Rosario Strano
We characterize the curves in $ P^3$ which are minimal in their biliaison class. Such curves are exactly the curves which do no admit an elementary descending biliaison. As a consequence we have that every curve in $P^3$ can be obtained from a minimal one by means of a finite sequence of ascending elementary biliaisons.
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Let $X$ be an integral projective scheme satisfying the condition $S_3$ of Serre and $H^1({\mathcal O}_X(n)) = 0$ for all $n \in {\mathbb Z}$. We generalize Rao's theorem by showing that biliaison equivalence classes of codimension two subschemes without embedded components are in one-to-one correspondence with pseudo-isomorphism classes of coherent sheaves on $X$ satisfying certain depth conditions. We give a new proof and generalization of Strano's strengthening of the Lazarsfeld--Rao property, showing that if a codimension two subscheme is not minimal in its biliaison class, then it admits …
This paper is the text of a lecture given at the Catania conference on Commutative Algebra and Algebraic Geometry, in honor of the 60th birthday of Silvio Greco. It analyses the correspondences between equivalence classes of objects related to the projective 3-space ${\bf P}^3_k$ over a field $k$ : - vector bundles ${\cal F}$ such that $H_*^2({\cal F})=0$ up to stable equivalence - coherent sheaves of projective dimension $\leq 1$ up to pseudo-equivalence - finite length graded $k[X,Y,Z,T]$-modules, modulo isomorphism - biliaison classes of locally Cohen-Macaulay curves. In particular, it lead…
We characterize the postulation character of arithmetically Gorenstein curves in ${\mathbb P}^4$. We give conditions under which the curve can be realized in the form $mH-K$ on some ACM surface. Finally, we strengthen a theorem of Watanabe by showing that any general arithmetically Gorenstein curve in ${\mathbb P}^4$ can be obtained from a line by a series of ascending complete-intersection biliaisons.
We study the concept of liaison addition for codimension two subschemes of an arithmetically Gorenstein projective scheme. We show how it relates to liaison and biliaison classes of subschemes and use it to investigate the structure of Gorenstein liaison equivalence classes, extending the known theory for complete intersection liaison of codimension two subschemes. In particular, we show that on the non-singular quadric threefold in projective 4-space, every non-licci ACM curve can be obtained from a single line by successive liaison additions with lines and CI-biliaisons.
The aim of this paper is to prove a generalization of a theorem of Rao for families of space curves, which caracterizes the biliaison classes of curves. First we introduce the concept of pseudo-isomorphism of coherent sheaves, which generalizes the concept of stable isomorphism. An N-type resolution for a family of curves $C$ over the local ring $A$, defined by an ideal $J$, is an exact sequence $0\to P\to N\to J\to 0$ where $N$ is a locally free sheaf on $P^3_A$ and $P$ is a direct sum of invertible sheaves $O_P(-n_i)$. We prove the two following results, when the residual field of $A$ is inf…
Here we calculate the Chern classes of ${\bar {\mathcal M}}_{g,n}$, the moduli stack of stable n-pointed curves. In particular, we prove that such classes lie in the tautological ring.
This is the second in a series of three papers in which we investigate the rational Chow ring of the stack consisting of nodal curves of genus. Here we define the basic classes: the classes of strata and the Mumford classes.
In this paper the number of $\mathbb{F}_q$-isomorphism classes of Legendre elliptic curves over the finite fields $\mathbb{F}_q$ is enumerated.
We study the codimension n locus of curves of genus 2 with n distinct marked Weierstrass points inside the moduli space of genus 2, n-pointed curves, for n <= 6. We give a recursive description of the classes of the closure of these loci inside the moduli space of stable curves. For n<= 4, we express these classes using a generating function over stable graphs indexing the boundary strata of moduli spaces of pointed stable curves. Similarly, we express the closure of these classes inside the moduli space of curves of compact type for all n. This is a first step in the study of the structure of…
We define tautological Chow classes on the moduli space of curves with linear series. In the case where the forgetful morphism to the moduli space of curves has relative dimension zero, we describe the images of these classes in the Chow group of Mgbar. As an application, we compute the (virtual) slopes of several different classes of divisors on Mgbar.
Recovering curves from isogeny classes of Prym varieties
本文提出从Prym簇同源类恢复代数曲线的猜想，并据此分析同源Jacobian及多个同源Prym的曲线对构造。
We prove that every curve on a rationally connected variety is algebraically equivalent to a (non-effective) integral sum of rational curves.
We study for rationally connected varieties $X$ the group of degree 2 integral homology classes on $X$ modulo those which are algebraic. We show that the Tate conjecture for divisor classes on surfaces defined over finite fields implies that this group is trivial for any rationally connected variety.
We study the nef cone of self-products of a curve. When the curve is very general of genus $g>2$, we construct a nontrivial class of self-intersection 0 on the boundary of the nef cone. Up to symmetry, this is the only known nontrivial boundary example that exists for all $g > 2$. When the curve is general, we identify nef classes that improve on known examples for arbitrary curves. We also consider self-products of more than two copies of the curve.
This paper deals with rational curves and birational contractions on irreducible holomorphically symplectic manifold. We survey some recent results about minimal rational curves, their deformations, extremal rays associated with these curves, and the geometry of the Kähler cone.
For a hyperelliptic curve of genus $g$, a divisor in general position of degree $g+1$ is given by polynomial equations. There is an action from an algebraic group on the representations of divisors by polynomials which fixes divisor classes. This structure reduces the question of rationality of divisor classes to rationality of polynomials which is more easy to control.
We prove the Integral Hodge Conjecture for curve classes on smooth varieties of dimension at least three with nef anticanonical divisor constructed as a complete intersection of ample hypersurfaces in a smooth toric variety. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric MMP, we prove that in each case, $H_2(X,\mathbb{Z})$ is generated by classes of rational curves.
We construct an infinite collection of universal -- independent of $(g,n)$ -- polynomials in the Miller-Morita-Mumford classes $κ_m\in H^{2m}( \overline{\cal M}_{g,n},\bq)$, defined over the moduli space of genus $g$ stable curves with $n$ labeled points. We conjecture vanishing of these polynomials in a range depending on $g$ and $n$.
Higher Gaussian maps for special classes of curves
本文针对平面曲线、乘积曲面中特定线性系统内的曲线以及Enriques曲面上充分丰沛线丛中的曲线，确定了其典范丛的高阶高斯映射的秩。
Brauer classes not split by genus one curves
证明存在域F上的Brauer类不能被任何F上的亏格一曲线分裂，回答了Clark和Saltman的问题。
We explain how to compute top-dimensional intersections of psi-classes on moduli spaces of m-stable curves. On the moduli spaces of Deligne-Mumford stable pointed curves of genus one, these intersection numbers are determined by two recursions, namely the string equation and the dilaton equation. We establish, for each integer m>0, an analogous pair of recursions which determine these intersection numbers on the moduli spaces of m-stable pointed curves of genus one.
We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.
We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invarian…
We study new effective curve classes on the moduli space of stable pointed rational curves given by the fixed loci of subgroups of the permutation group action. We compute their numerical classes and provide a strategy for writing them as effective linear combinations of F-curves, using Losev-Manin spaces and toric degeneration of curve classes.
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.
We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to certain polyhedral cones which we describe explicitly. Not all homology classes in the cone correspond to rational curves, but all classes sufficiently far from the walls of the cone do. The proof for $X_5$ is explicit, the proof for $X_6$ was carried out using computers.
In this paper, we consider the tautological ring containing the extended Brill-Noether algebraic classes on the normalization of the compactified Jacobian of a complex nodal projective curve (with one node). This smallest $\Q$-subalgebra of algebraic classes under algebraic equivalence, stable under extensions of the maps induced by multiplication maps, Pontrayagin product and Fourier transform, is shown to be generated by pullback of the Brill-Noether classes of the Jacobian of the normalized curve and some natural classes.
In [DKO] we constructed virtual fundamental classes $[[ Hilb^m_V ]]$ for Hilbert schemes of divisors of topological type m on a surface V, and used these classes to define the Poincare invariant of V: (P^+_V,P^-_V): H^2(V,Z) --> Λ^* H^1(V,Z) x Λ^* H^1(V,Z) We conjecture that this invariant coincides with the full Seiberg-Witten invariant computed with respect to the canonical orientation data. In this note we prove that the existence of an integral curve $C \subset V$ induces relations between some of these virtual fundamental classes $[[Hilb^m_V ]]$. The corresponding relations for the Poinca…
We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation. We also study the case of arbitrary codimension.
It is well known that the Hodge conjecture with rational coefficients holds for degree 2n-2 classes on complex projective n-folds. In this paper we study the more precise question if on a rationally connected complex projective n-fold the integral Hodge classes of degree 2n-2 are generated over $\mathbb Z$ by classes of curves. We combine techniques from the theory of singularities of pairs on the one hand and infinitesimal variation of Hodge structures on the other hand to give an affirmative answer to this question for a large class of manifolds including Fano fourfolds. In the last case, on…
Block and Göttsche have defined a $q$-number refinement of counts of tropical curves in $\mathbb{R}^2$. Under the change of variables $q=e^{iu}$, we show that the result is a generating series of higher genus log Gromov-Witten invariants with insertion of a lambda class. This gives a geometric interpretation of the Block-Göttsche invariants and makes their deformation invariance manifest.
We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i-g+1 genus 0 components. (The proof of this result is quite short and naive.) As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appears to be the geometric content behind these results; the r…
The Pandharipande-Thomas rationality conjecture for superpositive curve classes on projective complex 3-manifolds
本文通过第二作者发展的阿贝尔范畴枚举不变量理论和墙交叉公式，证明了对于射影复三维流形上的超正曲线类，带后代插入的Pandharipande-Thomas不变量生成函数的理性和极点猜想。
Movable curve classes and slope stability on Deligne-Mumford stacks
将光滑射影簇上关于可动曲线类和相干层的斜率稳定性的结果推广到具有射影粗模空间的光滑真DM栈上，并建立了Bogomolov-Gieseker不等式。
Gopakumar-Vafa invariants of fiber classes on Calabi-Yau 4-folds fibered over curves
论文证明了Cao-Maulik-Toda猜想，将光滑纤维化于曲线的Calabi-Yau四折面的纤维类Gopakumar-Vafa不变量与光滑纤维的Gopakumar-Vafa不变量联系起来，基于模空间的定向兼容性假设。
Let X be a smooth, complete, toric variety. We study those curves C in X that are contractible, in the sense that there exists an equivariant morphism with connected fibers, with source X, that contracts exactly the irreducible curves that are numerically equivalent to a multiple of C. When X is projective, we compare contractible and extremal curves, and we show that every curve in X is numerically equivalent to a linear combination with positive integral coefficients of contractible curves.
This note pursues the techniques of modified psi classes on the stack of stable maps (cf. [Graber-Kock-Pandharipande]) to give concise solutions to the characteristic number problem of rational curves in P^2 or P^1 x P^1 with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials.
Inside the symmetric product of a very general curve, we consider the codimension-one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension. We compute the classes of such divisors, and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve. By letting the moduli of the curve vary, we study more generally the classes of the related divisors inside the moduli space of stable pointed curves.
On a space of stable maps, the psi classes are modified by subtracting certain boundary divisors. The top products of modified psi classes, usual psi classes, and classes pulled back along the evaluation maps are called twisted descendants; it is shown that in genus 0, they admit a complete recursion and are determined by the Gromov-Witten invariants. One motivation for this construction is that all characteristic numbers (of rational curves) can be interpreted as twisted descendants; this is explained in the second part, using pointed tangency classes. As an example, some of Schubert's number…
Fundamental groups of the complements to reducible curves on smooth surfaces with restricted classes of components
本文分类光滑单连通射影曲面上满足对偶簇次数≤6等奇异曲线束族，进而描述不可约分量为超平面截影类且基本群可满射到秩>6自由群的曲面上曲线补集的基本群。
We construct explicit examples of algebraic cycles in \bar M_g (for large g congruent to 2 mod 4) and in M_2,20 (no bar) which are not in the tautological ring. In an appendix we give a general method for computing intersections in the tautological ring.
After a short exposition of the basic properties of the tautological ring of the moduli space of genus g Deligne-Mumford stable curves with n markings, we explain three methods of detecting non-tautological classes in cohomology. The first is via curve counting over finite fields. The second is by obtaining length bounds on the action of the symmetric group S_n on tautological classes. The third is via classical boundary geometry. Several new non-tautological classes are found.
Cohomology classes on moduli of curves from Theta Characteristics
本文利用 theta 特征的 Galois–Stiefel–Whitney 类，证明全实域上曲线与主极化阿贝尔簇模栈在指定次数存在非平凡上同调类，并计算亏格 3 情形的 Brauer 群拉回。
Thom (residual) polynomials in characteristic classes are used in the analysis of geometry of functional spaces. They serve as a tool in description of classes Poincaré dual to subvarieties of functions of prescribed types. We give explicit universal expressions for residual polynomials in spaces of functions on complex curves having isolated singularities and multisingularities, in terms of few characteristic classes. These expressions lead to a partial explicit description of a stratification of Hurwitz spaces.
Projective structures on compact real manifolds are classical objects in real differential geometry. Complex manifolds with a holomorphic projective structure on the other hand form a special class as soon as the dimension is greater than one. In the Kähler Einstein case, projective space, tori and ball quotients are essentially the only examples. They can be described purely in terms of Chern class conditions. We give a complete classification of all projective manifolds carrying a projective structure. The only additional examples are modular abelian families over quaternionic Shimura curves…
Universal Chern classes on the moduli of bundles
本文针对曲线稳定丛模空间非精细（秩与次数不互素）的情况，构造泛上同调类，证明曲线与丛模栈乘积上的泛丛的部分陈类可提升至曲线与稳定丛模空间的乘积上。
Chow classes of divisors on stacks of pointed hyperelliptic curves
本文在带尖点超椭圆曲线栈中计算通用 Weierstrass 除子与通用 $g^1_2$ 除子的周类，并用 Scavia 给出的除子类群基加以表达。
We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Co…
We give an effective iterative characterization of the classes of (smooth, rational) (-1)-curves on the blowup of the projective plane at general points. Such classes are characterized as having self-intersection -1, arithmetic genus 0, and intersecting every (-1)-curve of smaller degree nonnegatively.
The locus of genus-two curves with n marked Weierstrass points has codimension n inside the moduli space of genus-two curves with n marked points, for n<=6. It is well known that the class of the closure of the divisor obtained for n=1 spans an extremal ray of the cone of effective divisor classes. We generalize this result for all n: we show that the class of the closure of the locus of genus-two curves with n marked Weierstrass points spans an extremal ray of the cone of effective classes of codimension n, for n<=6. A related construction produces extremal nef curve classes in moduli spaces …
Motivic classes of fixed-generators Hilbert schemes of unibranch curve singularities and Igusa zeta functions
本文研究固定生成元数的单分支平面曲线 Hilbert 概形的动机类，给出公式、正性与拓扑不变性，并联系 Igusa zeta 函数。
This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for ample divisor classes. The second function captures the asymptotic geometry of curves analogously to the volume function for divisors. We prove that the two functions coincide, generalizing Zariski's classical result for surfaces to all varieties. Our result confirms the log concavity conjecture of the first named author for weighted mobility of curve classes in an unexpected way, via Legendr…
In this paper we prove a general theorem concerning the number of translation classes of curves of genus $g$ belonging to a fixed cohomology class in a polarized abelian variety of dimension $g$. For $g = 2$ we recover results of Göttsche and Bryan-Leung. For $g = 3$ we deduce explicit numbers for these classes.
This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre-Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves defined by the second named author, yielding some fundamental positivity results for curve classes. For varieties with special structures, the Zariski decomposition for curve classes admits an interesting geometric interpretation.
The subalgebra of the tautological ring of the moduli of curves of compact type generated by the kappa classes is studied. Relations, constructed via the virtual geometry of the moduli of stable maps, are used to prove universality results relating the kappa rings in genus 0 to higher genus. Predictions for kappa classes of the Gorenstein conjecture are proven.
We study the variety of rank $\leq k$ quadrics containing a general projective curve and show that it has the expected dimension in the range $g-d+r\leq 1$. By considering the loci where this expectation is not true, we construct new divisor classes in $\overline{\mathcal{M}}_{g,n}$. We use one of these classes to show that $\overline{\mathcal{M}}_{15,9}$ is of general type.
A method of constructing Cohomological Field Theories (CohFTs) with unit using minimal classes on the moduli spaces of curves is developed. As a simple consequence, CohFTs with unit are found which take values outside of the tautological cohomology of the moduli spaces of curves. A study of minimal classes in low genus is presented in the Appendix by D. Petersen.
Mehta and Seshadri have proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this note, we discuss the Mehta-Seshadri correspondence over an irreducible projective curve with at most nodes as singularities.
We study the cycle-valued reduced Gromov-Witten theory of a nonsingular projective K3 surface. For primitive curve classes, we prove that the correspondence induced by the reduced virtual fundamental class respects the tautological rings. Our proof uses monodromy over the moduli space of K3 surfaces, degeneration formulae and virtual localization. As a consequence of the monodromy argument, we verify an invariance property for Gromov-Witten invariants of K3 surfaces in primitive curve class conjectured by Oberdieck-Pandharipande.
We study the exceptional loci of birational (bimeromorphic) contractions of a hyperkähler manifold $M$. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the Kähler cone. Homology classes which can possibly be orthogonal to a wall of the Kähler cone of some deformation of $M$ are called MBM classes. We prove that all MBM classes of type (1,1) can be represented by rational curves, called MBM curves. All MBM curves can be contracted on an appropriate birational model of $M$, unless $b_2(M) \leq 5$. When $…
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