A. Arsie, C. Galati
The notion of geometric k-normality for curves is introduced in complete generality and is investigated in the case of nodal and cuspidal curves living on several types of surfaces. We discuss and suggest some applications of this notion to the study of Severi varieties of nodal curves on surfaces of general type and on the plane.
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We give some results on quadratic normality of reducible curves canonically embedded and partially extend this study to their projective normality.
In this paper we study the projective normality of certain Artin-Schreier curves $Y_f$ defined over a field $\F$ of characteristic $p$ by the equations $y^q+y=f(x)$, $q$ being a power of $p$ and $f\in \F[x]$ being a polynomial in $x$ of degree $m$, with $(m,p)=1$. Many $Y_f$ curves are singular and so, to be precise, here we study the projective normality of appropriate projective models of their normalizations.
We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve X of genus g. We relate it with the Clifford index of the base curve X. If d>=4g-2i-Cliff(X)+1, i>=3 and R is smooth, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree.
We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve of genus g. If d>=4g-2i+1, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree.
A well-known conjecture asserts that smooth threefolds $X\subset\{\mathbb P}^5$ are quadratically normal with the only exception of the Palatini scroll. As a corollary of a more general statement we obtain the following result, which is related to the previous conjecture: If $X\subset\{\mathbb P}^5$ is not quadratically normal, then its triple curve is reducible. Similar results are also given for higher dimensional varieties.
Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with $\frac{3g+3}{2}<°L\le 2g-5$. Then $L$ is normally generated if $°L>\max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple covering of genus $p$ curve $C'$ with $C\stackrelϕ\to C'$ and $D$ a divisor on $C'$ with $4p<°D< \frac{g-1}{6}-2p$. Then $K_C(-ϕ^*D)$ becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.
Recall that a smooth complex projective curve has a very ample canonical bundle when it is non-hyperelliptic, and according to a theorem of M. Noether the resulting embedding is projectively normal. A theorem of Petri further asserts that the homogeneous ideal is generated by quadrics if the curve is neither trigonal nor a smooth plane quintic. In this note, we prove an analogue of Noether's theorem for the symmetric square of the curve - namely, the canonical bundle of the symmetric square determines a projectively normal embedding exactly when the curve itself is neither hyperelliptic, trigo…
We give necessary and sufficient criteria for a smooth Enriques surface S in P^r to be scheme-theoretically an intersection of quadrics. Moreover we prove in many cases that, when S contains plane cubic curves, the intersection of the quadrics containing S is the union of S and the 2-planes spanned by the plane cubic curves. We also give a new (very quick) proof of the projective normality of S if the degree of S is at least 12.
For a smooth curve of genus $g$ embedded by a line bundle of degree at least $2g+3$ we show that the ideal sheaf of the secant variety is 5-regular. This bound is sharp with respect to both the degree of the embedding and the bound on the regularity. Further, we show that the secant variety is projectively normal for the generic embedding of degree at least $2g+3$. We also give a conjectural description of the resolutions of the ideals of higher secant varieties.
We investigate the projective normality of smooth, linearly normal surfaces of degree 9. All non projectively normal surfaces which are not scrolls over a curve are classified. Results on the projective normality of surface scrolls are also given. One of the reasons that brought us to look at this question is our desire to find examples for a long standing problem in adjunction theory. Andreatta followed by a generalization by Ein and Lazarsfeld posed the problem of classifying smooth n-dimensional varieties (X,L) polarized with a very ample line bundle L, such that the adjoint linear system |…
In this paper, we study the deformations of curves in the projective 3-space $\mathbb P^3$ (space curves), one of the most classically studied objects in algebraic geometry. We prove a conjecture due to J. O. Kleppe (in fact, a version modified by Ph. Ellia) concerning maximal families of space curves lying on a smooth cubic surface, assuming the quadratic normality of its general members. We also give a sufficient condition for curves lying on a cubic surface to be obstructed in $\mathbb P^3$ in terms of lines on the surface. For the proofs, we use the Hilbert-flag scheme of $\mathbb P^3$ as …
We give simple geometric proofs of Aprodu-Farkas-Papadima-Raicu-Weyman's theorem on syzygies of tangent developable surfaces of rational normal curves and Raicu-Sam's result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green's conjecture for general curves of genus $g$ over an algebraically closed field $\mathbf{k}$ with $\operatorname{char}(\mathbf{k}) = 0$ or $\operatorname{char}(\mathbf{k}) \geq \lfloor (g-1)/2 \rfloor$. We also show the arithmetic normality of tangent developable surfaces of arbitrary smooth projective curves of large degree.
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 use the geometric technique, developed by Weyman, to calculate the resolution of orbit closures of representations of Dynkin quivers with every vertex being source or sink. We use this resolution to derive the normality of such orbit closures. As a consequence we obtain the normality of certain orbit closures of type E.
We show that the linear syzygy spaces of elliptic normal curves, their secant varieties and of bielliptic canonical curves are spanned by geometric syzygies.
We study the family of irreducible curves with $δ$ nodes belonging to a free linear system $|C|$ with smooth general member on a surface $S$ such that $|K_S|$ is ample. Under the assumption that $C$ is numerically equivalent to $pK_S$, $p\ge 3$ rational, we give an upper bound on $δ$ ensuring that the family is smooth of codimension $δ$ in $|C|$ (plus another slightly different result of the same type). We give examples showing that the bound is sharp. Some related results concerning linear normality of nodal curves are proved.
The aim of this work is to provide a construction of generalized local symbols on algebraic curves as morphisms of group schemes. From a closed point of a complete, irreducible and non-singular curve $C$ over a perfect field $k$ as the only data, using theta groups over Picard schemes of curves, we offer a geometric construction that allows us to define generalizations of the tame symbol and the Hilbert norm residue symbol.
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.
Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli spaces of rational curves on $X$ using the perspective of Manin's Conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on $X$. We propose a Geometric Manin's Conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.
The arithmetic geometric mean algorithm for calculation of elliptic integrals of the first type was introduced by Gauss. The analog algorithm for Abelian integrals of genus 2 was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for Abelian integrals of genus 3.
In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in $\mathbb{P}^5$. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge's curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-…
In this paper, we define a generalized arithmetic-geometric mean $μ_g$ among $2^g$ terms motivated by $2τ$-formulas of theta constants. By using Thomae's formula, we give two expressions of $μ_g$ when initial terms satisfy some conditions. One is given in terms of period integrals of a hyperelliptic curve $C$ of genus $g$. The other is by a period integral of a certain Calabi-Yau $g$-fold given as a double cover of the $g$-dimensional projective space $\mathbf{P}^g$.
We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We focus on genus 3 hyperelliptic curves. Both geometric and arithmetic aspects are considered.
Cubic arithmetic-geometric mean and isogenies of Hesse elliptic curves
本文在特征非三的任意域上构造Hesse椭圆曲线的三次同源，对应三次算术几何平均的一步，并应用于特征三中普通椭圆曲线的Serre-Tate典范提升，同时探讨与超几何函数的联系。
The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.
Let $H$ be the Hilbert scheme of curves in complex projective $3$-space, with $d\geq 3$ and genus $g \leq (d-2)^2/4$. A complete, explicit description of the cone of curves and the ample cone of $H$ is given. From this, partial results on the group $\mathop{Aut}(H)$ are deduced.
We show that geometric syntomic cohomology lifts canonically to the category of Banach-Colmez spaces and study its relation to extensions of modifications of vector bundles on the Fargues-Fontaine curve. We include some computations of geometric syntomic cohomology Spaces: they are finite rank $Q_p$-vector spaces for ordinary varieties, but in the nonordinary case, these cohomology Spaces carry much more information, in particular they can have a non-trivial $C$-rank. This dichotomy is reminiscent of the Hodge-Tate period map for $p$-divisible groups.
Based on a recent result of Voisin [2001] we describe the last nonzero syzygy space in the linear strand of a canonical curve C of even genus g=2k lying on a K3 surface, as the ambient space of a k-2-uple embedded P^{k+1}. Furthermore the geometric syzygies constructed by Green and Lazarsfeld [1984] from g^1_{k+1}'s form a non degenerate configuration of finitely many rational normal curves on this P^{k+1}. This proves a natural generalization of Green's conjecture [1984], namely that the geometric syzygies should span the space of all syzygies, in this case.
Dynamic constructions of hyperbolisms of plane curves: an automated exploration of geometric loci
本文通过自动化探索，利用两种数学软件交互，研究平面曲线关于点和线的双曲线构造，涵盖四次及更高次曲线，并推导出Gerono双纽线的新构造，通过参数方程和Groebner基隐式化验证不可约性。
Building on the positive solution of Pillay's conjecture we present a notion of "intrinsic" reduction for elliptic curves over a real closed field K. We compare such notion with the traditional algebro-geometric reduction and produce a classification of the group of K-points of an elliptic curve E with three "real" roots according to the way E reduces (algebro-geometrically) and the geometric complexity of the "intrinsically" reduced curve.
We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) étale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric étale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to distinguish between two different ones. The aim of this note is to generalize the classical concept of curvature and torsion to so-called higher algebraic curvatures and to show that each analytic branch of a complex curve is already uniquely defined by them.
The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 defined over a function field of characteristic zero. We recover the known result for genus 2 curves and in many cases improve upon the known bound for genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved.
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…
We consider a variant of metrised graphs where the edge lengths take values in a commutative monoid, as a higher-rank generalisation of the notion of a tropical curve. Divisorial gonality, which Baker and Norine defined on combinatorial graphs in terms of a chip firing game, is extended to these monoid-metrised graphs. We define geometric gonality of a monoid-metrised graph as the minimal degree of a horizontally conformal, non-degenerate morphism onto an monoid-metrised tree, and prove that geometric gonality is an upper bound for divisorial gonality in the monoid-metrised case. We also show …
We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most nodes as singularities, form an important class of curves, as any projective algebraic curve is birational to a plane nodal curve.
We present a framework for constructing examples of smooth projective curves over number fields with explicitly given elements in their second K-group using elementary algebraic geometry. This leads to new examples for hyperelliptic curves and smooth plane quartics. Moreover, we show that most previously known constructions can be reinterpreted using our framework.
We transfer the algebro-geometric method of construction of solutions of the discrete KP equation to the finite field case. We emphasize role of the Jacobian of the underlying algebraic curve in construction of the solutions. We illustrate in detail the procedure on example of a hyperelliptic curve.
In this paper we study the dual codes of a wide family of evaluation codes on norm-trace curves. We explicitly find out their minimum distance and give a lower bound for the number of their minimum-weight codewords. A general geometric approach is performed and applied to study in particular the dual codes of one-point and two-point codes arising from norm-trace curves through Goppa's construction, providing in many cases their minimum distance and some bounds on the number of their minimum-weight codewords. The results are obtained by showing that the supports of the minimum-weight codewords …
Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the Baker-Akhiezer functions. We straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviors of the Baker-Akhiezer functions. We analyze the four-component AKNS soliton hier…
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