Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze

Carlos D’Andrea; Teresa Krick; Martín Sombra

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 4, page 549-627
  • ISSN: 0012-9593

Abstract

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We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion from the point of view of resultant theory and establish some of its basic properties, including its behavior with respect to intersections, projections and products. We obtain analogous results for the function field case, including a parametric Nullstellensatz.

How to cite

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D’Andrea, Carlos, Krick, Teresa, and Sombra, Martín. "Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze." Annales scientifiques de l'École Normale Supérieure 46.4 (2013): 549-627. <http://eudml.org/doc/272167>.

@article{D2013,
abstract = {We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion from the point of view of resultant theory and establish some of its basic properties, including its behavior with respect to intersections, projections and products. We obtain analogous results for the function field case, including a parametric Nullstellensatz.},
author = {D’Andrea, Carlos, Krick, Teresa, Sombra, Martín},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {multiprojective spaces; mixed heights; resultants; implicitization; arithmetic nullstellensatz},
language = {eng},
number = {4},
pages = {549-627},
publisher = {Société mathématique de France},
title = {Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze},
url = {http://eudml.org/doc/272167},
volume = {46},
year = {2013},
}

TY - JOUR
AU - D’Andrea, Carlos
AU - Krick, Teresa
AU - Sombra, Martín
TI - Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 4
SP - 549
EP - 627
AB - We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion from the point of view of resultant theory and establish some of its basic properties, including its behavior with respect to intersections, projections and products. We obtain analogous results for the function field case, including a parametric Nullstellensatz.
LA - eng
KW - multiprojective spaces; mixed heights; resultants; implicitization; arithmetic nullstellensatz
UR - http://eudml.org/doc/272167
ER -

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