arXiv subjects

Kapil Hari Paranjape

Publications and source records attributed to Kapil Hari Paranjape.

3 recordsLinked to original sources

arXiv2003

The Bogomolov--Pantev resolution, an expository account

This is a paper based on a talk given at the Warwick Symposium on Algebraic Geometry in 1996. The resolution of singularities given by F. Bogomolov and A. Pantev (arXiv:math.AG/9603019) is presented in a self-contained and "elementary" manner. The main result is that given a variety and a proper closed subvariety, there is a projective birational morphism from a smooth variety to the given variety so that the inverse image of the given closed subvariety and the exceptional locus are sub-divisors of a divisor with simple normal crossings.

math.AG↗

arXiv2003

Cycles over fields of transcendence degree one

We extend earlier examples provided by Schoen, Nori and Bloch to show that when a surface has the property that the kernel of its Albanese map is non-zero over the field of complex numbers, this kernel is non-zero over a field of transcendence degree one. This says that the conjecture of Bloch and Beilinson that this kernel is zero for varieties over number fields is precise in the sense that it is not valid for fields of transcendence degree one.

math.AG↗

arXiv2002

A Geometric characterization of Arithmetic Varieties

A result of Belyi can be stated as follows. Every curve defined over a number field can be expressed as a cover of the projective line with branch locus contained in a rigid divisor. We define the notion of geometrically rigid divisors in surfaces and then show that every surface defined over a number field can be expressed as a cover of the projective plane with branch locus contained in a geometrically rigid divisor in the plane. The main result is the characterisation of arithmetically defined divisors in the plane as geometrically rigid divisors in the plane.

math.AG↗