L. Chiantini, C. Madonna
We give a partial positive answer to a conjecture of Tyurin (\cite {Tyu}). Indeed we prove that on a general quintic hypersurface of $\Pj^4$ every arithmetically Cohen--Macaulay rank 2 vector bundle is infinitesimally rigid.
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Let $X$ be a smooth quintic hypersurface in $\mathbb{P}^3$, let $C$ be a smooth hyperplane section of $X$, and let $H=\mathcal{O}_X(C)$. In this paper, we give a necessary and sufficient condition for the line bundle given by a non-zero effective divisor on $X$ to be initialized and aCM with respect to $H$.
We study automorphism groups of smooth quintic threefolds. Especially, we describe all the maximal ones with explicit examples of target quintic threefolds. There are exactly $22$ such groups.
We study the geometry of quintic threefolds $X\subset \mathbb{P}^4$ with only ordinary triple points as singularities. In particular, we show that if a quintic threefold $X$ has a reducible hyperplane section then $X$ has at most $10$ ordinary triple points, and that this bound is sharp. We construct various examples of quintic threefolds with triple points and discuss their defect.
In this article, we study the geometric invariant theory (GIT) compactification of quintic threefolds. We study singularities, which arise in non-stable quintic threefolds, thus giving a partial description of the stable locus. We also give an explicit description of the boundary components and stratification of the GIT compactification.
We prove a conjecture of Jockers-Mayr and Garoufalidis-Scheidegger, relating genus zero quantum $K$-invariants and Gopakumar--Vafa invariants on the quintic threefold.
We prove the following statement, predicted by Clemens' conjecture: A generic quintic threefold contains only finitely many smooth rational curves of degree 12.
We study the Clifford type inequality for a particular type of curves $C_{2,2,5}$, which are contained in smooth quintic threefolds. This allows us to prove some stronger Bogomolov-Gieseker type inequalities for Chern characters of stable sheaves and tilt-stable objects on smooth quintic threefolds. Employing the previous framework by Bayer, Bertram, Macrì, Stellari and Toda, we construct an open subset of stability conditions on every smooth quintic threefold in $\mathbf{P}^4_{\mathbb C}$.
Our earlier proof of mirror formulas for genus 0 Gromov -- Witten invariants of Fano and Calabi -- Yau toric complete intersections is illustrated in the example of quintic 3-folds.
We give some necessary conditions for a smooth irreducible curve $C\subset \mathbb{P}^4$ to be isolated in a smooth quintic threefold, and also find a lower bound for $h^1(\mathcal{N}_{C/{\mathbb{P}^4}})$. Combining these with beautiful results in Castelnuovo theory, we prove certain non-existence results on smooth curves in smooth quintic threefolds. As an application, we can prove Knutsen's list of examples of smooth isolated curves in general quintic threefolds is complete up to degree 9.
We show that every morphism from a degree 5 hypersurface in 4-dimensional projective space to a nonsingular degree 3 hypersurface in 4-dimensional projective space is necessarily constant. In the process, we also classify morphisms from the projective plane to nonsingular cubic threefolds given by degree 3 polynomials.
The aim of this paper is to classify indecomposable rank 2 arithmetically Cohen-Macaulay (ACM) bundles on compete intersection Calabi-Yau (CICY) threefolds and prove the existence of some of them. New geometric properties of the curves corresponding to rank 2 ACM bundles (by Serre correspondence) are obtained. These follow from minimal free resolutions of curves in suitably chosen fourfolds (containing Calabi-Yau threefolds as hypersurfaces). Also the existence of an indecomposable vector bundle of higher rank on a CICY threefold of type (2,4) is proved.
In this paper, we call a sub-scheme of dimension one in $\mathbb{P}^3$ a curve. It is well known that the arithmetic genus and the degree of an aCM curve $D$ in $\mathbb{P}^3$ is computed by the $h$-vector of $D$. However, for a given curve $D$ in $\mathbb{P}^3$, the two invariants of $D$ do not tell us whether $D$ is aCM or not. In this paper, we give a classification of curves on a smooth quintic hypersurface in $\mathbb{P}^3$ which are not aCM.
Quintic del Pezzo threefolds in positive and mixed characteristic
该研究对任意基概形上的光滑五次del Pezzo三重曲面进行分类，描述其相关几何结构并发现特征2下的新现象，还给出Shafarevich猜想的精细化及整数环上的同构类结果。
We study type III contractions of Calabi-Yau threefolds containing a ruled surface over a smooth curve. We discuss the conditions necessary for the image threefold to by smoothable. We describe the change in Hodge numbers caused by this contraction and smoothing deformation. A generalization of a fomula for calculating Hodge numbers of hypersurfaces in $\mathbb{P}^4$ with ordinary double and triple points is presented. We use these results to construct new Calabi-Yau threefolds of Picard rank two arising from a family of quintic threefolds containing a cone.
We present explicit equations for the space of conics in the Fermat quintic threefold $X$, working within the space of plane sections of $X$ with two singular marked points. This space of two-pointed singular plane sections has a birational morphism to the space of bitangent lines to the Fermat quintic threefold, which in its turn is birational to a 625-to-1 cover of $\PP^4.$ We illustrate the use of the resulting equations in identifying special cases of one-dimensional families of conics in $X.$
In the paper ``Chirality change in string theory'', by Douglas and Zhou, the authors give a list of bundles on a quintic Calabi-Yau threefold. Here we prove the semistability of most of these bundles. This provides examples of string theory compactifications which have a different number of generations and can be connected.
The moduli stack of Mixed Spin P-fields (MSP) provides an effective algorithm to evaluate all genus Gromov-Witten invariants of quintic Calabi-Yau threefolds. This paper is to apply the algorithm in genus one case. We use the localization formula, the proposed algorithm in [CLLL1, CLLL2], and Zinger's packaging technique to compute the genus one Gromov-Witten invariants of quintic Calabi-Yau threefolds. New hypergeometric series identities are also discovered in the process.
Nodal quintic del Pezzo threefolds and their derived categories
本文为节点五次 del Pezzo 三维簇构造川又型半正交分解，将其导出范畴分解为有限维代数的导出范畴，通过双有理映射到二次曲面纤维化实现。
Stronger Bogomolov--Gieseker type inequality on quintic threefold
本文通过结合细化限制定理与Clifford型界限，为五次三次曲面的斜半稳定sheaf建立了更强的Bogomolov-Gieseker不等式，并暗示Toda的猜想不等式。
Motivated by the general question of existence of open A1-cylinders in higher dimensional pro-jective varieties, we consider the case of Mori Fiber Spaces of relative dimension three, whose general closed fibers are isomorphic to the quintic del Pezzo threefold V5 , the smooth Fano threefold of index two and degree five. We show that the total spaces of these Mori Fiber Spaces always contain relative A2-cylinders, and we characterize those admitting relative A3-cylinders in terms of the existence of certain special lines in their generic fibers.
The purpose of the article is to give a proof of a conjecture of Maulik and Pandharipande for genus 2 and 3. As a result, it gives a way to determine Gromov-Witten invariants of the quintic threefold for genus 2 and 3.
We prove the genus-one restriction of the all-genus Landau-Ginzburg/Calabi-Yau conjecture of Chiodo and Ruan, stated in terms of the geometric quantization of an explicit symplectomorphism determined by genus-zero invariants. This provides the first evidence supporting the higher-genus Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold, and exhibits the first instance of the "genus zero controls higher genus" principle, in the sense of Givental's quantization formalism, for non-semisimple cohomological field theories.
Let $X_0$ be a generic quintic threefold in projective space $\mathbf P^4$ over complex numbers and $C_0$ be an irreducible rational curve on $X_0$. Let $$c_0: \mathbf P^1\to C_0\subset X_0$$ be its normalization. In this paper, we show (1) $c_0$ must be an immersion, i.e. the differential $(c_0)_\ast: T_{t}\mathbf P^1 \to T_{c_0(t)} X_0$ is injective at each $t\in \mathbf P^1$, (2) the normal bundle of $c_0$ satisfies $$H^1(N_{c_0/X_0})=0.$$
We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in P^4, there are only finitely many smooth rational curves of degree 10, and each curve is embedded in F with normal bundle O(-1)^2. Moreover, in degree 10, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components in F.
We give a description of the relative Hilbert scheme of lines in the Dwork pencil of quintic threefolds. We describe the corresponding relative Hilbert scheme associated to the mirror family of quintic threefolds.
For all $m > 0$ we build a two-dimensional family of smooth manifolds of real dimension $3m + 2$ and use it to interpolate between the anticanonical family in the complex projective space of dimension $m + 1$ and its mirror dual. The main tool is the notion of self-dual manifold.
Quadratic types and the dynamic Euler number of lines on a quintic threefold
本文为五次三维流形上直线的局部贡献提供几何解释，定义动态欧拉数计算 A^1-欧拉数，并通过 Fermat 五次三维流形上 2875 条特殊直线的二次计数，得出一般五次三维流形上直线型的总和在 GW(k) 中为 1445<1> + 1430<-1>。
We study ACM bundles on cubic threefolds by using derived category techniques. We prove that the moduli space of stable Ulrich bundles of any rank is always non-empty by showing that it is birational to a moduli space of semistable torsion sheaves on the projective plane endowed with the action of a Clifford algebra. We describe this birational isomorphism via wall-crossing in the space of Bridgeland stability conditions, in the example of instanton sheaves of minimal charge.
We correct the definitions and descriptions of the integral structures in [U14]. The previous flat basis in [ibid] is characterized by the Frobenius solutions and integral in the first approximation by mean of the graded quotients of monodromy filtration, but not integral in the strict sense. In this article, we use the integral structure of Iritani in [I11] for A-model. Using this precise version, we study open mirror symmetry for quintic threefolds through log mixed Hodge theory, especially the recent result on Neron models for admissible normal functions with non-torsion extensions in the j…
We consider Calabi-Yau threefolds Y defined as smooth linear sections of the double cover of the quintic symmetric determinantal hypersurface in P^{14}. In our previous works, we have shown that these Calabi-Yau threefolds Y are naturally paired with Reye congruence Calabi-Yau threefolds X, and X and Y have several interesting properties from the view point of mirror symmetry and projective geometry. In this paper, we prove the derived equivalence between Y and X.
We classify indecomposable aCM bundles of rank $2$ on the del Pezzo threefold of degree $7$ and analyze the corresponding moduli spaces.
This is another proof of the same result in [9]. Let $X_0$ be a generic quintic hypersurface in $\mathbf P^4$ over $\mathbb C$ and $c_0$ a regular map $\mathbf P^1\to X_0$ that is generically one-to-one to its image. In this paper, we show (1) $c_0$ must be an immersion, i.e. the differential $(c_0)_\ast: T_{t}\mathbf P^1 \to T_{c_0(t)} X_0$ is injective at each $t\in \mathbf P^1$, (2) the normal bundle of $c_0$ satisfies $$H^1(N_{c_0/X_0})=0.$$
We calculate the matrix of the Frobenius map on the middle dimensional cohomology of the one parameter family that is related by mirror symmetry to the family of all quintic threefolds.
We consider a family of quintic threefolds specializing to a certain reducible threefold. We describe the space of genus zero stable morphisms to the central fiber (as defined by J. Li). As an elementary application of an extension of the analysis, we prove the existence of rigid stable maps of arbitrary genus and sufficiently high degree to very general quintics.
In this paper we derive a list of all the possible indecomposable normalized rank--two vector bundles without intermediate cohomology on the prime Fano threefolds and on the complete intersection Calabi Yau threefolds, say $V$, of Picard number $ρ=1$. For any such bundle $\E$, if it exists, we find the projective invariants of the curves $C \subset V$ which are the zero-locus of general global sections of $\E$. In turn, a curve $C \subset V$ with such invariants is a section of a bundle $\E$ from our lists. This way we reduce the problem for existence of such bundles on $V$ to the problem for …
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。
Let $X_0$ be a generic quintic threefold in projective space $\mathbf P^4$ over the complex numbers. For a fixed natural number $d$, let $R_d(X_0)$ be the open sub-scheme of the Hilbert scheme, parameterizing irreducible rational curves of degree $d$ on $X_0$. In this paper, we show that (1) $R_d(X_0)$ is smooth and of expected dimension, \par (2) Combining the Calabi-Yau condition on $X_0$, we further show that it consists of immersed rational curves. (3) Parts (1) and (2) imply a statement of Clemens' conjecture: if $C_0\in R_d(X_0)$ and $c_0:\mathbf P^1\to C_0$ is the normalization, the \pa…
In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold which is modular, in the sense that the L-function of its middle-dimensional cohomology is associated to a classical modular form of weight 4 and level 55.
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。
By the results of the author and Chiantini in Math.AG/0110102, on a general quintic threefold $X \subset {\mathbf P}^4$ the minimum integer $p$ for which there exists a positive dimensional family of irreducible rank $p$ vector bundles on $X$ without intermediate cohomology is at least three. In this paper we show that $p \leq 4$, by constructing series of positive dimensional families of rank 4 vector bundles on $X$ without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class $Ext^1(E,F)$, for a suitable choice of the rank 2 ACM bundl…
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…
Open/closed Correspondence and Extended LG/CY Correspondence for Quintic Threefolds
本文证明五次三维流形的Walcher盘势可表示为扩展五次GLSM的中心荷，由此建立开/闭对应，并阐明开LG/CY对应与开镜像对称同扩展GLSM的壁穿越及镜像对称的相容性。
The celebrated Mirror Theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of its mirror dual orbifold. In this article, we establish a mirror-dual statement. Namely, the B model of the Fermat quintic threefold is shown to be equivalent to the A model of its mirror, and hence establishes the mirror symmetry as a true duality.
The general quintic hypersurface in ${\mathbb P}^4$ is the most famous example of a Calabi--Yau threefold for which mirror symmetry has been investigated in detail. There is a description of the mirror as a hypersurface in a certain weighted projective space. In this note we present a model for the mirror which is again (the resolution of) a quintic hypersurface in ${\mathbb P}^4$. We also deal with the special members in the respective families. They lead to rigid Calabi--Yau threefolds with interesting arithmetical properties. In the last section we try to find a similarly nice model for the…
The paper investigates some aspects of the geometry and the arithmetic of a non-rigid Calabi-Yau threefold. Particular emphasis is given to the study of its L-function L(H^3,s) and the Galois representation.
Given a smooth non-hyperelliptic prime Fano threefold X, we prove the existence of all rank 2 ACM vector bundles on X by deformation of semistable sheaves. We show that these bundles move in generically smooth components of the corresponding moduli space. We give applications to pfaffian representations of quartic threefolds in P^4 and cubic hypersurfaces of a smooth quadric of P^5.
The derived category of coherent sheaves on a general quintic threefold is a central object in mirror symmetry. We show that it can be embedded into the derived category of a certain Fano elevenfold. Our proof also generates related examples in different dimensions.
Let $X$ be any smooth prime Fano threefold of degree $2g-2$ in $\mathbb{P}^{g+1}$, with $g \in \{3,\ldots,10,12\}$. We prove that for any integer $d$ satisfying $\left\lfloor \frac{g+3}{2} \right\rfloor \leq d \leq g+3$ the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree $d$ in $X$ is nonempty and has a component of dimension $d$, which is furthermore reduced except for the case when $(g,d)=(4,3)$ and $X$ is contained in a singular quadric. Consequently, we deduce that the moduli space of rank--two slope--stable $ACM$ bundles $\mathcal{F}_d$ on $X$ such that $\det(\ma…
Using an explicit resolution of the diagonal for the variety V_5, we provide cohomological characterizations of the universal and quotient bundles. A splitting criterion for bundles over V_5 is also proved. The presentation of semistable aCM (i.e. with no intermediate cohomology) bundles is shown, together with a resolution-theoretic classification of low rank aCM bundles.
We rederive a relation between the genus-one GW-invariants of a quintic threefold in $\Pf$ and the genus-zero and genus-one GW-invariants of $\Pf$. In contrast to the more general derivation in a separate paper, the present derivation relies on a widely believed, but still unproven, statement concerning rigidity of holomorphic curves in Calabi-Yau threefolds. On the other hand, this paper's derivation is more direct and geometric. It requires a bit more effort, but relies on less outside work.
In this paper, we study non-commutative projective schemes whose associated non-commutative graded algebras are finite over their centers. We study their moduli spaces of stable sheaves, and construct a symmetric obstruction theory in the Calabi-Yau-3 case. This allows us to define Donaldson-Thomas type invariants. We also discuss the simplest examples, called quantum Fermat quintic threefolds.
This paper is a continuation of author's previous work arXiv:1911.07949, where we defined Donaldson-Thomas invariants of quantum Fermat threefolds. In this paper, we study the generic quantum Fermat threefold. We give explicit local models for Hilbert schemes of points as quivers with potential, and compute degree zero Donaldson-Thomas invariants. The result is expressed in terms of certain colored plane partitions.
We prove that the incidence scheme of rational curves of degree 11 on quintic threefolds is irreducible. This implies a strong form of the Clemens conjecture in degree 11. Namely, on a general quintic threefold $F$ in $\mathbb{P}^4$, there are only finitely many smooth rational curves of degree 11, and each curve $C$ is embedded in $F$ with normal bundle $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$. Moreover, in degree 11, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on $F$.
We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new $q$-expansion principle for functions on the moduli space of Calabi-Yau manifolds, and the ``mirror symmetry'' phenomenon recently observed by string theorists.
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 derive a closed formula for the generating function of genus two Gromov-Witten invariants of quintic 3-folds and verify the corresponding mirror symmetry conjecture of Bershadsky, Cecotti, Ooguri and Vafa.
We carry out the explicit computations that are used to write down the integrable hierarchy associated with the quintic Calabi-Yau threefold. We also do the calculations for the geometric structures emerging in the Gromov-Witten theory of the quintic, such as the Frobenius manifold structure and the special Kähler structure.
Let k be an algebraically closed field of characteristic 0. A del Pezzo threefold F with maximal Picard number is isomorphic to P^1xP^1xP^1, where P^1 is the projective line over k. In the present paper we completely classify locally free sheaves of rank 2 with vanishing intermediate cohomology over such an F. Such a classification extends similar results proved by E. Arrondo and L. Costa regarding del Pezzo threefolds with Picard number 1.
This is the part II of our series of two papers, "Clemens' conjecture: part I", "Clemens' conjecture: part II". Continuing from part I, in this paper we turn our attention to general quintic threefolds. In a universal quintic threefold X, we construct a family of quasi-regular deformation B_b such that the generic member in this family is non-deviated, but some special member is deviated. By the result from part I, this is impossible unless there is no one parameter family of smooth rational curves in a generic quintic threefold.
We prove the following form of the Clemens conjecture in low degree. Let $d\le9$, and let $F$ be a general quintic threefold in $\IP^4$. Then (1)~the Hilbert scheme of rational, smooth and irreducible curves of degree $d$ on $F$ is finite, nonempty, and reduced; moreover, each curve is embedded in $F$ with normal bundle $Ø(-1)\oplusØ(-1)$, and in $\IP^4$ with maximal rank. (2)~On $F$, there are no rational, singular, reduced and irreducible curves of degree $d$, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3)~On $F$, there are no connected, reduced and reducible …
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