scholarly journals Algebraicity of normal analytic compactifications of ℂ2 with one irreducible curve at infinity

2016 ◽  
Vol 10 (8) ◽  
pp. 1641-1682 ◽  
Author(s):  
Pinaki Mondal
Keyword(s):  
Irreducible Curve

scholarly journals Connected Numbers and the Embedded Topology of Plane Curves

2018 ◽  
Vol 61 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Taketo Shirane
Keyword(s):  
Plane Curve ◽  
Smooth Curve ◽  
Plane Curves ◽  
Irreducible Curve ◽  
Galois Cover ◽  
Splitting Number

AbstractThe splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

scholarly journals Numerical computation of the genus of an irreducible curve within an algebraic set

2011 ◽  
Vol 215 (8) ◽  
pp. 1844-1851 ◽  
Author(s):  
Daniel J. Bates ◽  
Chris Peterson ◽  
Andrew J. Sommese ◽  
Charles W. Wampler
Keyword(s):  
Numerical Computation ◽  
Irreducible Curve ◽  
Algebraic Set

scholarly journals Higher Nash blowups

2007 ◽  
Vol 143 (6) ◽  
pp. 1493-1510 ◽  
Author(s):  
Takehiko Yasuda
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Negative Integer ◽  
Curve Singularity ◽  
Irreducible Curve ◽  
Numerical Monoid

AbstractFor each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n=1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.

On the Hyperplane Sections Through two Given Points of an Algebraic Variety

1970 ◽  
Vol 22 (1) ◽  
pp. 128-133 ◽  
Author(s):  
Wei-Eihn Kuan
Keyword(s):  
Simple Proof ◽  
Hyperplane Section ◽  
Rational Points ◽  
Infinite Field ◽  
Important Special Case ◽  
Irreducible Curve ◽  
Irreducible Variety ◽  
Dimension 2 ◽  
Hyperplane Sections ◽  
Special Case

1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.

A Note About Linear Systems on Curves

1984 ◽  
Vol 27 (3) ◽  
pp. 371-374
Author(s):  
Allen Tannenbaum
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Minimal Degree ◽  
Maximal Dimension ◽  
Degree Relative ◽  
Irreducible Curve ◽  
The Given

AbstractInverting the Castelnuovo bound in two ways, we show that for given integers p ≥ 0, d > 1, n > 1, we can find a smooth irreducible curve of genus p which contains a linear system of degree d and of maximal dimension relative to the given data p and d, and a smooth irreducible curve of genus p which contains a linear system of dimension n and of minimal degree relative to the data p and n.

scholarly journals Differential operators on a hypersurface

1986 ◽  
Vol 103 ◽  
pp. 67-84 ◽  
Author(s):  
Balwant Singh
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Differential Operators ◽  
Coordinate Ring ◽  
Irreducible Curve ◽  
Additional Hypothesis

We study differential operators on an affine algebraic variety, especially a hypersurface, in the context of Nakai’s Conjecture. We work over a field k of characteristic zero. Let X be a reduced affine algebraic variety over k and let A be its coordinate ring. Let be the A-module of differential operators of A over k of order ≤ n. Nakai’s Conjecture asserts that if is generated by for every n ≥ 2 then A is regular. In 1973 Mount and Villamayor [6] proved this in the case when X is an irreducible curve. In the general case no progress seems to have been made on the conjecture, except for a result of Brown [2], where the assertion is proved under an additional hypothesis. An interesting result proved by Becker [1] and Rego [8] says that Nakai’s Conjecture implies the Conjecture of Zariski-Lipman, which is still open in the general case and which asserts that if the module of k-derivations of A is A-projective then A is regular.

scholarly journals The osculating of a certain curve in [4]

1971 ◽  
Vol 17 (3) ◽  
pp. 277-280 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Special Interest ◽  
Algebraic Surfaces ◽  
Analogous Problem ◽  
Irreducible Curve ◽  
Linearly Independent ◽  
Canonical Curve

The equation of the osculating plane at a point on the complete irreducible curve of intersection of two algebraic surfaces in [3] was found by Hesse (5, p. 283); the plane, having to contain the tangent of the curve, belongs to the pencil spanned by the tangent planes of the two surfaces, and it is a question of determining which plane of the pencil to choose. The equation also appears in the books of Salmon (6, p. 378) and Baker (1, p. 206). The analogous problem for the osculating solid at a point on the complete irreducible curve of intersection of three algebraic primals, or threefolds, in [4] does not appear to have been considered. The simplest instance is the octavic curve C of intersection of three quadrics, and this has the special interest of being a canonical curve; moreover the quadrics are of the same order, and so can be replaced by any three linearly independent members of the net which they determine, a replacement of which it may be prudent to take advantage with a view to simplifying the algebra. It is a question of determining which solid to choose among the tangent solids to the quadrics of the net at a point on C, but while Hesse's methods serve to carry one a certain distance there seems no obvious way of pushing them to a conclusion. It is then natural, with a view to reaching a conclusion, to choose a net of quadrics that, through having some particular property, is more amenable.

Dimension of the Moduli Space of a Germ of Curve in ℂ2

2020 ◽  
Author(s):  
Yohann Genzmer
Keyword(s):  
Complex Plane ◽  
Moduli Space ◽  
Irreducible Curve

Abstract In this article, we prove a formula that computes the generic dimension of the moduli space of a germ of irreducible curve in the complex plane. It is obtained from the study of the Saito module associated to the curve, which is the module of germs of holomorphic $1$-forms leaving the curve invariant.

scholarly journals Osculating primes to curves of intersection in 4-space, and to certain curves in n-space

1973 ◽  
Vol 18 (4) ◽  
pp. 325-338 ◽  
Author(s):  
R. H. Dye
Keyword(s):  
Linear Combination ◽  
Complete Intersection ◽  
Tangent Line ◽  
Simple Point ◽  
Irreducible Curve

An irreducible curve in S4, projective 4-space, may arise as the complete intersection of three given irreducible threefolds. At a simple point P on such a curve there is an osculating solid, and we would like to have its equation. This solid, necessarily containing the tangent line to the curve at P, belongs to the net spanned by the tangent solids at P to the threefolds. We seek the appropriate linear combination of the known equations for these tangent solids.

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