Michela Artebani
The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a K3 surface with a non symplectic action of Z_4. In this paper we study the geometry of the one dimensional family of K3 surfaces associated to the locus of plane quartics with five nodes.
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This short note contains an example of a 4-dimensional family of K3 surfaces having finite-dimensional motive. Some consequences are presented, for instance the verification of a conjecture of Voisin (concerning 0-cycles on the self-product) for K3 surfaces in this family.
We construct a new family of minimal smooth surfaces of general type with K^2=7 and p_g= 0. We show that for a surface in this family, its canonical divisor is ample and its bicanonical morphism is birational. We prove that these surfaces satisfy Bloch's conjecture.
We give a 1-dimensional family of classical and supersingular Enriques surfaces in characteristic 2 covered by the supersingular K3 surface with Artin invariant 1. Moreover we show that there exist 30 nonsingular rational curves and ten non-effective (-2)-divisors on these Enriques surfaces whose reflection group is of finite index in the orthogonal group of the Neron-Severi lattice modulo torsion.
In this note we construct an unlimited family of irregular algebraic surfaces of general type with canonical map of degree $ 8 $, irregularity $ 1 $ and arbitrarily large geometric genus such that the image of the canonical map is not a surface of minimal degree.
We complete the classification of smooth surfaces swept out by a 1-dimensional family of plane curves that do not form a fibration. As a consequence, we characterize manifolds swept out by a 1-dimensional family of hypersurfaces that do not form a fibration.
A few facts concerning the phrase "the automorphism groups become larger at special points of the moduli of K3 surfaces" are presented. It is also shown that the automorphism groups are of infinite order over a dense subset in any one-dimensional non-trivial family of projective K3 surfaces.
In this paper we describe a construction method for numerical Godeaux surfaces based on homological algebra. We show the existence of an 8-dimensional locally complete family of simply connected numerical Godeaux surfaces.
We describe a new method of constructing rational surfaces with given invariants in P^4 and present a family of degree 11 rational surfaces of sectional genus 11 with 2 six-secants that we found with this method.
We define and study a family of cubic surfaces in the projectivized tangent bundle over a four dimensional projective space associated to the root system D_5. The 27 lines are rational over the base and we determine the classifying map to the moduli space of marked cubic surfaces. This map has degree two and we use it to get short proofs for some results on the Chow group of the moduli space of marked cubic surfaces.
Let $p \geqslant 3$ be a prime number and let $n \geqslant 0$ be an integer such that $p-1$ divides $n.$ In this short note we construct a family of $(p,n)$-gonal Riemann surfaces of maximal genus $2np+(p-1)^2$ with more than one $(p,n)$-gonal group.
The degeneration of a family of rational surface automorphisms
研究有理表面族的自同构退化，发现极限映射在特殊纤维上为恒等映射，通过吹开后得到诱导的双有理映射，其动力学度数为16。
We study a family of surfaces of general type with $p_g=q=2$ and $K^2=7$, originally constructed by Cancian and Frapporti by using the Computer Algebra System MAGMA. We provide an alternative, computer-free construction of these surfaces, that allows us to describe their Albanese map and their moduli space.
We study the construction of complex minimal smooth surfaces $S$ of general type with $p_g(S)=0$ and $K_S^2=7$. Inoue constructed the first examples of such surfaces, which can be described as Galois $\mathbb{Z}_2\times\mathbb{Z}_2$-covers over the four-nodal cubic surface. Later the first named author constructed more examples as Galois $\mathbb{Z}_2\times\mathbb{Z}_2$-covers over certain six-nodal del Pezzo surfaces of degree one. In this paper we construct a two-dimensional family of minimal smooth surfaces of general type with $p_g=0$ and $K^2=7$, as Galois $\mathbb{Z}_2\times\mathbb{Z}_2$…
Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus $g \ge 2$ different from $3, 6, 12, 15$ and 30, with exactly $4g$ automorphisms form an equisymmetric one-dimensional family, denoted by $\mathcal{F}_g.$ In this paper, for every prime number $q \ge 5,$ we explore further properties of each Riemann surface $S$ in $\mathcal{F}_q$ as well as of its Jacobian variety $JS.$
In this paper, we study the period map of a certain one-parameter family of quartic K3 surfaces with an S5-action. We construct automorphic forms on the period domain as the pull-backs of theta constants of genus 2 by a modular embedding. Using these automorphic forms, we give an explicit presentation of the inverse period map.
For a smooth quasi-projective surface $X$ and an integer $n\ge 3$, we show that the universal family $Z^n$ over the Hilbert scheme $\text{Hilb}^{n}(X)$ of $n$ points has non $\mathbb{Q}$-Gorenstein, rational singularities, and that the Samuel multiplicity $μ$ at a closed point on $Z^n$ can be computed in terms of the dimension of the socle. We also show that $μ\le n$.
We study a family of surfaces of general type with $p_g=q=2$ and $K^2=7$, originally constructed by C. Rito. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus $\mathcal{M}$ in the moduli space of the surfaces of general type. In particular we prove that $\mathcal{M}$ is an open subset, and it has three connected components, two dimensional, irreducible and generically smooth.
Higher Chow cycles on a family of Kummer surfaces
本文在二维Kummer曲面族上构造了类型为(2,1)的高阶Chow环，并证明其不可分解部分生成秩至少18的子群，方法结合有限群作用与Picard-Fuchs算子。
We construct a new family of minimal surfaces of general type with $p_g=q=2$ and $K^2=6$, whose Albanese map is a quadruple cover of an abelian surface with polarization of type $(1,3)$. We also show that this family provides an irreducible component of the moduli space of surfaces with $p_g=q=2$ and $K^2=6$. Finally, we prove that such a component is generically smooth of dimension 4 and that it contains the 2-dimensional family of product-quotient examples previously constructed by the first author. The main tools we use are the Fourier-Mukai transform and the Schrödinger representation of t…
In 1899, Hutchinson presented a way to obtain a three-parameter family of Hessians of cubic surfaces as blowups of Kummer surfaces. We show that this family consists of those Hessians containing an extra class of conic curves. Based on this, we find the invariant of a cubic surface C in pentahedral form that vanishes if its Hessian is in Hutchinson's family, and we give an explicit map between cubic surfaces in pentahedral form and blowups of Kummer surfaces.
We study a family of surfaces of general type that arises from the intersections of two translates of the theta divisor on a principally polarized complex abelian fourfold. In particular we determine the Néron-Severi lattices of these surfaces and show that for a general abelian fourfold the monodromy group of the associated variation of Hodge structures is a subgroup of index at most two in the Weyl group W(E_6), whereas for a Jacobian variety it degenerates to a subgroup of the symmetric group S_6.
We show that the generic fiber of a family of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking the base, such a family actually factors through an $\mathbb{A}^{1}$-fibration over a certain scheme, induced by the MRC-fibration of a relative smooth projective model of the family. For affine threefolds fibered by irrational $\mathbb{A}^{1}$-ruled surfaces, this induced $\mathbb{A}^{1}$-fib…
The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is the symmetric group on four elements. There are three pairs of three-dimensional reflexive polytopes with this symmetry group, up to isomorphism. We identify a natural one-parameter family of K3 surfaces corresponding to each of these pairs, show that the symmetric group on four elements acts symplectically on members of these families, and show that a general K3 surface in each family has Picard rank 19. The properties of two of these families have been analyzed in the literature using oth…
We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs $(X,εR)$ over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.
Automorphisms of a family of surfaces with $p_g=q=2$ and $K^2=7$
本文计算了由 C. Rito 构造的一族具有 $p_g=q=2$ 和 $K^2=7$ 的一般型曲面的自同构群，并探讨了该结果对 Mumford-Tate 猜想的推论。
We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K_1 of a mirror family of quartic K3 surfaces. The resulting multivalued function does not satisfy the hypergeometric differential equation of the periods and we conclude that the cycle is indecomposable for most points in the mirror family. The occurring inhomogenous Picard-Fuchs equation are related to Painlevé VI type differential equations.
Let X be an affine T-variety. We study two different quotients for the action of T on X: the toric Chow quotient X/_CT and the toric Hilbert scheme H. We introduce a notion of the main component H_0 of H which parameterizes general T-orbit closures in X and their flat limits. The main component U_0 of the universal family U over H is a preimage of H_0. We define an analogue of a universal family W_X over the main component of the X/_CT. We show that the toric Chow morphism restricted on the main components lifts to a birational projective morphism from U_0 to W_X. The variety W_X also provides…
Let S be a minimal complex surface of general type with p_g=0 such that the bicanonical map of S is not birational and let Z be the bicanonical image. In [M.Mendes Lopes, R.Pardini, "Enriques surfaces with eight nodes", Math. Zeit. 241 (4) (2002), 673-683] it is shown that either: i) Z is a rational surface, or ii) K^2_S=3, the bicanonical map is a degree two morphism and Z is birational to an Enriques surface. Up to now no example of case ii) was known. Here an explicit construction of all such surfaces is given. Furthermore it is shown that the corresponding subset of the moduli space of sur…
We compute the deformations in the sense of generalized complex structures of the standard classical complex structure on a primary Kodaira surface and we prove that the obtained family of deformations is a smooth locally complete family depending on four complex parameters. This family is the same as the extended deformations (in the sense of Kontsevich and Barannikov) in degree two, obtained by Poon using differential Gerstenhaber algebras.
We consider algebraic actions of a cyclic group of order p on a K3 surface defined over an algebraically closed field of characteristic p. We classify possible loci of fixed points as well as possible quotient surfaces.
Let $f:X\to C$ be a family of semistable K3 surfaces with non-empty set $S$ of singular fibres having infinite local monodromy. Then, when the so called Arakelov-Yau inequality reaches equality, we prove that $C\setminus S$ is a modular curve and the family comes essentially from a family of elliptic curves through a so called Nikulin-Kummer construction. In particular, when $C=\BBb P^1$, the family of elliptic curves must be one of Beauville's 6 examples where Arakelov inequality reaches equality.
Minimal algebraic surfaces of general type $X$ such that $K^2_X=2χ(\mathcal{O}_X)-6$ or $K^2_X=2χ(\mathcal{O}_X)-5$ are called Horikawa surfaces. In this note we study $\mathbb{Z}^2_2$-actions on Horikawa surfaces. The main result is that all the connected components of Gieseker's moduli space of canonical models of surfaces of general type with invariants satisfying these relations contain surfaces with $\mathbb{Z}^2_2$-actions.
Birational geometry of actions on del Pezzo surfaces
本文完成有限群在del Pezzo表面的正则一般自由作用的分类，解决了若干等变理查德几何中的开放问题。
We complete the proof of a theorem we announced and partly proved in [Math. Nachr. 271 (2004), 69-90, math.AG/0111299]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in that paper. It describes a natural cycle that enumerates the curves in the family with precisely $r$ ordinary nodes. The second part is proved here. It asserts that, for $r\le 8$, the class of this cycle is given by a computable universal polynomial in the pushdowns to the parameter space of products of the Chern classes of the family.
The usual Gromov-Witten invariants are zero for Kähler surfaces with $p_g\geq 1$. In this paper we use analytic methods to define Family Gromov-Witten Invariants for Kähler surfaces. We prove that these are well-defined invariants of the deformation class of the Kähler structure.
By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a classification of additive actions on complete toric surfaces.
We compute entropies in a large family of K3 surface automorphisms in (P^1)^3. In keeping with a result by Xie, we find that the entropies vary in a lower semi-continuous manner as the Picard ranks of the K3 surfaces vary.
We study the extension of a hyperelliptic K3 surface to a Fano 6-fold. This determines a family of surfaces of general type with p_g=1, K^2=2 and hyperelliptic canonical curve, where each surface is a weighted complete intersection inside a Fano 6-fold. Finally, we use these hyperelliptic surfaces to determine an 8-parameter family of Godeaux surfaces with torsion Z/2.
The aim of this note is to study local and global Seshadri constants for a family of smooth surfaces with prescribed polarization. We shall first observe that given $α$ being smaller than the square root of the degree of polarization, the set of local Seshadri constants in the range $(0, α]$ is finite. This in particular implies that the square root of the degree of polarization is the only possible accumulation point of the set of local Seshadri constants. Next we shall remark the Zariski closedness of the set of points whose local Seshadri constants are in any given interval $(0, a]$. As app…
Shafarevich's hyperbolicity conjecture asserts that a family of curves over a quasi-projective 1-dimensional base is isotrivial unless the logarithmic Kodaira dimension of the base is positive. More generally it has been conjectured by Viehweg that the base of a smooth family of canonically polarized varieties is of log general type if the family is of maximal variation. In this paper, we relate the variation of a family to the logarithmic Kodaira dimension of the base and give an affirmative answer to Viehweg's conjecture for families parametrized by surfaces.
We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $\mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every compact differentiable surface endowed with an action of the circle $S^1$ admits a unique smooth rational real quasi-projective model up to $\mathbb{S}^1$-equivariant birational diffeomorphism.
The aim of this article is to classify the pairs (S, G), where S is a smooth minimal surface of general type with p_g=0 and K^2=7, G is a subgroup of the automorphism group of S and G is isomorphic to the group $\mathbb{Z}_2^2$. The Inoue surfaces with K^2=7, which are finite Galois $\mathbb{Z}_2^2$-covers of the 4-nodal cubic surface, are the first examples of such pairs. More recently, the author constructed a new family of such pairs. They are finite Galois $\mathbb{Z}_2^2$-covers of certain 6-nodal Del Pezzo surfaces of degree one. We prove that the base of the Kuranishi family of deformat…
Starting with the Dolgachev surface E(1)_{2,3} we construct an infinite family of distinct exotic copies of the rational surface E(1), each of which admits a handlebody decomposition without 1- and 3- handles, and we draw these handlebodies.
Additive actions on projective surfaces with a finite number of orbits
本文研究了具有有限个轨道的加性作用在射影曲面上的存在性，描述了带有 du Val 奇点的射影曲面，并提供了成对非同构加性作用的 1-参数族示例，回答了 Hassett 和 Tschinkel 的问题。
A 9-dimensional family of K3 surfaces with finite dimensional motive
该研究利用与四次三维立方体的关联，证明了一类二次曲面三重覆盖K3曲面的Kimura有限维motive，并确立其Kuga-Satake Hodge猜想与Voisin零闭链猜想，进而得到带3阶非辛自同构K3曲面模空间2个9维不可约分支的对应结果。
Group actions and irrationality in surface families
本文研究曲面族中非有理度的特化问题，证明了当一般纤维的非有理度由群作用商诱导时相应的特化结论。
We describe the Hodge theory of brilliant families of K3 surfaces. Their characteristic feature is a close link between the Hodge structures of any two fibres over points in the Noether-Lefschetz locus. Twistor deformations, the analytic Tate-Safarevic group, and one-dimensional Shimura special cycles are covered by the theory. In this setting, the Brauer group is viewed as the Noether-Lefschetz locus of the Brauer family or as the specialization of the Noether-Lefschetz loci in a family of approaching twistor spaces. Passing from one algebraic twistor fibre to another, which by construction i…
The Hodge numerical invariants of a variation of Hodge structure over a smooth quas--projective variety are a measure of complexity for the global twisting of the limit mixed Hodge structure when it degenerates. These invariants appear in inequalities which they may have correction terms, called Arakelov inequalities. One may investigate the correction term to make them into equalities, also called Arakelov equalities. We investigate numerical Arakelov type (in)equilities for a family of surfaces fibered by curves. Our method uses Arakelov identities in a weight 1 and also in a weight 2 variat…
In this paper we determine all finite groups G that can act on some compact Riemann surface M with the property that if H is any non-trivial subgroup of G, then the orbit surface M/H is the Riemann sphere. The idea is to look at the induced action on the vector space of holomorphic differentials on M (in the positive genus case) and then use the old-known (Wolf) classification of groups admitting fixed point-free linear actions. A description of the corresponding group actions is given in terms of Fuchsian representations.
We classify equivariantly Gorenstein log del Pezzo surfaces with boundaries at infinity and with finite group actions such that the quotient surface modulo the finite group has Picard number one. We also determine the corresponding finite groups. Better figures are available upon request.
Let us consider the rank 14 lattice $P=D_4^3\oplus < -2> \oplus < 2>$. We define a K3 surface S of type P with the property that $P\subset {\rm Pic}(S) $, where ${\rm Pic}(S) $ indicates the Picard lattice of S. In this article we study the family of K3 surfaces of type P with a certain fixed multipolarization. We note the orthogonal complement of P in the K3 lattice takes the form $$ U(2)\oplus U(2)\oplus (-2I_4). $$ We show the following results: \item{(1)} A K3 surface of type P has a representation as a double cover over ${\bf P}^1\times {\bf P}^1$ as the following affine form in (s,t,w) s…
In this note the geography of minimal surfaces of general type admitting $\mathbb{Z}_2^2$-actions is studied. More precisely, it is shown that Gieseker's moduli space $\mathfrak{M}_{K^2,χ}$ contains surfaces admitting a $\mathbb{Z}_2^2$-action for every admissible pair $(K^2, χ)$ such that $2χ-6\leq K^2\leq 8χ-8$ or $K^2=8χ$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification $\overline{\mathfrak{M}}_{K^2,χ}$ of Gieseker's moduli space $\mathfrak{M}_{K^2,χ}$ for every admissible pair $(K^2, χ)$ such that $2χ-6\leq K^2\…
We prove a big monodromy result for a smooth family of complex algebraic surfaces of general type, with invariants p_g=q=1 and K^2=3, that has been introduced by Catanese and Ciliberto. This is accomplished via a careful study of degenerations. As corollaries, when a surface in this family is defined over a finitely generated extension of Q, we verify the semisimplicity and Tate conjectures for the Galois representation on the middle \ell-adic cohomology of the surface.
Using Calegari's result on the Fontaine-Mazur conjecture, we study the irreducibility of pure, regular, rank 3 weakly compatible systems of self-dual l-adic representations. As a consequence, we prove that the Tate conjecture holds for a family of elliptic surfaces defined over Q with geometric genus bigger than 1.
In this paper, one of a series devoted to the classification, the moduli spaces and the discovery of new surfaces of general type with geometric genus p_g= 0, we generalize a classical construction method due to Burniat (and revisited by Inoue), constructing surfaces with K^2 = 3 and a with a new fundamental group of order 16.
We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fail the Hasse principle, by building on previous work of la Bretèche and Browning.
Picard-Fuchs system for family of Kummer surfaces as subsystem of GKZ hypergeometric system
本文利用 Kummer 曲面上的 Jacobi 椭圆纤维化和相应的 GKZ 超几何系统，为所有主极化阿贝尔曲面确定了秩为五的三变量 Picard-Fuchs 系统的简单表达式。
Let $\mathbb{P}$ denote the weighted projective space with weights $(1,1,1,3)$ over the rationals, with coordinates $x,y,z,$ and $w$; let $\mathcal{X}$ be the generic element of the family of surfaces in $\mathbb{P}$ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field $\mathbb{Q}(t)$. In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the fa…
In this article we study the period map for a family of $K3$ surfaces which is given by the anticanonial divisor of a toric variety. We determine the period differential equation and its monodromy group. Moreover we show the exact relation between our period differential equation and the unifomizing differential equation of the Hilbert modular orbifold for the field $\mathbb{Q}(\sqrt{5})$.
We study a class of algebraic surfaces of degree 3n in the complex projective space with only ordinary double points. They are obtained by using bivariate polynomials with complex coefficients related to the generalized cosine associated to the affine Weyl group of the root system A2.
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