Rationality of bundles of conics with degenerate curve of degree five and even theta-characteristic

1984 ◽  
Vol 24 (4) ◽  
pp. 449-452 ◽  
Author(s):  
I. A. Panin
Keyword(s):  
Degenerate Curve ◽  
Theta Characteristic

scholarly journals Thomae-Weber Formula: Algebraic Computations of Theta Constants

2019 ◽  
Author(s):  
Turku Ozlum Celik
Keyword(s):  
Hyperelliptic Curve ◽  
Algebraic Method ◽  
Hyperelliptic Curves ◽  
Fourth Power ◽  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Singular Quadric ◽  
Level 2 ◽  
Theta Characteristic

Abstract We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated with a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic curves of genus 4, in particular, such curves lying on a singular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, we obtain a complete level 2 structure of the curves by studying their theta characteristic divisors via exceptional divisors of the del Pezzo surfaces as the structure is required for the method.

Degenerate surfaces in higher space

1933 ◽  
Vol 29 (1) ◽  
pp. 88-94 ◽  
Author(s):  
L. Roth
Keyword(s):  
General Type ◽  
General Character ◽  
Space Curves ◽  
Single Curve ◽  
General Surface ◽  
Previous Note ◽  
Surface Of General Type ◽  
Degenerate Curve

1. In a previous note the author has examined the systems of tangent planes to a degenerate surface in S3 consisting of n planes, regarded as the limit of a general surface of the same order. It is well known that a pair of space curves which are the limiting form of a non-degenerate curve must have a certain number of intersections; hence, if a surface in higher space degenerates into a number of surfaces, these must intersect in curves of various orders. In the present paper we consider the nature of the envelope to a surface consisting of two general surfaces of S4 having in common a single curve of general character, the degenerate surface being regarded as the limit of some surface of general type. The same conclusions hold for a similar degenerate surface in Sr (r>4).

A Vanishing Theorem for the Twisted Normal Bundle of Curves in ,

2019 ◽  
Vol 63 (1) ◽  
pp. 1-5
Author(s):  
E. Ballico
Keyword(s):  
Normal Bundle ◽  
Necessary Condition ◽  
Vanishing Theorem ◽  
Degenerate Curve

AbstractWe prove the existence of a smooth and non-degenerate curve $X\subset \mathbb{P}^{n}$, $n\geqslant 8$, with $\deg (X)=d$, $p_{a}(X)=g$, $h^{1}(N_{X}(-1))=0$, and general moduli for all $(d,g,n)$ such that $d\geqslant (n-3)\lceil g/2\rceil +n+3$. It was proved by C. Walter that, for $n\geqslant 4$, the inequality $2d\geqslant (n-3)g+4$ is a necessary condition for the existence of a curve with $h^{1}(N_{X}(-1))=0$.

scholarly journals THE CURVE OF LINES ON A PRIME FANO THREE-FOLD OF GENUS 8

2010 ◽  
Vol 21 (12) ◽  
pp. 1561-1584
Author(s):  
F. FLAMINI ◽  
E. SERNESI
Keyword(s):  
Natural Embedding ◽  
Theta Characteristic

We show that a general prime Fano three-fold X of genus 8 can be reconstructed from the pair (Γ, L), where Γ is its Fano curve of lines and L = OΓ(1) is the theta-characteristic which gives a natural embedding Γ ⊂ ℙ5.

On curves with a theta-characteristic whose space of sections has dimension 4

1994 ◽  
Vol 215 (1) ◽  
pp. 655-665 ◽  
Author(s):  
Elisabetta Colombo
Keyword(s):  
Theta Characteristic

Boundary value problems for first-order mixed complex equations with degenerate curve

2012 ◽  
Vol 91 (3) ◽  
pp. 545-558
Author(s):  
Guo Chun Wen
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Mixed Complex ◽  
First Order ◽  
Degenerate Curve

scholarly journals Degenerate curve attacks: extending invalid curve attacks to Edwards curves and other models

2018 ◽  
Vol 12 (3) ◽  
pp. 217-225 ◽  
Author(s):  
Samuel Neves ◽  
Mehdi Tibouchi
Keyword(s):  
Edwards Curves ◽  
Degenerate Curve

scholarly journals Space curves, X-ranks and cuspidal projections

2021 ◽  
Author(s):  
Edoardo Ballico
Keyword(s):  
Lower Bounds ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Geometric Genus ◽  
Space Curves ◽  
Unique Pair ◽  
Degenerate Curve

AbstractLet $$X\subset \mathbb {P}^3$$ X ⊂ P 3 be an integral and non-degenerate curve. We say that $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X has X-rank 3 if there is no line $$L\subset \mathbb {P}^3$$ L ⊂ P 3 such that $$q\in L$$ q ∈ L and $$\#(L\cap X)\ge 2$$ # ( L ∩ X ) ≥ 2 . We prove that for all hyperelliptic curves of genus $$g\ge 5$$ g ≥ 5 there is a degree $$g+3$$ g + 3 embedding $$X\subset \mathbb {P}^3$$ X ⊂ P 3 with exactly $$2g+2$$ 2 g + 2 points with X-rank 3 and another embedding without points with X-rank 3 but with exactly $$2g+2$$ 2 g + 2 points $$q\in \mathbb {P}^3$$ q ∈ P 3 such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X . For all integers $$g\ge 5$$ g ≥ 5 , $$x\ge 0$$ x ≥ 0 we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, $$\deg (X) = x+g+3$$ deg ( X ) = x + g + 3 and having at least $$x+2g+2$$ x + 2 g + 2 points of X-rank 3.

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