Antonio Laface, Luca Ugaglia
In this note we deal with rational curves in P^3 which are images of a line by means of a finite sequence of cubo-cubic Cremona transformations. We prove that these curves can always be obtained applying to the line a sequence of such transformations increasing at each step the degree of the curve. As a corollary we get a result about curves that can give speciality for linear systems of P^3.
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We prove an elementary method to classify, up to rigid isotopy, all nodal degree 4 real rational curves in $\mathbb{RP}^2$ that have only real double points. We show how to associate a chord diagram to a nodal real degree 4 planar rational curve and prove that such curves are defined up to rigid isotopy by their chord diagrams.
Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$. In this paper we consider elementary transformations of linear pfaffian representations of $C$. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on $C$ with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of $C$ can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications t…
We give an elementary proof of the group law for elliptic curves using explicit formulas.
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.
Let W be the germ of a smooth complex surface around an exceptional curve and let E be a rank 2 vector bundle on W. We study the cohomological properties of a finite sequence $E_i, 1 \leq i \leq t$ of rank 2 vector bundles canonically associated to E. We calculate numerical invariants of E in terms of the $E_i.$ If S is a compact complex smooth surface and E is a rank 2 bundle on the blow- up of S at a point, we show that all values of $c_2(E)-c_2(π_* E^{\vee\vee})$ that are numerically possible are actually attained.
Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.
The full automorphism group of a certain elementary abelian $p$-cover of the Hermitian curve in characteristic $p>0$ is determined. It is remarkable that the order of Sylow $p$-groups of the automorphism group is close to Nakajima's bound in terms of the $p$-rank. Weierstrass points, Galois points, Frobenius nonclassicality, and arc property are also investigated.
We compute the stable reduction of the Lubin-Tate space $X(π^2)$ in the equal characteristic case, on the basis of Coleman-McMurdy's ideas. Namely, in this paper, we actually construct a stable covering of $X(π^2).$ This paper also includes intersection multiplicity datum of the stable model of $X(π^2)$. Our method is very explicit and elementary.
Torsion of elliptic curves with rational $j$-invariant over the maximal elementary abelian 2-extension of $\mathbb{Q}$
本文针对定义在有理数域所有二次扩张合成域上、j不变量为非0或1728的有理数的椭圆曲线，完成了对其可能挠子群结构的分类。
Lifting Elementary Abelian Covers of Curves
本文针对特征p域上曲线的初等阿贝尔p-群伽罗瓦覆盖的提升问题，给出依赖p-循环子覆盖提升分支轨迹的组合判据，研究分支点在特殊纤维的合并规律，并分析多类(ℤ/2)³-覆盖的提升情形。
If a real symmetric matrix of linear forms is positive definite at some point, then its determinant is a hyperbolic hypersurface. In 2007, Helton and Vinnikov proved a converse in three variables, namely that every hyperbolic plane curve has a definite real symmetric determinantal representation. The goal of this paper is to give a more concrete proof of a slightly weaker statement. Here we show that every hyperbolic plane curve has a definite determinantal representation with Hermitian matrices. We do this by relating the definiteness of a matrix to the real topology of its minors and extendi…
The group structure on the rational points of elliptic curves plays several important roles, in mathematics and recently also in other areas such as cryptography. However, the famous proofs for the group property (in particular, for its associative law) require somewhat advanced mathematics and therefore are not easily accessible by non-mathematician. On the other hand, there have been attempts in the literature to give an elementary proof, but those rely on computer-aided calculation for some part in their proofs. In this paper, we give a self-contained proof of the associative law for this o…
In this note is we exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of class field theory.
By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface, a projective homogeneous space admits a rational point if the elementary obstruction vanishes.
In this work we use elementary methods to discuss the question of the minimal number of points with bad reduction over the projective line for elliptic curves E/k(T) which are non-constant resp. have non-constant j-invariant.
Let X be a smooth Fano variety of dimension at least 4. We show that if X has an elementary birational contraction sending a divisor to a curve, then the Picard number of X is smaller or equal to 5.
In this short note we give an elementary combinatorial argument, showing that the Conjecture of J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández, A. Némethi follows from the results of M. Borodzik and C. Livingston in the case of rational cuspidal curves with two critical points.
We exploit an elementary specialization technique to study some properties of rational curves on index $n-1$ Fano $n$-folds. We prove a simple formula for counting rational curves passing through a suitable number of points in the case $n=3$. The arguments have immediate deformation theoretic consequences which translate to properties of the normal bundles of such curves.
An Elementary Formal Proof of the Group Law on Weierstrass Elliptic Curves in any Characteristic
本文在Lean定理证明器中形式化了任意特征域上Weierstrass椭圆曲线的非奇异点，并给出纯代数证明，证明其加法构成阿贝尔群，避免了高级代数几何或繁琐计算。
Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will give an alternative and elementary proof of Lelli Chiesa's Theorem in the case of $r= 2$. More precisely we will prove that that under certain condition the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system $|L|$. Using Lelli Chiesa's theorem for $r \ge 3$ we also extend Lelli Chiesa's Theorem in the case of $r= 2$ in weaker condition.
We discuss a non-computational elementary approach to a well-known criterion of divisibility by 2 in the group of rational points on an elliptic curve.
The first part of Hilbert's sixteenth problem deals with the classification of the isotopy types realizable by real plane algebraic curves of a given degree $m$. For $m = 9$, the classification of the $M$-curves is still wide open. Let $C_9$ be an $M$-curve of degree 9 and $O$ be a non-empty oval of $C_9$. If $O$ contains in its interior $α$ ovals that are all empty, we say that $O$ together with these $α$ ovals forms a nest. The present paper deals with the $M$-curves with three nests. Let $α_i, i = 1, 2, 3$ be the numbers of empty ovals in each nest. We prove that at least one of the $α_i$ i…
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.
Elementary links from prime Fano threefolds along two lines
本文研究了素Fano三维流形沿两条不相交直线的吹胀操作，建立了与Fano三维流形的初等链接，并探讨了不同亏格下的几何构造与反向链接。
In this note we give a summary of some elementary results in the theory of super Riemann surfaces (SUSY curves).
In this, largely expository, note, we show how the simplicial structure of the moduli spaces of stable rational curves with marked points allows to produce explicit equations for these spaces. The key argument is an elementary combinatorial statement about the sets of trees with marked leaves.
These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected varieties. At the end, some connections with symplectic geometry are considered.
We consider the locus of irreducible nonsingular rational curves of degree d Pn, n>2, meeting a generic collection of linear subspaces. When this locus is 0 (resp 1)- dimensional, we compute (recursively) its degree (resp. geometric genus). The method is completely elementary and similar to that of (alg-geom/9704004, alg-geom/9708013), where the case n=2 was considered.
In this paper we study the space $Ω(m)$, of holomorphic $m$-(poly)differentials of a function field of a curve defined over an algebraically closed field of characteristic $p>0$ when $G$ is cyclic or elementary abelian group of order $p^n$; we give bases for each case when the base field is rational, introduce the Boseck invariants and give an elementary approach to the $G$ module structure of $Ω(m)$ in terms of Boseck invariants. The last computation is achieved without any restriction on the base field in the cyclic case, while in the elementary abelian case it is assumed that the base field…
This work deals with an algebro-geometric theory of solutions of the classical Yang-Baxter equation based on torsion free coherent sheaves of Lie algebras on Weierstrass cubic curves.
We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is "elementary" in the sense that it does not assume the finiteness of any Shafarevich-Tate group. On the other hand, using Kolyvagin's construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show that there are infinitely many curves of every index over every number field.
We give an exposition of the theory of adjoints and conductors for curves on nonsingular surfaces, emphasizing the case of plane curves, for which the presentation is particularly elementary. This is closely related to Max Noether's ``AF+BG'' theorem, which is proved for curves with arbitrary multiple components.
This note contains an elementary discussion of the Arakelov intersection theory of elliptic curves. The main new results are a projection formula for elliptic arithmetic surfaces and a formula for the "energy" of an isogeny between Riemann surfaces of genus 1. The latter formula provides an answer to a question originally posed by Szpiro.
We shall give an elementary and rigorous proof of the Thomae formula for ${\bf Z}_N$ curves which was discovered by Bershadsky and Radul. Instead of using the determinant of the Laplacian we use the traditional variational method which goes back to Riemann, Thomae, Fuchs. In the proof we made explicit the algebraic expression of the chiral Szegö kernels and proves the vanishing of zero values of derivatives of theta functions with ${\bf Z}_N$ invariant $1/2N$ characteristics.
Applying the Second Main Theorem we deal with the algebraic degeneracy of entire holomorphic curves from the complex plane into a complex algebraic normal variety of positive log Kodaira dimension that admits a finite proper morphism to a semi-abelian variety. We will also discuss applications to the Kobayashi hyperbolicity problem.
The logarithmic connections studied in the paper are direct images of regular connections on line bundles over genus-2 double covers of the elliptic curve. We give an explicit parametrization of all such connections, determine their monodromy, differential Galois group and the underlying rank-2 vector bundle. The latter is described in terms of elementary transforms. The question of its (semi)-stability is addressed.
These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties through the example of elliptic curves. The reason for working with elliptic curves is that most constructions are elementary and explicit in this case. All four approaches to moduli spaces of curves -- complex analytic, topological, algebro-geometric, and number theoretic -- are con…
Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Conjecture and the Slope Conjecture (which relate the Jones polynomial of a knot and its parallels to the $\SL(2,\BC)$ character variety and to slopes of incompressible surfaces). Our discussion predicts that the tropical curve is dual to a Newton subdivision of the $A$-polynomial o…
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