Ali Bajravani
In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill-Noether varieties. Among other results, it is shown that the focal variety of a general, canonical and nonhyperelliptic curve of genus 6 is a hypersurface.
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In this article we apply the classical method of focal loci of families to give a lower bound for the genus of curves lying on general surfaces. First we translate and reprove Xu's result that any curve C on a general surface in P^3 of degree d>4 has geometric genus g > 1 + deg(C)(d - 5)/2. Then we prove a similar lower bound for the curves lying on a general surface in a given component of the Noether-Lefschetz locus in P^3 and on a general projectively Cohen-Macaulay surface in P^4.
This paper surveys the authors recent work on two variable elliptic genus of singular varieties. The last section calculates a generating function for the elliptic genera of symmetric products. This generalizes the classical results of Macdonald and Zagier.
Borisov and Libgober recently proved a conjecture of Dijkgraaf, Moore, Verlinde, and Verlinde on the elliptic genus of a Hilbert scheme of points on a surface. We show how their result can be used together with our work on complex genera of generalised Kummer varieties to deduce a formula for the elliptic genus of a generalised Kummer variety.
We discuss the basic properties of various versions of two variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the elliptic genera of Witten's phases on $N=2$ theories.
We prove the irreducibility of universal Severi varieties parametrizing irreducible, reduced, nodal hyperplane sections of primitive K3 surfaces of genus g, with 3 \le g \le 11, g \neq 10.
We consider the moduli space $A_{pol}(n)$ of (non-principally) polarised abelian varieties of genus $g\geq3$ with coprime polarisation and full level-$n$ structure. Based upon the analysis of the Tits building in math/0405321, we give an explicit lower bound on $n$ that is sufficient for the compactified moduli space to be of general type if one further explicit condition is satisfied.
Given a smooth and projective curve C and a smooth and projective toric variety X, we first describe a compactification of the space of morphisms from C to X representing a fixed homology class, and after we study the intersection theory on this variety.
Integrality of genus zero Gopakumar-Vafa type invariants of semi-positive varieties
本文通过代数几何方法，将半正簇上亏格零 Gopakumar-Vafa 型不变量与量子 K-不变量建立联系，并借助后者的整性给出该整性猜想的另一证明。
Bi-algebraic geometry of strata of differentials in genus zero
本文研究亏格零微分层中满足周期代数关系的代数子簇，证明了Ax-Schanuel与André-Oort型定理，并给出双代数簇的等价刻画及其叶状化性质，同时构造了无穷多新线性簇。
For a nonsingular projective curve $C$ of genus 3 defined over an algebraically closed field of characteristic $p > 2$, we give a necessary and sufficient condition that the Jacobian variety $J(C)$ has a decomposed Richelot isogeny outgoing from it and we determine the structures of decomposed ones.
In this note we describe a unique linear embedding of a prime Fano 4-fold F of genus 10 into the Grassmannian G(3,6). We use this to construct some moduli spaces of bundles on linear sections of F. In particular the moduli space of bundles with Mukai vector (3,L,3) on a generic polarized K3 surface (S,L) of genus 10 is constructed as a double cover of the projective plane branched over a smooth sextic.
An inequality for the Delta-genus of toric varieties
本文针对维数小于5的极化环面簇，利用伴随丛的消没数建立Delta亏格的下界，并证明在伴随丛非消没时Delta亏格至少为n-1。
We construct two pencils of bielliptic curves of genus three and genus five. The first pencil is associated with a general abelian surface with a polarization of type $(1,2)$. The second pencil is related to the first by an unramified double cover, the Prym variety of which is canonically isomorphic to the Jacobian of a very general curve of genus two. Our results are obtained by analyzing suitable elliptic fibrations on the associated Kummer surfaces and rational double covers among them.
In this paper, we construct certain algebraic correspondences between genus three curves and certain type of Calabi-Yau threefolds which is double coverings of three dimensional projective space. Via this correspondences, the first cohomology groups of the curves can be embedded into the third cohomology groups of the Calabi-Yau three folds. Moreover we prove that the cokernel of this inclusion of variations ofHodge structures can not be a factor of any variations of Hodge structures comming from polarized abelian schemes.
We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g=3 into $ SL(2,C), and also of the moduli space of twisted representations. The case of genus g=1,2 has already been done in [http://arxiv.org/abs/1106.6011]. We follow the geometric technique introduced in [http://arxiv.org/abs/1106.6011], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.
The focal locus $Σ_X$ of an affine variety $X$ is roughly speaking the (projective) closure of the set of points $O$ for which there is a smooth point $x \in X$ and a circle with centre $O$ passing through $x$ which osculates $X$ in $x$. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map $ε$ between the Euclidean normal bundle $N_X$ and the projective ambient space ($ε$ sends the normal vector $O-x$ to its endpoint $O$), and in this paper we address two general problems : 1) Characterize the "degenerate" case where the focal locus is not a hypersurface 2) …
Modular differential equations of minimal orders of the elliptic genus of Calabi--Yau varieties
本文研究Calabi-Yau簇椭圆亏格的模微分方程，证明严格六维情形最小阶为四，并为六维超Kähler簇构造7阶方程，刻画非一般解的结构。
A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to write down affine glue equations for the overlaps. Taking multi-projective completions for all of these then yields a complete multi-projective two-dimensional variety whose points are in one-to-one correspondence with degree zero divisor classes on the curve (i.e. the Picard group). This multi-projective variety then becomes a complete projective two-dimension…
We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except for the case of genus $9$, where we assume $n \ge 5$. Furthermore, we prove that Mukai varieties of genus $g \in \{7,8,9,10\}$ and dimension $n \ge 5$ contain cylinders if they have a $\mathsf{k}$-point. Finally, we prove that the embedding $X \hook…
We compute the Hodge-Deligne polynomials of the moduli spaces of representations of the fundamental group of a complex surface into SL(2,C), for the case of small genus g, and allowing the holonomy around a fixed point to be any matrix of SL(2,C), that is Id, -Id, diagonalisable, or of either of the two Jordan types. For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the Hodge-Deligne polynomial under fibrations.
We discuss variations of Hodge structures on abelian varieties that arise from intersecting translates of theta divisors with a special focus on the case of abelian varieties of dimension 4
We show how to calculate the Euler characteristic of a local system associated to an irreducible representation of the symplectic group of genus 3 on the moduli space of curves of genus 3 and the moduli space of principally polarized abelian varieties of dimension 3.
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.
Constructing and characterizing prime $\mathbb{Q}$-Fano threefolds of genus one and with six $1/2(1,1,1)$-singularites via key varieties
针对至多具有1/2(1,1,1)奇点的素Q-Fano三维簇的分类问题，构造了两类亏格1且有6个该奇点的三维簇，通过关键簇Σ实现，证明其与Σ的线性截面对应关系，推进分类研究。
The main outcome of this paper is that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth symplectic 4-fold obtained a deformation of the Hilbert square of a K3 surface of genus 8. After publishing it in Trans. Am. Math. Soc. 353, No.4, 1455-1468 (2001), it was noted to us by Eyal Markman that in Theorem 3.17 we conclude without proof that VSP(F,10) should be the 4-fold of lines on another cubic 4-fold. We correct this in the e-print "Addendum to K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds" (math.AG/061…
Syzygies of secant varieties of curves of genus 2
本文利用Boij-Söderberg理论，通过计算Betti图的特殊位置确定贡献的纯图，完整计算了亏格2曲线的割线簇的Betti图，解决了权重k+1线性链长度未知的问题。
We study various generalisations of rationally connected varieties, allowing the connecting curves to be of higher genus. The main focus will be on free curves $f:C\to X$ with large unobstructed deformation space as originally defined by Kollár, but we also give definitions and basic properties of varieties $X$ covered by a family of curves of a fixed genus $g$ so that through any two general points of $X$ there passes the image of a curve in the family. We prove that the existence of a free curve of genus $g \geq 1$ implies the variety is rationally connected in characteristic zero and initia…
In this note, which is an addendum to the e-print math.AG/9810121, we prove that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth symplectic 4-fold, which is deformation equivalent to the Hilbert square of a K3 surface of genus 8 but different from the family of lines on a cubic 4-fold. This provides a new geometric construction of a compact complex symplectic fourfold, different from a Hilbert square of a K3 surface, a generalized Kummer 4-fold, the variety of lines on a cubic 4-fold and the recent examples of O'Grady (see Duke M…
We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties $S_K$ corresponding to the group $\mbox{GSp}_{4,F}$ over a totally real field $F$, along with the relative Chow motives $^λ\mathcal{V}$ of abelian type over $S_K$ obtained from irreducible representations $V_λ$ of $\mbox{GSp}_{4,F}$. We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight $λ$ which characterises the absence of the middle weights 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, th…
Hodge integrals and $λ_{g}$ conjecture with target varieties
本文提出带靶簇Hodge积分的$\lambda_{g}$猜想，建立其与Virasoro猜想的关系，并在多类情形下证明该猜想，导出后代Gromov-Witten不变量的新型普适约束。
We obtain a formula for the degrees of the varieties parameterizing complex algebraic curves of any divisor class and genus on P^2_6, the projective plane blown-up at 6 generic points. Moreover, the formula computes the degrees of the varieties parameterizing curves on P^2_6 which additionally satisfy certain tangency conditions to a fixed exceptional divisor on P^2_6. Our formula contains as special cases the degrees of the analogous varieties parameterizing curves on P^2_q, for q=1,...,5, and on the quadric (P^1)^2. It is an extension of the Caporaso-Harris formula counting curves of any deg…
An old conjecture of Voisin describes how zero-cycles on a variety $X$ should behave when pulled-back to the self-product $X^m$ for $m$ larger than the geometric genus of $X$. Using complete intersections of quadrics, we give examples of varieties in any dimension and with arbitrarily high geometric genus that verify Voisin's conjecture.
We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal surface (degree, class, class of its hyperplane section, sectional genus and degrees of the nodal and cuspidal curve). We study in particular the congruences of chords to a smooth curve and the congruences of bitangents or flexes to a smooth surface. We find that they possess unexpected components in their focal surface, and conjecture that they are the only on…
Arithmetic finiteness of Mukai varieties of genus 7
该研究针对亏格7的素Fano三维簇及高维Mukai簇的算术有限性展开，得到Torelli定理的算术精细化、Shafarevich型有限性的维数差异结果，并明确了定义在整数环上的这类簇的存在维数范围。
Mukai varieties are Fano varieties of Picard number one and coindex three. In genus seven to ten they are linear sections of some special homogeneous varieties. We describe the generic automorphism groups of these varieties. When they are expected to be trivial for dimensional reasons, we show they are indeed trivial, up to three interesting and unexpected exceptions in genera 7, 8, 9, and codimension 4, 3, 2 respectively. We conclude in particular that a generic prime Fano threefold of genus g has no automorphisms for 7 $\le$ g $\le$ 10. In the Appendix by Y. Prokhorov, the latter statement i…
Using degeneration techniques, we prove the correspondence of tropical curve counts and log Gromov-Witten invariants with general incidence and psi-class conditions in toric varieties for genus zero curves and all non-superabundant higher-genus situations. We also relate the log invariants to the ordinary ones, in particular explaining the appearance of negative multiplicities in the descendant correspondence result of Mark Gross.
We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of these invariants. We then apply our results to curves over imperfect fields. In particular, we establish a genus change formula and prove the boundedness of non-smooth regular curves of genus one. We also compute our invariants for some explicit examples.
We perform a systematic study of Gushel-Mukai varieties---quadratic sections of linear sections of cones over the Grassmannian Gr(2,5). This class of varieties includes Clifford general curves of genus 6, Brill-Noether general polarized K3 surfaces of genus 6, prime Fano threefolds of genus 6, and their higher-dimensional analogues. We establish an intrinsic characterization of normal Gushel-Mukai varieties in terms of their excess conormal sheaves, which leads to a new proof of the classification theorem of Gushel and Mukai. We give a description of isomorphism classes of Gushel-Mukai varieti…
Let M be a two cusped hyperbolic 3-manifold and let M(r) be the result of r Dehn filling of a fixed cusp of M. We study canonical components of the SL(2,C) character varieties of M(r). We show that the gonality of these sets is bounded, independent of the filling parameter. We also obtain bounds, depending on r, for the genus of these sets. We compute the gonality for the double twist knots, demonstrating canonical components with arbitrarily large gonality.
Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces
本文证明了基变换型Enriques曲面上任意亏格曲线的Severi簇可数族非空，并构造了有理椭圆曲面上特殊超丰对数Severi簇的例子。
Orbifold elliptic genus and elliptic genus of singular varieties are introduced and relation between them is studied. Elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties introduced earlier. Orbifold elliptic genus is given in terms of the fixed point sets of the action. We show that the generating function for this orbifold elliptic genus $\sum Ell_{orb}(X^n,Σ_n)p^n$ for symmetric groups $Σ_n$ acting on $n$-fold products coincides with the one proposed by Dijkgraaf,…
Let K be a field. Let A be a central simple algebra over K and let X be the associated Brauer-Severi variety over K. It has recently been asked if there exists a genus 1 curve C over K such that K(C) splits A. In other words, is there a genus one curve C over K with a morphism to X? In this short note, we explicitly construct such a curve in the case where X has dimension at most 4 (equivalently, when A has degree at most 5).
Partial Hasse invariants for genus zero curves in Hilbert modular varieties
本文研究了Hilbert模形式中零 genus 曲线的部分Hassé不变量的特征零提升，并通过计算Teichmüller曲线得到类似Deuring的公式。
The aim of this note is to use elementary methods to give a large class of examples of projective varieties $ Y \subseteq \mathbb{P}^d_k$ over a field $k$ with the property that $Y$ is not isomorphic to a hypersurface $H\subseteq \mathbb{P}^N_k$ in projective space $\mathbb{P}^N_k$ with $N:=dim(Y)+1$. We apply this construction to the study of the arithmetic genus $p_a(Y)$ of $Y$ and the problem of determining if $p_a(Y)$ is a birational invariant of $Y$ in general. We give an infinite number of examples of pairs of projective varieties $(Y, Y')$ in any dimension $dim(Y)=dim(Y')\geq 4$ where $…
Irreducible components of moduli spaces of maps to smooth projective toric varieties in genus 0
本文通过阿贝尔锥的不可约分支研究，给出了亏格0到光滑射影环簇的映射模空间不可约分支的组合描述，并首次描述了非射影空间情形的可光滑轨迹。
Given a family of Galois coverings of the projective line we give a simple sufficient condition ensuring that the closure of the image of the family via the period mapping is a special (or Shimura) subvariety in A_g. By a computer program we get the list of all families in genus up to 8 satisfying our condition. There is no family in genus 8, all of them are in genus at most 7. These examples are related to a conjecture of Oort. Among them we get the cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot, Rohde, Moonen and others) and the abelian non-cyclic examples foun…
Seiberg-like duality for resolutions of determinantal varieties
该研究针对行列式簇的PAX与PAXY两种自然消解的零亏格Gromov-Witten理论，通过箭图突变关联对应箭图丛，证明二者生成函数满足聚类变量替换，还得到量子Thom-Porteous公式与箭图丛的类Seiberg对偶结论。
Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym-variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to do Chabauty- and Brauer-Manin type calculations for curves of genus 5 with an unramified involution. As an application, we determine the rational points on a smooth plane quartic with no particular geometric properties and give examples of curves of genus 3 and 5 violating the Hass…
To every double cover ramified in two points of a general trigonal curve of genus g, one can associate an étale double cover of a tetragonal curve of genus g+1. We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties.
In recent years there has been an interest in constructing examples of closed Riemann surfaces whose jacobian varieties are isogenous to a product of many elliptic factors and some other jacobian varieties. The first ones, provided by Ekedahl and Serre, are examples for which the isogenous decomposition has all factors being elliptic curves. It is well known that given two elliptic curves $E_{1}$ and $E_{2}$, there is a closed Riemann surface $X$ of genus two, with equations in terms of the elliptic curves, and whose jacobian variety $JX$ is isogenous to $E_{1} \times E_{2}$. In this paper, gi…
In this paper we study the gonality of the normalizations of curves in the linear system $|H|$ of a general primitively polarized complex $K3$ surface $(S,H)$ of genus $p$. We prove two main results. First we give a necessary condition on $p, g, r, d$ for the existence of a curve in $ |H|$ with geometric genus $g$ whose normalization has a $g^ r_d$. Secondly we prove that for all numerical cases compatible with the above necessary condition, there is a family of \emph{nodal} curves in $|H|$ of genus $g$ carrying a $g^1_k$ and of dimension equal to the \emph{expected dimension} $\min\{2(k-1),g\…
We give an inductive proof that the generalized Severi varieties -- the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and several mobile points -- are irreducible.
We investigate the problem of counting tropical genus g curves in g-dimensional tropical abelian varieties. For g = 2, 3, we prove that the tropical count matches the count provided by Göttsche, Bryan-Leung, and Lange-Sernesi in the complex setting.
We describe the spaces of minimal rank last syzygies for the Mukai Varieties of sectional genus 6,7 and 8. Based on this we show: 1. The first geometric syzygies of a general canonical curve of genus 6 form a non degenerate configuration of 5 lines in P^4. 2. The first geometric syzygies of a general canonical curve of genus 7 form a non degenerate, linearly normal, ruled surface of degree 84 on a spinor variety S in P^15. 3. The second geometric syzygies of a general canonical curve of genus 8 form a non degenerate configuration of 14 conics on a 2-uple embedded P^5 in P^20. This proves a nat…
In this note we give a brief review of the construction of a toric variety $\mathcal{V}$ coming from a genus $g \geq 2$ Riemann surface $Σ^g$ equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey in \cite{JH1}. In \cite{T1} A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety $DM_g$ -- the so-called Delzant model of moduli space -- for each genus $g.$ We conclude this note with some basic facts about the moment polytopes of the varieties $\mathcal{V}.$ In particular, we show that the…
We present a recursive algorithm computing all the genus-zero Gromov-Witten invariants from a finite number of initial ones, for Fano varieties with generically tame semi-simple quantum (and small quantum) (p, p)- type cohomology, whose first Chern class is a strictly positive combination of effective integral basic divisors.
Looking in positive characteristic for failures of the Bertini-Sard theorem, we determine, up to birational equivalence, the separable proper morphisms of smooth algebraic varieties in characteristic two, whose fibres are non-smooth curves of arithmetic genus two.
Let X be a general proper and smooth curve of genus 2 (resp. of genus 3) defined over an algebraically closed field of characteristic p. When 3\leq p \leq 7, the action of Frobenius on rank 2 semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order 2 line bundle over X. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian varieties of dimension 2) that appear as the Prym varieties associated to double étale …
Gromov-Witten theory of abelian varieties in families and modular forms
本文为系列研究首篇，将椭圆曲线格罗莫夫-维滕理论拟模性推广至高维，经重言投影后证明亏格2情形的相关猜想，引入商格罗莫夫-维滕不变量并给出其显式公式。
Let $X$ be a regular geometrically integral variety over an imperfect field $K$. Unlike the case of characteristic $0$, $X':=X\times_{\mathrm{Spec}\,K}\mathrm{Spec}\,K'$ may have singular points for a (necessarily inseparable) field extension $K'/K$. In this paper, we define new invariants of the local rings of codimension $1$ points of $X'$, and use these invariants for the calculation of $δ$-invariants (, which relate to genus changes,) and conductors of such points. As a corollary, we give refinements of Tate's genus change theorem and the Patakfalvi-Waldron Theorem. Moreover, when $X$ is a…
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