Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze

Details Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze Heights of varieties in multiprojective spaces and arithmetic Nullstellensatze

2011-03-23

arxiv.org/abs/1103.4561v1

Mathematics

Algebraic Geometry

73 pages, 2 figures

Scientific paper

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 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.

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