Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze

D'ANDREA, CARLOS; KRICK, TERESA; SOMBRA, MARTÍN

ANNALES SCIENTIFIQUES DE L4ECOLE NORMALE SUPERIEURE

SOC MATHEMATIQUE FRANCE

Lugar: Paris; Año: 2013 vol. 46 p. 549 - 627

0012-9593

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    heights} of multiprojective varieties. 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.