Generators of Ideals Defining Certain Surfaces in Projective Space

1996 ◽  
Vol 48 (3) ◽  
pp. 585-595 ◽  
Author(s):  
Sandeep H. Holay
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Ample Divisor ◽  
Plane Curves ◽  
Closed Field ◽  
Divisor Class ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Generating Sets ◽  
Blowing Up

AbstractWe consider the surface obtained from the projective plane by blowing up the points of intersection of two plane curves meeting transversely. We find minimal generating sets of the defining ideals of these surfaces embedded in projective space by the sections of a very ample divisor class. All of the results are proven over an algebraically closed field of arbitrary characteristic.

scholarly journals Algebraic Barth-Lefschetz theorems

1996 ◽  
Vol 142 ◽  
pp. 17-38 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Algebraic Variety ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Basic Properties ◽  
Positive Integers ◽  
Lefschetz Theorems ◽  
Algebraic Scheme

We shall work over a fixed algebraically closed field k of arbitrary characteristic. By an algebraic variety over k we shall mean a reduced algebraic scheme over k. Fix a positive integer n and e = (e0, el,…, en) a system of n + 1 weights (i.e. n + 1 positive integers e0, el,…, en). If k[T0, Tl,…, Tn] is the polynomial k-algebra in n + 1 variables, graded by the conditions deg(Ti) = ei i = 0, 1,…, n, denote by Pn(e) = Proj(k[T0, T1,…, Tn]) the n-dimensional weighted projective space over k of weights e. We refer the reader to [3] for the basic properties of weighted projective spaces.

scholarly journals Isomorphisms between complements of projective plane curves

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Mattias Hemmig
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curves ◽  
Complex Numbers ◽  
Closed Field ◽  
Algebraically Closed Field

In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if there exists a line that intersects a unicuspidal curve $C \subset \mathbb{P}^2$ only in its singular point, then any other curve whose complement is isomorphic to $\mathbb{P}^2 \setminus C$ must be projectively equivalent to $C$. This generalizes a result of H. Yoshihara who proved this result over the complex numbers. Moreover, we study properties of multiplicity sequences of irreducible curves that imply that any isomorphism between the complements of these curves extends to an automorphism of $\mathbb{P}^2$. Using these results, we show that two irreducible curves of degree $\leq 7$ have isomorphic complements if and only if they are projectively equivalent. Finally, we describe new examples of irreducible projectively non-equivalent curves of degree $8$ that have isomorphic complements.

Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces

2002 ◽  
Vol 45 (3) ◽  
pp. 349-354 ◽  
Author(s):  
Marc Coppens
Keyword(s):  
Projective Space ◽  
Linear Systems ◽  
Exceptional Divisor ◽  
Closed Field ◽  
Invertible Sheaf ◽  
Algebraically Closed Field ◽  
Structure Sheaf ◽  
Blowing Up ◽  
Dimensional Projective Space ◽  
Ample Invertible Sheaf

AbstractLet Pn be the n-dimensional projective space over some algebraically closed field k of characteristic 0. For an integer t ≥ 3 consider the invertible sheaf O(t) on Pn (Serre twist of the structure sheaf). Let , the dimension of the space of global sections of O(t), and let k be an integer satisfying 0 < k ≤ N − (2n + 2). Let P1,…,Pk be general points on Pn and let π : X → Pn be the blowing-up of Pn at those points. Let Ei = π−1(Pi) with 1 ≤ i ≤ k be the exceptional divisor. Then M = π*(O(t)) ⊗ OX(−E1 — … — Ek) is a very ample invertible sheaf on X.

scholarly journals A remark on the Grothendieck-Lefschetz theorem about the Picard group

1978 ◽  
Vol 71 ◽  
pp. 169-179 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Algebraic Variety ◽  
Picard Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Lefschetz Theorem

Let K be an algebraically closed field of arbitrary characteristic. The term “variety” always means here an irreducible algebraic variety over K. The notations and the terminology are borrowed in general from EGA [4].

scholarly journals Elements Generating a Proper Normal Subgroup of the Cremona Group

2020 ◽  
Author(s):  
Serge Cantat ◽  
Vincent Guirardel ◽  
Anne Lonjou
Keyword(s):  
Normal Subgroup ◽  
Projective Plane ◽  
Infinite Order ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cremona Group ◽  
Birational Transformations ◽  
Proper Normal Subgroup

Abstract Consider an algebraically closed field ${\textbf{k}}$, and let $\textsf{Cr}_2({\textbf{k}})$ be the Cremona group of all birational transformations of the projective plane over ${\textbf{k}}$. We characterize infinite order elements $g\in \textsf{Cr}_2({\textbf{k}})$ having a power $g^n$, $n\neq 0$, generating a proper normal subgroup of $\textsf{Cr}_2({\textbf{k}})$.

scholarly journals On Equivariant Vector Bundles on an Almost Homogeneous Variety

1975 ◽  
Vol 57 ◽  
pp. 65-86 ◽  
Author(s):  
Tamafumi Kaneyama
Keyword(s):  
Vector Bundles ◽  
Multiplicative Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Algebraic Torus ◽  
Homogeneous Variety

Let k be an algebraically closed field of arbitrary characteristic. Let T be an n-dimensional algebraic torus, i.e. T = Gm × · · · × Gm n-times), where Gm = Spec (k[t, t-1]) is the multiplicative group.

Dimension of noncommutative plane curves

2019 ◽  
Vol 18 (03) ◽  
pp. 1950057
Author(s):  
Søren Jøndrup
Keyword(s):  
Proper Ideal ◽  
Plane Curves ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Noncommutative Plane

In this paper, we prove that an algebra of the form [Formula: see text] is never right (or left) artinian in case [Formula: see text] is a proper ideal and [Formula: see text] is an uncountable, algebraically closed field of characteristic [Formula: see text].

scholarly journals Every Curve of Genus not Greater Than Eight Lies on a K3 Surface

2008 ◽  
Vol 190 ◽  
pp. 183-197 ◽  
Author(s):  
Manabu Ide
Keyword(s):  
Smooth Curve ◽  
Ample Divisor ◽  
Closed Field ◽  
K3 Surface ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Smooth Locus ◽  
Double Coverings

Let C be a smooth irreducible complete curve of genus g ≥ 2 over an algebraically closed field of characteristic 0. An ample K3 extension of C is a K3 surface with at worst rational double points which contains C in the smooth locus as an ample divisor.In this paper, we prove that all smooth curve of genera. 2 ≤ g ≤ 8 have ample K3 extensions. We use Bertini type lemmas and double coverings to construct ample K3 extensions.

scholarly journals Which rational double points occur on del Pezzo surfaces?

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Claudia Stadlmayr
Keyword(s):  
Positive Characteristic ◽  
Del Pezzo Surfaces ◽  
Closed Field ◽  
Arbitrary Degree ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Arbitrary Characteristic ◽  
Classical Work ◽  
Del Pezzo

We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${\rm char}(k)=p \geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.

On the complement of a nef and big divisor on an algebraic variety

1996 ◽  
Vol 120 (3) ◽  
pp. 411-422 ◽  
Author(s):  
Francesco Russo
Keyword(s):  
Algebraic Variety ◽  
Cartier Divisor ◽  
Ample Divisor ◽  
Closed Field ◽  
Finitely Generated ◽  
Complete Variety ◽  
Algebraically Closed Field

Let X be an algebraic (complete) variety over a fixed algebraically closed field k. To every Cartier divisor D on X, we can associate the graded k-algebra . As is known, for a semi-ample divisor D, R(X, D) is a finitely generated k-algebra (see [21] or [9]), while this property is no longer true for arbitrary nef and big divisors (see [21]).

Sign in / Sign up

Export Citation Format

Share Document