, unirational varieties.
We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle.
The bielliptic locus in the Hilbert scheme of canonical curves is unirational
本文证明了亏格 $g \geq 11$ 的典范曲线 Hilbert 概型中双椭圆曲线轨迹的单有理性，并以此给出了亏格 $g \geq 11$ 曲线模空间中双椭圆轨迹单有理性的另一证明。
We prove that the moduli space of cubic fourfolds $\mathcal{C}$ contains a divisor $\mathcal{C}_{42}$ whose general member has a unirational parametrization of degree 13. This result follows from a thorough study of the Hilbert scheme of rational scrolls and an explicit construction of examples. We also show that $\mathcal{C}_{42}$ is uniruled.
We prove that the moduli space of polarized $K3$ surfaces of genus eleven with $n$ marked points is unirational when $n\leq 6$ and uniruled when $n\leq7$. As a consequence, we settle a long standing but not proved assertion about the unirationality of $\cal{M}_{11,n}$ for $n\leq6$. We also prove that the moduli space of polarized $K3$ surfaces of genus eleven with $9$ marked points has non-negative Kodaira dimension.
In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many further unirationality results for special moduli spaces of rational curves on quadric hypersurfaces and del Pezzo surfaces are obtained via explicit birational methods.
We find certain relations between flag Hilbert schemes of points on plane curves and moduli spaces of one-dimensional plane sheaves. We show that some of these moduli spaces are unirational.
The geometry of the moduli space of odd spin curves
该研究明确了亏格g<12时奇自旋曲线模空间S_g⁻为单直纹簇、g≥12时为一般型，且亏格g<9时S_g⁻是单有理的，还对比了亏格8时其与偶自旋对应空间的类型差异。
We prove that the moduli spaces A_3(D) of polarized abelian threefolds with polarizations of types D=(1,1,2), (1,2,2), (1,1,3) or (1,3,3) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree 2 or 3 and on the study of the corresponding period mappings associated with holomorphic differentials with trace 0. In particular we prove the unirationality of the Hurwitz space H_{3,A}(Y) which parameterizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to A^{-1}.
We study the moduli space of minimal surfaces of general type with $K_S^2 = 1$ and $p_g = 2$ and show that it is irreducible, has dimension $28$ and admits a compactification which is unirational.
We prove that the moduli space of smooth primitively polarized $\mathrm{K3}$ surfaces of genus 33 is unirational.
We describe birational models and decide the rationality/unirationality of moduli spaces AA_{d} (and AA^{lev}_{d}) of (1,d)-polarized abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi-Yau threefolds with Euler characteristic zero.
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 $n = 1$.
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.
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 show that supersingular K3 surfaces in characteristic $p\geq5$ are related sequences of very special correspondences. This is not enough to conclude that they are unirational. As a byproduct, we exhibit a fibration structure on the moduli space of rigidified K3 crystals. We also establish Shioda-Inose type isogeny theorems for K3 surfaces with Picard rank $ρ\geq19$ in positive characteristic.
The Noether inequality for threefolds and three moduli spaces with minimal volumes
本文建立了几何亏格5≤p_g≤10的一般型射影三维簇的诺特不等式，解决三维簇诺特不等式所有剩余情形，并证明三个特定极小体积典范三维簇的粗模空间是不可约单有理簇且成员单连通。
Let $X$ be an irreducible smooth complex projective curve of genus at least two. Let $N$ be a connected component of the moduli space of semistable principal ${\rm PGL}_r({\mathbb C})$- bundles over $X$; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of $N$ is trivial.
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.
We discuss the Picard group of moduli space $\mathcal{K}_g$ of quasi-polarized K3 surfaces of genus $g\leq 12$ and $g\neq 11$. In this range, $\mathcal{K}_g$ is unirational and a general element in $\mathcal{K}_g$ is a complete intersection with respect to a vector bundle on a homogenous space, by the work of Mukai. In this paper, we find generators of the Picard group $Pic_\mathbb{Q}(\mathcal{K}_g)$ using Noether-Lefschetz theory. This verifies the Noether-Lefschetz conjecture on moduli of K3 surfaces in these cases.
This is the sequel to arXiv:math/0001089. In this paper, we complete the promised description of moduli of abelian surfaces of low degree, covering the cases of degree (1,12), (1,14), (1,16), (1,18) and (1,20). In each case, we describe birational models for the moduli space of abelian surfaces with polarization of the given degree and either a level structure, or what we call a partial level structure. In every case, the moduli space is rational or unirational. In additional, several new Calabi-Yau threefolds fibred in abelian surfaces are described.
We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, i.e. $U\oplus \langle -2k \rangle$-polarized K3 surfaces. Such moduli spaces are proved to be of general type for $k\geq 220$. The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead to the unirationality of these moduli spaces for $k < 11$ and for 19 other isolated values up to $k=64$.
Smooth primitively polarized $\mathrm{K3}$ surfaces of genus 36 are studied. It is proved that all such surfaces $S$, for which there exists an embedding $\mathrm{R} \hookrightarrow \mathrm{Pic}(S)$ of some special lattice $\mathrm{R}$ of rank 2, are parameterized up to an isomorphism by some 18-dimensional unirational algebraic variety. More precisely, it is shown that a general $S$ is an anticanonical section of a (unique) Fano 3-fold with canonical Gorenstein singularities.
In this paper we study the minimal surfaces of general type with $p_g=q=1$ and $K^2=4$ whose Albanese general fibre has genus 2, classifying those such that the direct image (under the Albanese morphism) of the bicanonical sheaf is sum of line bundles. We find 8 unirational families, all of dimension strictly bigger than the expected one. These families are pairwise disjoint irreducible components of the moduli space of minimal surfaces of general type.
Let A be the moduli space of principally polarized abelian varieties of dimension 4 over an algebraically closed field k of characteristic different from 2,3. It is proved that the universal principally polarized abelian variety over A, as well as the universal theta divisor over A, are unirational varieties.
Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties
本文证明了Gushel-Mukai簇Kuznetsov分量上Bridgeland稳定性条件的存在性，描述了偶数维情形下的模空间结构，并据此构造了K3型超凯勒簇的新无限系列族及Hodge理论等价判据。
In this paper, we prove the moduli spaces of genus zero stable log maps to a large class of wonderful compactifications are irreducible and unirational.
We carry out a complete birational classification of the degree g universal Jacobian P_g over the moduli space of curves, highlighting the transition cases g=10, 11. The universal Jacobian is unirational when g<10, has Kodaira dimension zero for g=10 and Kodaira dimension 19 for g=11. For g>11, the variety P_g has Kodaira dimension 3g-3, that is, the maximum allowed by Iitaka's easy addition formula for fibre spaces. In particular, we disprove the expectation that P_g and M_g have the same Kodaira dimension for all genera.
A note on some moduli spaces of Ulrich Bundles
本文研究Hirzebruch曲面上三维scroll的Ulrich向量丛模空间，证明了两类模空间的一般光滑性与单有理性，其中相伴模空间还满足不可约性，推进了Ulrich丛模空间的几何性质研究。
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…
Using a new description of Keum Naie surfaces and their fundamental group, we prove the following main result: Let S be a smooth complex projective surface which is homotopically equivalent to a Keum - Naie surface. Then S is a Keum - Naie surface. The connected component of the Gieseker moduli space corresponding to Keum - Naie surfaces is irreducible, normal, unirational of dimension 6.
We establish a structure result for the universal abelian variety over the moduli space A_5, in terms of discriminant curves of conic bundles over a del Pezzo surface. In particular, this gives a very simple unirational parametrization of the boundary divisor of the moduli space A_6 of principally polarized abelian varieties of dimension 6. Using this we derive a lower bound on the slope of all effective divisors on A_6.
This paper investigates the geometry of smooth canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces. In particular, an explicit real valued function f(x) is obtained such that if $X$ is a smooth canonically polarized surface defined over an algebraically closed field of characteristic p>0 such that $K_X^2 <f(p)$, then $X$ is unirational and the order of its algebraic fundamental group is at most two. As a con…
We prove that the irreducible components of the moduli space of polarized Calabi-Yau pairs are projective.
For a hypersurface in complex projective space $X\subset \PP^n$, we investigate the singularities and Kodaira dimension of the Kontsevich moduli spaces $\Kbm{0,0}{X,e}$ parametrizing rational curves of degree $e$ on $X$. If $d+e \leq n$ and $X$ is a general hypersurface of degree $d$, we prove that $\Kbm{0,0}{X,e}$ has only canonical singularities and we conjecture the same is true for the coarse moduli space $\kbm{0,0}{X,e}$.This investigation is motivated by the question of which Fano hypersurfaces are unirational.
In this paper we consider moduli spaces of polarized and numerically polarized Enriques surfaces. The moduli spaces of numerically polarized Enriques surfaces can be described as open subsets of orthogonal modular varieties of dimension 10. One of the consequences of our description is that there are only finitely many isomorphism classes of moduli spaces of polarized and numerically polarized Enriques surfaces. We use modular forms to prove for a number of small degrees that the Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative. Finally we prove that…
We will calculate the essential dimension of the moduli stack of polarized K3 surfaces, including the positive char case.
We prove the infinitesimal Torelli theorem for general minimal complex surfaces X's with the first Chern number 3, the geometric genus 1, and the irregularity 0 which have non-trivial 3-torsion divisors. We also show that the coarse moduli space for surfaces with the invariants as above is a 14-dimensional unirational variety.
In this note it is shown that associating to a principally polarized abelian variety its de Rham cohomology defines a faithfully flat morphism of the moduli space of principally polarized abelian varieties in positive characteristics to the moduli space of symplectic $F$-Zips as defined in \cite{MW}.
For a smooth canonically embedded curve $C$ of genus $9$ together with a pencil $|L|$ of degree $6$, we study the relative canonical resolution of $C\subset X\subset \mathbb{P}^8$, where $X$ is the scroll swept out by the pencil $|L|$. We show that the second syzygy bundle in this resolution of $C\subset X$ is unbalanced. The proof reveals a new geometric connection between the universal Brill--Noether variety $\mathcal{W}^1_{9,6}$ and a moduli space $\mathcal{F}^\mathfrak{h}$ of lattice polarized $K3$ surfaces (for a certain rank $3$ lattice $\mathfrak{h}$). As a by-product we prove the unira…
We study the global structure of moduli spaces of quasi-isogenies of polarized p-divisible groups introduced by Rapoport and Zink. Using the corresponding results for non-polarized p-divisible groups from a previous paper, we determine their dimensions and their sets of connected components and of irreducible components.
We discuss topics on the geometry of the moduli space of curves. We present a short proof of the Harris-Mumford theorem on the Kodaira dimension of the moduli space which replaces the computations on the stack of admissible covers by a simple study of tautological Koszul bundles on M_g. We also present a streamlined self-contained account of Verra's recent work on the unirationality of M_g. Finally, we discuss a proof that the moduli space of curves of genus 22 is of general type. Written for Surveys in Differential Geometry.
We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.
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