2011/03/23 by Carlos D'Andrea, Teresa Krick, D'Andrea, Carlos +3 · 1 citation
Mathematics · #11G50 #13P15 #14Q20 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #math.AC #math.AG #math.NT #msc:11G50 #msc:13P15 #msc:14Q20
paper · pdf · doi:10.48550/arxiv.1103.4561
73 pages, 2 figures. To appear in Annales Scientifiques de l'ENS
arxiv created 2012/10/20 · arxiv updated 2012/10/23
We present bounds for the degree and the height of the polynomials arising in some central problems in effective algebraic geometry including the implicitation 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.