A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of $X$. We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$. Under two geometric assumptions on the web-structure, the pairwise non-integrability condition and the bracket-generating condition, we prove that the local differential geometry determines the global algebraic geometry of $X$, up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when $X \subset {\bf P}^N$ is a Fano submanifold of Picard number 1, and the family of lines covering $X$ becomes a web. In this special case, we have a stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X, X' \subset {\bf P}^N, \dim X' \geq 3,$ are two such Fano manifolds of Picard number 1, then any surjective morphism $f: X \to X'$ is an isomorphism.
We classify noninvertible, holomorphic selfmaps of the projective plane that preserve an algebraic web. In doing so, we obtain interesting examples of critically finite maps.
We classify webs of minimal degree rational curves on surfaces and give a criterion for webs being hexagonal. In addition, we classify Neron-Severi lattices of real weak del Pezzo surfaces. These two classifications are related to root subsystems of E8.
In the complex setting, let $F(x,y,y')=0$ be an analytic or algebraic differential equation with $y'$-degree $d$. We deal with the qualitative study of such equations through the geometry of the planar $d$-web generated by the generic family of integral curves. Infinitesimal symmetries of these configurations are discussed by using meromorphic connections methods "à la Cartan-Spencer".
We prove that a generic second order differential equation in the projective plane has no algebraic solutions when the bidegree is big enough. We also proof an analogous result for webs on $\mathbb{P}}^{2}$.
New relations between algebraic geometry, information theory and Topological Field Theory are developed. One considers models of databases subject to noise i.e. probability distributions on finite sets, related to exponential families. We prove explicitly that these manifolds have the structure of a pre-Frobenius manifold, being a pre-structure appearing in the process of axiomatisation of Topological Field Theory. On one hand, this allows us to develop relations to algebraic geometry, by proving explicitly that a statistical pre-Frobenius manifold forms an algebraic variety over $\mathbb{Q}$ …
We describe the global structure of holomorphic webs in codimension one, and in particular their singularity (caustic). Various concepts are introduced, which have no interest locally near a regular point, such as the type, the reducibility, the quasi-smoothness, the CI property (complete intersection), the dicriticity... We prove for instance that the algebraicity of a web globally defined on a complex projective space may be readen on its caustic (dicriticity), at least if each irreducible component is CI, and the web quasi-smooth. .
Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional relation among the first integrals of the foliations defining the web reminiscent of Abel's addition theorem in classical algebraic geometry. The abelian relations of a given web form a finite dimensional vector space with dimension (the rank of the web) bounded by Castelnuovo number p(n,k) where n is the dimension of the ambient space and k is the number of foliations defining the web. A fundamental p…
We define the polar curves and the polar family associated to a projective web and obtain some results about the geometry of the generic element of this family. We also deal with the particular case of foliations and prove the constancy of the topological embedded type of the generic polar.
In arXiv:1302.3142, it has been proved that for r>1, n>1 and d>(r+1)(n-1)+1, a d-web of type (r,n) with maximal rank is algebraizable in the classical sense, except maybe when n>2 and d = (r+2)(n-1)+1. In the present paper, one considers this particular case. Under these hypotheses on r, n and d, one constructs some examples of `exceptional algebraic webs': these are generalized algebraic webs of maximal rank that aren't algebraizable in the classical sense.
Notions of directional Chebyshev constant and transfinite diameter have recently been studied on certain algebraic curves in $\mathbb{C}^2$. The theory is extended here to curves in $\mathbb{C}^N$ for arbitrary $N$. The results are analogous but require more methods from computational algebraic geometry.
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.
We explain how, given a plane algebraic curve $\mathcal{C}\colon f(x,y) = 0$, $x_1 \in \mathbb{C}$ not a singularity of $y$ w.r.t. $x$, and $\varepsilon > 0$, we can compute $δ> 0$ such that $|y_j(x_1) - y_j(x_2)| < \varepsilon$ for all holomorphic functions $y_j(x)$ which satisfy $f(x, y_j(x)) = 0$ in a neighbourhood of $x_1$ and for all $x_2$ with $|x_1 - x_2| < δ$. Consequently, we obtain an algorithm for reliable homotopy continuation of plane algebraic curves. As an example application, we study continuous deformation of closed discrete Darboux transforms. Moreover, we discuss a scheme fo…
In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs $(P,τ)$, where $P$ is a compact Riemann surface with a finite number of holes and punctures and $τ:P\to P$ is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.
For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.
We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.
We study conchoids to algebraic curve from the perspective of algebraic geometry, analyzing their main algebraic properties. We introduce the formal definition of conchoid of an algebraic curve by means of incidence diagrams. We prove that, with the exception of a circle centered at the focus and taking $d$ as its radius, the conchoid is an algebraic curve having at most two irreducible components. In addition, we introduce the notions of special and simple components of a conchoid. Moreover we state that, with the exception of lines passing through the focus, the conchoid always has at least …
In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface. We use the moment map as a tool for studying the topology of real algebraic curves and their complexifications.
We establish for smooth projective real curves the equivalent of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.
Two sets of conditions are presented for the compactness of a real plane algebraic curve, one sufficient and one necessary, in terms of the Newton polygon of the defining polynomial.
We formulate a refined theory of linear systems, using the methods of a previous paper, "A Theory of Branches for Algebraic Curves", and use it to give a geometric interpretation of the genus of an algebraic curve. Using principles of duality, we prove generalisations of Plucker's formulae for algebraic curves. The results hold for arbitrary characteristic of the base field L, with some occasional exceptions when characteristic(L)=2, which we observe in the course of the paper.
The coamoeba of any complex algebraic plane curve $V$ is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in $(\mathbb{C}^*)^2$ is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in $(\mathbb{C}^*)^2$, the complex algebraic plane curves whose coamoebas are of maximal area (counted with multiplicity) are defined over $\mathbb{R}$, and their real loci are Harnack curves possibly with ordinary real isolated double points (c.f. \cite{MR-00}). In addition, we charac…
Let X,Y be algebraic curves in P^n over C. We give an effective description of the join J(X,Y)\in P^n of X and Y in terms of local parametrizations of X and Y.
Let $\mathcal{X}$ be an algebraic curve of genus $g$ defined over an algebraically closed field $K$ of characteristic $p \geq 0$, and $q$ a prime dividing $|\mbox{Aut}(\mathcal{X})|$. We say that $\mathcal{X}$ is a $q$-curve. Homma proved that either $q \leq g+1$ or $q = 2g+1$, and classified $(2g+1)$-curves. In this note, we classify $(g+1)$-curves, and fully characterize the automorphism groups of $q$-curves for $q= 2g+1, g+1$. We also give some partial results on $q$-curves for $q = g, g-1$.
This article shows that on a generic principally polarized abelian variety of dimension five the $\mathbb{Q}$-vector space of algebraic equivalences among the 27 Abel-Prym curves has dimension 20.
Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a patchworking theorem which relates plane tropical curves with complex and real algebraic curves having prescribed multiple points. It can be used to compute Welschinger invariants of non-toric Del Pezzo surfaces.
Let $X \subset \mathbb{C}^{2n}$ be an $n$-dimensional algebraic variety. We define the algebraic version of the generic symmetry defect set (Wigner caustic) of $X$. Moreover, we compute its singularities for $X_d$ being a generic curve of degree $d$ in $\mathbb{C}^2$.
We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic).
We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the compl…
A projective non-singular plane algebraic curve of degree d<=4 is called maximally symmetric if it attains the maximum order of the automorphism groups for complex non-singular plane algebraic curves of degree d. For d<=7, all such curves are known. Up to projectivities, they are the Fermat curve for d=5,7, see \cite{kmp1,kmp2}, the Klein quartic for d=4, see \cite{har}, and the Wiman sextic for d=6, see \cite{doi}. In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every d>=8 showing that the Fermat …
In this paper, we introduce the notion of infinity branches as well as approaching curves. We present some properties which allow us to obtain an algorithm that compares the behavior of two implicitly defined algebraic plane curves at the infinity. As an important result, we prove that if two plane algebraic curves have the same asymptotic behavior, the Hausdorff distance between them is finite.
We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems.
A criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves is presented. A new example of a plane curve with non-collinear Galois points as an application is described. Furthermore, a new characterization of the Fermat curve in terms of non-collinear Galois points is presented.
Given a real curve, we study special linear systems called "very special" for which the dimension does not satisfy a Clifford type inequality. We classify all these very special linear systems when the gonality of the curve is small.
We present a treatment of the algebraic description of the Jacobian of a generic genus two plane curve which exploits an SL2(k) equivariance and clarifes the structure of E.V.Flynn's 72 defining quadratic relations. The treatment is also applied to the Kummer variety.
We present relations between cycles with rational coefficients modulo algebraic equivalence on the Jacobian of a curve. These relations depend on the linear systems the curve admits. They are obtained in the tautological ring, the smallest subspace containing (an embedding of) the curve and closed under the basic operations of intersection, Pontryagin product and the pullback and pushdown induced by homotheties.
In this paper, we develop the theory of flashes of an algebraic curve. We show that the theory is birationally invariant in a sense which we will make more precise below. We also show how the theory provides a foundation for the method of asymptotic degenerations, a particular class of degenerations of plane projective algebraic curves. In particular, we consider the geometrical technique in relation to the Severi problem of degenerating nodal curves to lines in general position, and correct Severi's original proof of his conjecture.
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theory allows to describe explicitly an image of a plane algebraic curve under rational transformation and to determine the number of common zeroes of two polynomials in two variables on a plane algebraic cu…
We prove a Curve Selection Lemma valid in infinite-dimensional algebraic geometry and apply it, together with a structure theorem of Grinberg-Kahzdan and Drinfeld, to obtain a Curve Selection Lemma in arc spaces.
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some topological features of surfaces (reality, compactness, connectivity) and to the problem of plotting.
Motivated by Xing's method [7], we show that there exist [n,k,d] linear Hermitian codes over F_{q^2} with k+d>=n-3 for all sufficiently large q. This improves the asymptotic bound of Algebraic-Geometry codes from Hermitian curves given in [9,10].
The purpose of this paper is two-fold. We first prove a series of results, concerned with the notion of Zariski multiplicity, mainly for non-singular algebraic curves. These results are required in the paper "A Theory of Branches for Algebraic Curves",(*), where, following Severi, we introduced the notion of the "branch" of an algebraic curve. Secondly, we use results from the cited paper, (*), in order to develop a refined theory of g_{n}^{r} on an algebraic curve. The refinement depends critically on relacing the notion of a point with a branch. This allows us to construct a theory of diviso…
