Elementární spůsob vyšetřování křivek v rovině. [II.]
Enumerative geometry of divisorial families of rational curves
We compute the number of irreducible rational curves of given degree with 1 tacnode in or 1 node in meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree passing through given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.
Euclidean algorithm and Kummer covers with many points.
Famiglie di curve sulle superfici algebriche
Families of smooth curves on surface singularities and wedges
Following the study of the arc structure of singularities, initiated by J. Nash, we give criteria for the existence of smooth curves on a surface singularity (S,O) and of smooth branches of its generic hypersurface section. The main applications are the following: the existence of a natural partition of the set of smooth curves on (S,O) into families, a description of each of them by means of chains of infinitely near points and their associated maximal cycle and the existence of smooth curves on...
Génération des lignes et des surfaces du second degré, d'après Jacobi
Genre de courbes lisses tracées sur certaines surfaces rationnelles de
Geometry of arithmetically Gorenstein curves in P4.
We characterize the postulation character of arithmetically Gorenstein curves in P4. We give conditions under which the curve can be realized in the form mH - K on some ACM surface. Finally, we complement a theorem by Watanabe by showing that any general arithmetically Gorenstein curve in P4 with arbitrary fixed postulation character can be obtained from a line by a series of ascending complete-intersection biliaisons.
Geometry of Puiseux expansions
We consider the space Curv of complex affine lines t ↦ (x,y) = (ϕ(t),ψ(t)) with monic polynomials ϕ, ψ of fixed degrees and a map Expan from Curv to a complex affine space Puis with dim Curv = dim Puis, which is defined by initial Puiseux coefficients of the Puiseux expansion of the curve at infinity. We present some unexpected relations between geometrical properties of the curves (ϕ,ψ) and singularities of the map Expan. For example, the curve (ϕ,ψ) has a cuspidal singularity iff it is a critical...
Gonality and Clifford index for real algebraic curves.
Let X be a smooth connected projective curve of genus g defined over R. Here we give bounds for the real gonality of X in terms of the complex gonality of X.
Gonality for stable curves and their maps with a smooth curve as their target
Here we study the deformation theory of some maps f: X → ℙr , r = 1, 2, where X is a nodal curve and f|T is not constant for every irreducible component T of X. For r = 1 we show that the “stratification by gonality” for any subset of [...] with fixed topological type behaves like the stratification by gonality of M g.
Gorenstein liaison of some curves in P4.
Despite the recent advances made in Gorenstein liaison, there are still many open questions for the theory in codimension ≥ 3. In particular we consider the following question: given two curves in Pn with isomorphic deficiency modules (up to shift), can they be evenly Gorenstein linked? The answer for this is yes for curves in P3, due to Rao, but for higher codimension the answer is not known. This paper will look at large classes of curves in P4 with isomorphic deficiency modules and show that...
Halphen gaps and good space curves
In questo articolo dimostriamo l'esistenza di curve "buone e generali" di grado e genere che giacciono su di una superfice liscia di grado , per ogni , , e in un certo intervallo vicino al genere massimo.
Hvězdicové mnohoúhelníky
The article presents some possibilities to create non-convex star polygons from regular polygons. The text includes exercises about the construction of the star polygons and exercises inciting to study their attributes and their using in the everyday life of the pupil. The subject of the star polygons deepens the basic curriculum in plane geometry in RVP and it is a suitable motivation complement in teaching mathematics at the second stage of elementary school as well as at grammar school.
Hyperelliptic plane curves of type .
Indice de Clifford des intersections complètes de l'espace
Inflection points on real plane curves having many pseudo-lines.
La conjecture de Green générique
Une courbe projective et lisse de genre , non hyperelliptique, admet un plongement canonique dans un espace projectif . Un résultat classique affirme que l’idéal gradué des équations de dans est engendré par ses éléments de degré , sauf si admet certains systèmes linéaires très particuliers. Mark Green en a proposé il y a vingt ans une vaste généralisation, qui décrit la résolution minimale de en fonction de l’existence de systèmes linéaires spéciaux sur . Claire Voisin vient de...
Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n'est (presque) jamais réduit
Linear systems over ℙ¹×ℙ¹ with base points of multiplicity bounded by three
We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general position. As an application, we obtain a classification of special linear systems on ℙ¹×ℙ¹ with multiplicities not exceeding 3.