scholarly journals Theta Surfaces

2020 ◽  
Author(s):  
Daniele Agostini ◽  
Türkü Özlüm Çelik ◽  
Julia Struwe ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions ◽  
Minkowski Sum ◽  
Zero Set ◽  
Analytic Surface ◽  
Numerical Algebraic Geometry ◽  
Abelian Integrals ◽  
Space Curves ◽  
Quartic Curves ◽  
Special Plane

Abstract A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

scholarly journals Algebraic Geometry and Theta Functions

Nature ◽  
1930 ◽  
Vol 125 (3160) ◽  
pp. 775-775
Author(s):  
H. T. H. P.
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions

scholarly journals Injective linear series of algebraic curves on quadrics

2020 ◽  
Vol 66 (2) ◽  
pp. 231-254
Author(s):  
Edoardo Ballico ◽  
Emanuele Ventura
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Algebraic Curves ◽  
Linear Series ◽  
Central Importance ◽  
Arbitrary Characteristic ◽  
Space Curves

Abstract 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).

scholarly journals Identifiability and numerical algebraic geometry

PLoS ONE ◽  
2019 ◽  
Vol 14 (12) ◽  
pp. e0226299
Author(s):  
Daniel J. Bates ◽  
Jonathan D. Hauenstein ◽  
Nicolette Meshkat
Keyword(s):  
Algebraic Geometry ◽  
Numerical Algebraic Geometry

Numerical irreducible decomposition over a number field

2018 ◽  
Vol 17 (10) ◽  
pp. 1850195
Author(s):  
Timothy M. McCoy ◽  
Chris Peterson ◽  
Andrew J. Sommese
Keyword(s):  
Algebraic Geometry ◽  
Number Field ◽  
Polynomial System ◽  
Field Extension ◽  
General Point ◽  
Numerical Algebraic Geometry ◽  
Irreducible Decomposition ◽  
Algebraic Set ◽  
Field Of Definition ◽  
Witness Set

Let [Formula: see text] be a set of elements in the polynomial ring [Formula: see text], let [Formula: see text] denote the ideal generated by the elements of [Formula: see text], and let [Formula: see text] denote the radical of [Formula: see text]. There is a unique decomposition [Formula: see text] with each [Formula: see text] a prime ideal corresponding to a minimal associated prime of [Formula: see text] over [Formula: see text]. Let [Formula: see text] denote the reduced algebraic set corresponding to the common zeroes of the elements of [Formula: see text]. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text]. This corresponds to producing a witness set for [Formula: see text] for each [Formula: see text] together with the degree and dimension of [Formula: see text] (a point in a witness set for [Formula: see text] can be considered as a numerical approximation for a general point on [Formula: see text]). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let [Formula: see text] be a number field (i.e. a finite field extension of [Formula: see text]) and let [Formula: see text] be a set of elements in [Formula: see text]. We show how to extend techniques from numerical algebraic geometry to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text].

scholarly journals What is numerical algebraic geometry?

2017 ◽  
Vol 79 ◽  
pp. 499-507 ◽  
Author(s):  
Jonathan D. Hauenstein ◽  
Andrew J. Sommese
Keyword(s):  
Algebraic Geometry ◽  
Numerical Algebraic Geometry
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