Luis Fuentes Garcia, Manuel Pedreira Perez
We give a geometrical characterization of the ideal of quadrics containing a canonical curve with an involution. This implies to study involutions of rational normal scrolls and Veronese surfaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we classify the minimal surfaces of general type with $χ=5$, $K^{2}=9$ whose canonical map is composed with an involution. We obtain 6 families, whose dimensions in the moduli space are 28, 27, 33, 32, 31 and 32 respectively. Among them, the family of surfaces having a genus 2 fibration forms an irreducible component of $\mathfrak{M}_{χ=5, K^{2}=9}$.
In this paper we classify completely all regular minimal surfaces with K^2=8, p_g=4 whose canonical map is composed with an involution. We obtain six unirational families of respective dimensions 28,28,32,33,38,34. The last two are irreducible components of the moduli space of minimal surfaces with K^2=8, p_g=4. These families hit three different topological types.
Given a smooth curve X of genus g we compute de dimension of the family of curves C which have an involution over X. Moreover we distinguish when the curve C is hyperelliptic.
The configuration of theta characteristics and vanishing thetanulls on a hyperelliptic curve is completely understood. We observe in this note that analogous results hold for the s-invariant theta characteristics on any curve C with an involution s . As a consequence we get examples of non hyperelliptic curves with a high number of vanishing thetanulls.
Galois lines for the quotient curve of the Hermitian curve by an involution
研究involution作用下Hermitian曲线商曲线在有限域上的Galois线排列，确定三种平面模型的Galois点。
We study the pencils of minimal degree on the smooth curves lying on a K3 surface X which carries a fixed-point free involution. Generically, the gonality of these curves is totally governed by the genus 1 fibrations of X
In this paper we consider plane quartics with to involutions. We compute the Dixmier invariants, the bitangents and the Matrix representation problem of these curves, showing that they have symbolic solutions for the last two questions.
Involutions of curves in abelian surfaces and their Jacobians
该研究针对复阿贝尔曲面上型为$(1,d)$的极化线丛线性系统中的亏格二曲线平展覆盖，给出了曲线对合不动点计数及雅可比簇同源分解的相关结果。
Involutions on hyperelliptic curves and Prym maps
本研究从Prym理论视角探究带额外对合的光滑超椭圆曲线几何，证明了正亏格超椭圆曲线上超椭圆$\n\mathbb{Z}_2^2$-覆叠的Prym映射的单射性。
In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron--Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibr…
Birational involutions of the real projective plane fixing an irrational curve
本文详细阐述了关于实射影平面上固定一条无理曲线的双有理对合的研究，基于2022年克拉科夫会议报告。
We present a systematic study of involutions on the moduli space of $G$-Higgs bundles over an elliptic curve $X$, where $G$ is complex reductive affine algebraic group. The fixed point loci in the moduli space of $G$-Higgs bundles on $X$, and in the moduli space of representations of the fundamental group of $X$ into $G$, are described. This leads to an explicit description of the moduli spaces of pseudo-real $G$-Higgs bundles over $X$.
Lusztig defined certain involutions on the equivariant K-theory of Slodowy varieties and gave a characterization of certain bases called canonical bases. In this paper, we give a conjectural generalization of these involutions and K-theoretic canonical bases to conical symplectic resolutions which have good Hamiltonian torus actions and state several conjectures related to them which we check for toric hyper-Kahler manifolds. We also propose an elliptic analogue of these bar involutions. As a verification of our proposal, we explicitly construct elliptic lifts of K-theoretic canonical bases an…
Hyperelliptic genus 3 curves with involutions and a Prym map
本文刻画了含两个额外对合的亏格3超椭圆曲线，给出显式方程，并证明在固定椭圆商曲线时Prym映射为2:1，同时构造了不与对偶同构的极化阿贝尔曲面族。
For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane which leave invariant a cubic pencil.
In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles' Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter's construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions.
Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an Artin-Schreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous to the product of these three elliptic curves. In this paper we exhibit explicit models for genus 3 curves with many involutions, and we compute explicit equations for the elliptic quotients. We then character…
We consider the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over the complex numbers, and study involutions defined by tensoring the vector bundle with an element $α$ of order 2 in the Jacobian of the curve, combined with multiplication of the Higgs field by $\pm 1$. We describe the fixed points of these involutions in terms of the Prym variety of the covering of $X$ defined by $α$, and give an interpretation in terms of the moduli space of representations of the fundamental group.
Sign involutions on para-abelian varieties
本研究探讨仿阿贝尔簇扭形上的符号对合，给出其存在的等价条件，证明对应商的Picard概形的性质，并分析一维情形下亏格一曲线的相关嵌入问题。
In the sl\_n case, A. Berenstein and A. Zelevinsky studied the Schützenberger involution in terms of Lusztig's canonical basis, [3]. We generalize their construction and formulas for any semisimple Lie algebra. We use for this the geometric lifting of the canonical basis, on which an analogue of the Schützenberger involution can be given. As an application, we construct semitoric degenerations of Richardson varieties, following a method of P. Caldero, [6]
We first demonstrate how duality for the fibres of the so-called Hitchin fibration works for the Langlands dual groups Sp(2m) and SO(2m+1). We then show that duality for G2 is implemented by an involution on the base space which takes one fibre to its dual. A formula for the natural cubic form is given and shown to be invariant under the involution.
We give two characterizations of Jacobians of curves with involution having fixed points in the framework of two particular cases of Welter's trisecant conjecture. The geometric form of each of these characterizations is the statement that such Jacobians are exactly those containing a shifted Abelian subvariety whose image under the Kummer map is orthogonal to an explicitly given vector.
Inversion of the Abel--Prym map for real curves with involutions
本文研究具有自映射的实曲线，详细阐述了普里姆θ函数的对称性，填补了现有文献中的空白。
Sign-reversing involutions in moduli spaces of curves
本文利用符号反转自映射解决模曲线模空间几何中的两个计算问题，给出 genus 零多色空间上 ψ 类交积的组合公式，并研究图形模空间的热带欧拉特征。
Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism σthat fixes pointwise all the order two points of Pic}^0(X), then we prove that X is hyperelliptic with σbeing the unique hyperelliptic involution. As a corollary, if a nontrivial automorphisms σ' of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with σ' being its hyperelliptic involution.
Numerical Godeaux Surfaces with many disjoint $(-2)$-curves and Applications
本文证明数值Godeaux曲面的互不相交(-2)-曲线至多6条且界最优，将该结果应用于细化两类p_g=0的光滑极小一般型曲面对合的分类。
The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus $Ł_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a birational parametrization of $Ł_g$ via dihedral invariants and show how these invariants can be used to determine the field of moduli of points $\p \in Ł_g$. We conjecture that for $\p\in \H_g$ with $|\Aut(\p)| > 2$ the field of moduli is a field of definition and prove this conjecture for any point $\p\in Ł_g$ such that the Klein 4-group is embedded in the red…
We consider the Jacobian Kummer surface $X$ of a genus two curve $C$. We prove that the Hutchinson-Weber involution on $X$ degenerates if and only if the Jacobian $J(C)$ is Comessatti. Also we give several conditions equivalent to this, which include the classical theorem of Humbert. The key notion is the Weber hexad. We include explanation of them and discuss the dependence between the conditions of main theorem for various Weber hexads. It results in "the equivalence as dual six". We also give a detailed description of relevant moduli spaces. As an application, we give a conceptual proof of …
We show how to construct counter-examples to the Hasse principle over the field of rational numbers on Atkin-Lehner quotients of Shimura curves and on twisted forms of Shimura curves by Atkin-Lehner involutions. A particular example is the quotient of the Shimura curve X attached to the indefinite rational quaternion algebra of discriminant 23*107 by the Atkin-Lehner involution w107. The quadratic twist of X by Q(sqrt{-23}) with respect to this involution is also a counter-example to the Hasse principle over Q.
We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) $X$ which is equipped with an involution $\imath$ preserving the holomorphic volume form. We define integers $n_{g,h}(β) $ which give a virtual count of the number of genus $g$ curves $C$ on $X$, in the class $β\in H_{2}(X)$, which are invariant under $\imath$, and whose quotient $C/\imath$ has genus $h$. We give two definitions of $n_{g,h}(β) $ which we conjecture to be equivalent: one in terms of a version of Pandharipande-Thomas theory and one in terms of a version of Maulik-Toda theory. We compute our in…
Hyperelliptic Stable Curves
本文给出了亏格≥2的超椭圆稳定曲线的内在刻画，通过分析对合作用及节点分解，得到超椭圆性的显式结构描述，并证明了对合的唯一性，适用于任意特征，精确捕捉了特征2中的特殊行为。
Multiplicative Higgs bundles and involutions
本文将曲线上的乘法$G$-希格斯丛理论推广到带对合$θ$的约化代数群对$(G,θ)$情形，引入取值于相关对称簇或其等变嵌入的对应丛概念，并研究其作为$θ$诱导的模空间对合的不动点的性质。
In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g > 2$, then a generic smooth curve of genus $2$ is nonhyperelliptic. In this article we investigate the situation of normal surface singularities, so we fix a resolution graph $\mathcal{T}$ and a generic singularity with resolution $\tX$ corresponding to it in the sense of \cite{NNII}. We consider an integer effective cycle $Z$ on the resolution $\tX$ and investigate the existence of a complete linear se…
Inversion of the Abel--Prym map in presence of an additional involution
本文研究了存在额外对合时Abel--Prym映射的反演问题，提出了第一型对合对的条件，并利用Prym theta函数定义了Jacobi反演的类比，给出了Hitchin和Kovalewski系统等曲线实例。
With every smooth, projective algebraic curve $\tilde{C}$ with involution $σ:\tilde{C}\longrightarrow \tilde{C}$ without fixed points is associated the Prym data which consists of the Prym variety $P:=(1-σ)J(\tilde{C})$ with principal polarization $Ξ$ such that $2Ξ$ is algebraically equivalent to the restriction on $P$ of the canonical polarization $Θ$ of the Jacobian $J(\tilde{C})$. In contrast to the classical Torelli theorem the Prym data does not always determine uniquely the pair $(\tilde{C},σ)$ up to isomorphism. In this paper we introduce an extension of the Prym data as follows. We con…
We propose two systems of "intrinsic" signs for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class $-2K$, but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.
In this paper we study Koszul cohomology and the Green and Prym-Green conjectures for canonical and Prym-canonical binary curves. We prove that if property $N_p$ holds for a canonical or a Prym-canonical binary curve of genus $g$ then it holds for a generic canonical or Prym-canonical binary curve of genus $g+1$. We also verify the Green and Prym-Green conjectures for generic canonical and Prym-canonical binary curves of low genus ($6\leq g\leq 15$, $g\neq 8$ for Prym-canonical and $3\leq g\leq 12$ for canonical).
We construct families of quartic and cubic hypersurfaces through a canonical curve, which are parametrized by an open subset in a Grassmannian and a Flag variety respectively. Using G. Kempf's cohomological obstruction theory, we show that these families cut out the canonical curve and that the quartics are birational (via a blowing-up of a linear subspace) to quadric bundles over the projective plane, whose Steinerian curve equals the canonical curve.
We establish GIT semistability of the 2nd Hilbert point of every Gieseker-Petri general canonical curve by a simple geometric argument. As a consequence, we obtain an upper bound on slopes of general families of Gorenstein curves. We also explore the question of what replaces hyperelliptic curves in the GIT quotients of the Hilbert scheme of canonical curves.
We discuss GIT for canonically embedded genus four curves and the connection to the Hassett-Keel program. A canonical genus four curve is a complete intersection of a quadric and a cubic, and, in contrast to the genus three case, there is a family of GIT quotients that depend on a choice of linearization. We discuss the corresponding VGIT problem and show that the resulting spaces give the final steps in the Hassett-Keel program for genus four curves.
The canonical ideal and the deformation theory of curves with automorphisms
该研究以典范理想为工具开展曲线形变理论研究，将带自同构曲线的提升问题转化为线性表示提升问题，为相关形变问题研究提供了新的转化思路。
We introduce the problem of GIT stability for syzygy points of canonical curves with a view toward a GIT construction of the canonical model of the moduli space of stable curves. As the first step in this direction, we prove semi-stability of the first syzygy point for a general canonical curve of odd genus.
We give a simple proof of Voisin's Theorem for general canonical curves of even genus. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves.
We compute equations for a Hurwitz curve of genus 17 and we conclude that the canonical ideal of any Hurwitz curve of genus 14 or 17 is generated by quadrics.
We prove that a generic canonically or bicanonically embedded smooth curve has semistable m-th Hilbert points for all m. We also prove that a generic bicanonically embedded smooth curve has stable m-th Hilbert points for all m \geq 3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with G_m-action, namely the canonically embedded balanced ribbon and the canonically embedded balanced double A_{2k+1}-curve. In the bicanonical case, we prove finite Hilbert stability of special hyperelliptic curves, namely Wiman curves. Finally, we giv…
It is shown that the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.
This paper is a sequel to arXiv:1109.4986, where we proved that a general smooth curve of odd genus, canonically or bicanonically embedded, has semistable finite Hilbert points. Here, we prove that a generic canonically embedded curve of even genus has semistable finite Hilbert points. More precisely, we prove that a generic canonically embedded trigonal curve of even genus has semistable finite Hilbert points. Furthermore, we show that the analogous result fails for bielliptic curves. Namely, the Hilbert points of bielliptic curves are asymptotically semistable but become non-semistable below…
Tetragonal Canonical Curves
研究四边形典范曲线的正常丛Harder-Narasimhan filtration结构及其计算方法。
We prove that there is no g for which the canonical embedding of a general curve of genus g lies on the Segre embedding of any product of three or more projective spaces.
We study the log canonical models of the moduli space MBar_{0,n} of pointed stable genus zero curves with respect to the standard log canonical divisors K+aD, where D denotes the boundary. In particular we show that, as a formal consequence of a conjecture by Fulton regarding the ample cone of MBar_{0,n}, these log canonical models are equal to certain of Hassett's weighted pointed curve spaces.
This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univa…
Green's Conjecture states the following : syzygies of the canonical model of a curve are simple up to the p^th stage if and only if the Clifford index of C is greater than p. We prove that the generic curve of genus g satisfies Green's conjecture.
We describe explicitly all hyperelliptic limits of quadrics through smooth canonical curves of genus $g$ in ${\mathbb P}^{g-1}$. Also, we construct an open embedding of the blow up of a ${\rm PGL}_g$-bundle over the moduli space of curves of genus $g$ along the hyperelliptic locus into the blow up of the canonical Hilbert scheme of ${\mathbb P}^{g-1}$ along the closure of the locus of canonical ribbons, which are certain double thickenings of rational normal curves introduced and studied by Bayer and Eisenbud.
In this paper we show that the relative canonical sheaf of a relatively minimal fibration of curves over a curve is semi-ample ; in fact, its m-tensored product is base point free for any m > 1. We use Koszul cohomology with it to prove that the relative canonical ring of the fibration is generated in degree up to five.
In this article we study genus 5 curves with a fixed point free involution. We give two geometrical caracterisations of these curves amoung all genus 5 curves. One of these was conjectured by Arbarello, Cornalba, Griffiths and Harris in their book on curves. The second one is given by some particular embeddings corresponding to the points of the Prym variety associated to the situation.
We prove Fujita-type basepoint-freeness for projective quasi-log canonical curves and surfaces.
Log canonical thresholds of high multiplicity reduced plane curves
研究高乘积简化平面曲线的log canonical阈值，确定小于2/(d-1)的可能值，并计算度数小于6的曲线阈值。
Log Canonical Thresholds for Plane Curves in Arbitrary Characteristic
本文通过估值理论将复数域上平面曲线的log canonical阈值公式推广到任意特征情况。
A tetragonal canonical curve is the complete intersection of two divisors on a scroll. The equations can be written in `rolling factors' format. For such homogeneous ideals we give methods to compute infinitesimal deformations. Deformations can be obstructed. For the case of quadratic equations on the scroll we derive explicit base equations. They are used to study extensions of tetragonal curves.
The Harder-Narasimhan Filtration of a Trigonal Canonical Curve
本研究利用三角典范曲线位于有理正规面scroll上的性质，计算了一般三角典范曲线及亏格6典范曲线法丛的Harder-Narasimhan滤过，为代数几何中相关丛的结构研究提供了具体结果。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。