Indranil Biswas, Jishnu Biswas, G. V. Ravindra
We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.
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For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero.
We prove that a nodal quartic threefold $X$ containing no planes is $Q$-factorial provided that it has not more than 12 singular points, with the exception of a quartic with exactly 12 singularities containing a quadric surface. We give some geometrical constructions related to the latter quartic.
The quartic hypersurfaces in P^4 invariant under the standard representation of S_6 form a linear pencil. We prove that a general member of this pencil is not rational.
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 $t=\frac{7}{10}$.
We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi-Yau threefolds, including their Hodge numbers and deformation theory.
The $\mathbb{Q}$-factoriality of a nodal quartic 3-fold implies its non-rationality. We prove that a nodal quartic 3-fold with at most 8 nodes is $\mathbb{Q}$-factorial, and we show that a nodal quartic 3-fold with 9 nodes is not $\mathbb{Q}$-factorial if and only if it contains a plane. However, there are non-rational non-$\mathbb{Q}$-factorial nodal quartic 3-folds in $\mathbb{P}^4$. In particular, we prove the non-rationality of a general non-$\mathbb{Q}$-factorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.
A remark on rational quartic curves in prime Fano threefolds of degree $22$
利用度22的素Fano三维流形$V_{22}$与五次del Pezzo三维流形$V_5$之间的Sarkisov联系，证明$V_{22}$中有理四次曲线的Hilbert方案到射影空间$\mathbb{P}^4$存在一个一般2对1的有理映射。
We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all $S_6$-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman-Edge pencil. As an application, we check that $S_6$-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class grou…
Quartic curves in the quintic del Pezzo threefold
本文证明五次 del Pezzo 三维簇 X5 上有理四次曲线的 Hilbert 概形 H4(X5) 同构于 X5 上直线的 Hilbert 概形上的 Grassmannian 丛，因此光滑且不可约。
Degenerations of elliptic quartics and K-moduli of Fano threefolds
本文研究经两次爆破构造的法诺三维簇的K-模空间，将其同构于带CM线丛线性化的VGIT商空间，分类了该形变族的K-(半/多)稳定成员，相关证明用到希尔伯特概型几何等新要素。
It is well-known that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an elliptic curve. It is also known that a factorial nodal quartic threefold is birationally rigid and its group of birational selfmaps is generated by biregular ones and certain birational involutions. We prove that all relations between these involutions are implied by standard relations on elliptic curves, complete …
Moduli space of genus one curves on quartic and quintic del Pezzo threefolds
本文研究四次和五次 del Pezzo 三维簇上光滑亏格一曲线的空间，描述了 Kontsevich 模空间的不可约分支并分类了从一般椭圆曲线出发的态射空间，验证了复数域上所有四次和五次 del Pezzo 三维簇的几何 Manin 猜想。
We study the mixed Hodge structure on the third homology group of a threefold which is the double cover of projective three-space ramified over a quartic surface with a double conic. We deal with the Torelli problem for such threefolds.
We prove the Shafarevich conjecture for Fano threefolds of Picard rank 1, index 1 and degree 4.
In this paper, we prove a Clifford type inequality for the curve $X_{2,2,2,4}$, which is the intersection of a quartic and three general quadratics in $\mathbb{P}^5$. We thus prove a stronger Bogomolov-Gieseker inequality for characters of stable vector bundles and stable objects on $X_{2,4}$. Applying the scheme proposed by Bayer, Bertram, Macrì, Stellari and Toda, we can construct an open subset of Bridgeland stability conditions on $X_{2,4}$.
We study noncompact Calabi-Yau threefolds, their moduli spaces of vector bundles and deformation theory. We present Calabi-Yau threefolds that have infinitely many distinct deformations, constructing them explicitily, and describe the effect that such deformations produce on moduli spaces of vector bundles.
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 $g\geq 2$.
K-moduli of Fano threefolds in family 3.10
研究3.10家族中光滑Fano三维流形的K-多项稳定极限，揭示其几何结构与稳定性条件。
In this paper we study ACM vector bundles $\E$ of rank $k \geq 3$ on hypersurfaces $X_r \subset\Pj^4$ of degree $r \geq 1$. We consider here mainly the case of degree $r = 4$, which is the first unknown case in literature. Under some natural conditions for the bundle $\E$ we derive a list of possible Chern classes $(c_1,c_2,c_3)$ which may arise in the cases of rank $k=3$ and $k=4$, when $r=4$. For some cases among these we give the corresponding examples, the existence of all the other cases remaining under question.
We study the Hilbert scheme $\mathcal{H}$ of twisted cubics on a special smooth Gushel-Mukai threefolds $X_{10}$. We show that it is a smooth irreducible projective threefold if $X_{10}$ is general among special Gushel-Mukai threefolds, while it is singular if $X_{10}$ is not general. We construct an irreducible component of a moduli space of Bridgeland stable objects in the Kuznetsov component of $X_{10}$ as a divisorial contraction of $\mathcal{H}$. We also identify the minimal model of Fano surface $\mathcal{C}(X_{10}')$ of conics on a smooth ordinary Gushel-Mukai threefold with moduli spac…
K-moduli of Fano threefolds of rank 2 and degree 28
本文针对2.21族的光滑K-多稳定法诺三维簇，找出其所有K-多稳定退化情形，描述对应模空间分支，属于代数几何中模空间的研究内容。
We give uniformizations of the Klein quartic curve and the Fermat septic curve as Shimura curves parametrizing Abelian $6$-folds with endomorphisms $\ZZ[\z_7]$.
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.
We shall show the existence of 15 automorphic forms of weight 8 on the moduli space of marked Hessian quartic surfaces of cubic surfaces. These automorphic forms can be interpreted in terms of the coefficients of the Sylvester form of a general cubic surface.
We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.
A Finiteness Theorem for Quartic K3-Fibred Calabi--Yau Threefolds in Scrolls
本文针对堆卷中以四次K3曲面为纤维的Calabi-Yau三维簇，应用Reid-Shepherd-Barron-Tai准则分类了n=1,2,3的情况，共得到14个形变族，补充了相关显式分类数据。
It is known that the moduli space of plane quartic curves is birational to an arithmetic quotient of a 6-dimensional complex ball. In this paper, we shall show that there exists a 15-dimensional space of meromorphic automorphic forms on the complex ball which gives a birational embedding of the moduli space of plane quartics with level 2 structure into 14-dimensional projective space. This map coincides with the one given by Coble by using Gopel invariants.
A note on geometry of Bridgeland Moduli spaces on Gushel-Mukai threefolds
本文研究古谢尔-穆凯三维簇的库兹涅佐夫分量上半稳定对象的布里奇兰德模空间，证明其正规性，并在特定数值条件下证明对应模空间是整的。
We prove several classification results for the components of the moduli space of rational curves on a smooth Fano threefold. In particular, we prove a conjecture of Batyrev on the growth of the number of components as the degree increases. The key to our approach is Geometric Manin's Conjecture which predicts the number of components parameterizing free curves.
The quartic threefold is symplectically irrational
本文证明光滑四次三维流形是辛非有理的，即不能通过一系列辛爆破、爆缩和形变与射影空间相关联，从而恢复了Iskovskikh-Manin的经典代数非有理结果，证明中使用了Iritani的工作建立了沿辛爆破的量子上同调分解定理。
As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic elements are, respectively, sheaves supported on rational quartic curves, on elliptic quartic curves, or on planar quartic curves. The main idea of the proof is to relate the moduli space with the Hilbert scheme of curves by wall crossing. We present all stable sheaves contained in the intersections of the three irreducible components. We also classify stable sheav…
A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves.
In the case of the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics, the subvariety of sheaves that are not locally free on their support is connected, singular, and has codimension 2.
We study two rational Fano threefolds with an action of the icosahedral group $\mathfrak{A}_5$. The first one is the famous Burkhardt quartic threefold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are $\mathfrak{A}_5$-Fano varieties that are $\mathfrak{A}_5$-birationally superrigid. This gives two new embeddings of the group $\mathfrak{A}_5$ into the space Cremona group.
The Abel-Jacobi map of the family of elliptic quintics lying on a general cubic threefold is studied. It is proved that it factors through a moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c_1=0, c_2=2, whose general point represents a vector bundle obtained by Serre's construction from an elliptic quintic. The elliptic quintics mapped to a point of the moduli space vary in a 5-dimensional projective space inside the Hilbert scheme of curves, and the map from the moduli space to the intermediate Jacobian is étale. As auxiliary results, the irreducibil…
We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2,3,5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological obstruction that measures if a given twist is birational to a moduli space of abelian varieties. This obstruction has implications for the rational points on these varieties. As a result, we see that all possible 3-level structures can be realized by abelian surfaces, whereas Kummer 3-level structures that group-theoretically may be admissible, may not be realizable…
Derived categories of quartic double fivefolds
该研究通过扭曲的Calabi-Yau三重线构造了四次双五重线的导出范畴，并验证了Kuznetsov的有理性猜想和Reid的非交换幻想。
We study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.
We study semistable sheaves of rank $2$ with Chern classes $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-fold $V_5$ of Picard number $1$, degree $5$ and index $2$. We show that the moduli space of such sheaves has a component that is isomorphic to $\mathbb{P}^5$ by identifying it with the moduli space of semistable quiver representations. This provides a natural smooth compactification of the moduli space of minimal instantons, as well as Ulrich bundles of rank $2$ on $V_5$.
A moduli space of stable sheaves on a cubic threefold
本文研究光滑三次三维簇X上特定陈示性类的Gieseker半稳定层的模空间，证明其非空、光滑、不可约且维数为8，还将其与X的Kuznetsov分支中的Bridgeland模空间联系并给出集合论描述。
A global description of the fine Simpson moduli spaces of $1$-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Drézet and Maican and show that the Simpson moduli space $M=M_{4m\pm 1}(\mathbb P_2)$ is a blow-down of a blow-up of a projective bundle over a smooth moduli space of Kronecker modules. An easy computation of the Poincaré polynomial of $M$ is presented.
We study semi-stable sheaves of rank $2$ with Chern class $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-folds $V_4$ of Picard number $1$, degree $4$ and index $2$. We show the moduli space of such sheaves is isomorphic to the moduli space of semi-stable rank $2$ even degree vector bundles on a genus $2$ curve. This provides a natural smooth compatification of the moduli space of Ulrich bundles of rank $2$ on $V_4$.
On the local geometry of the moduli space of $(2,2)$-threefolds in ${\mathcal A}_9$
研究 ${\mathbb P}^2 \times {\mathbb P}^2$ 中 $(2,2)$-三维簇中间雅可比模空间的局部几何，证明 Siegel 度量第二基本形式与乘法映射的复合为非零全纯截面，并描述其核。
Higher dimensional moduli spaces on Kuznetsov components of Fano threefolds
本文研究 Fano 三重形 Kuznetsov 分量中稳定对象的高维模空间，建立非空性判据，分析三次三重形模空间及其 Abel–Jacobi 纤维，并证明相关 Lagrangian 子簇猜想。
K-Moduli of Fano Threefolds of Family 3.3
本文显式刻画了Fano三维簇族3.3的K-模空间，通过降低体积界限并利用格极化K3曲面方法，将其等同于GIT商的Kirwan爆破。
We study the moduli space of triples $(C, L_1, L_2)$ consisting of quartic curves $C$ and lines $L_1$ and $L_2$. Specifically, we construct and compactify the moduli space in two ways: via geometric invariant theory (GIT) and by using the period map of certain lattice polarized $K3$ surfaces. The GIT construction depends on two parameters $t_1$ and $t_2$ which correspond to the choice of a linearization. For $t_1=t_2=1$ we describe the GIT moduli explicitly and relate it to the construction via $K3$ surfaces.
The intermediate Jacobian map, which associates to a smooth cubic threefold its intermediate Jacobian, does not extend to the GIT compactification of the space of cubic threefolds, not even as a map to the Satake compactification of the moduli space of principally polarized abelian fivefolds. A much better "wonderful" compactification of the space of cubic threefolds was constructed by the first and fourth authors --- it has a modular interpretation, and divisorial normal crossing boundary. We prove that the intermediate Jacobian map extends to a morphism from the wonderful compactification to…
We decompose each moduli space of semistable sheaves on the complex projective plane with support of dimension one and degree four into locally closed subvarieties, each subvariety being the good or geometric quotient of a set of morphisms of locally free sheaves modulo a reductive or a nonreductive group. We find locally free resolutions of length one of all these sheaves and describe them.
Chow and cohomology rings of moduli stacks of plane quartics
本文研究了四次曲线模栈Hacking模栈的Chow环和上同调环，通过堆叠理论加权吹胀构造了一个光滑proper Deligne-Mumford堆栈，解决了KSBA和K-模空间之间的Calabi-Yau墙交叉问题，并计算了该模栈的Poincaré多项式，证明了循环类映射在有理系数下是同构的，并确定了其Chow环的生成元和关系。
We describe a natural open stratum in the moduli space of smooth real pointed quartic curves in the projective plane. This stratum consists of real isomorphism classes of pairs $(C, p)$ with $p$ a real point on the curve $C$ such that the tangent line at $p$ intersects the curve in two distinct points besides $p$. We will prove that this stratum consists of $20$ connected components. Each of these components has a real toric structure defined by an involution in the Weyl group of type $E_7$.
K-moduli of Fano threefolds and genus four curves
本文研究沿$\mathbb{P}^3$中$(2,3)$完全交曲线吹胀得到的法诺三维簇的K-模空间，证明其是$\overline{M}_4$的Hassett--Keel程序模型之一，分类了该形变族中所有K-（半/多）稳定成员，采用模连续性方法并引入新成分。
Motivated by the computation of the BPS-invariants on a local Calabi-Yau threefold suggested by S. Katz, we compute the Chow ring and the cohomology ring of the moduli space of stable sheaves of Hilbert polynomial $4m+1$ on the projective plane. As a byproduct, we obtain the total Chern class and Euler characteristics of all line bundles, which provide a numerical data for the strange duality on the plane.
The Abel-Jacobi maps of the families of elliptic quintics and rational quartics lying on a smooth cubic threefold are studied. It is proved that their generic fiber is the 5-dimensional projective space for quintics, and a smooth 3-dimensional variety birational to the cubic itself for quartics. The paper is a continuation of the recent work of Markushevich-Tikhomirov, who showed that the first Abel-Jacobi map factors through the moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers $c_1=0, c_2=2$ obtained by Serre's construction from elliptic quintics, and…
We modify the standard perfect symmetric obstruction theory for moduli spaces of simple perfect complexes, to the situation of complexes on abelian threefolds with fixed determinant and Fourier-Mukai determinant. As outcome we attach nontrivial Donaldson-Thomas invariants to moduli spaces for complexes modulo twist and translation.
On K-moduli of Fano threefolds with degree 28 and Picard rank 4
本文研究次数28、Picard秩4的Fano三维簇的K-模，通过构造奇异环面K-多稳定Fano三维簇的代数形变，证明对应光滑Fano三维簇的K-模不可约分支为有理曲面。
K-stable Fano threefolds of rank 2 and degree 28
本文证明了除明确例外情况外，所有Picard秩2且次数为28的光滑Fano三维流形均为K-多稳定，并对光滑二次三维流形中有理正规四次曲线的法丛进行了分类。
Arithmetic aspects of the Burkhardt quartic threefold
本文研究 Burkhardt 四次三维簇的算术性质，证明其在特征非 3 域上有理、计算有限域 zeta 函数，并显式给出 genus 2 泛曲线模型及 j-平面与 3 阶子群、Hesse 铅笔的关系。
We study the moduli space of pairs $(X,H)$ consisting of a cubic threefold $X$ and a hyperplane $H$ in $\mathbb P^4$. The interest in this moduli comes from two sources: the study of certain weighted hypersurfaces whose middle cohomology admit Hodge structures of $K3$ type and, on the other hand, the study of the singularity $O_{16}$ (the cone over a cubic surface). In this paper, we give a Hodge theoretic construction of the moduli space of cubic pairs by relating $(X,H)$ to certain "lattice polarized" cubic fourfolds $Y$. A period map for the pairs $(X,H)$ is then defined using the periods o…
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}.
Zero-cycles on Moduli Spaces of Twisted Sheaves and Applications to Double EPW Quartics
本文研究了扭曲sheaves模空间上的极小循环，并将其应用于双EPW四次曲线，通过扩展Vial和Martin-Vial的结果，证明了有效极小循环在特定条件下的等价性，以及扭曲Beauville-Voisin类与普通Beauville-Voisin类的一致性。
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