Hilbert's theorem 90 forK 2, with application to the Chow groups of rational surfaces

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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On classes in the classification of curves on rational surfaces with respect to logarithmic plurigenera

Singularities in Geometry and Topology 2011 ◽

10.2969/aspm/06610093 ◽

2019 ◽

Author(s):

Hirotaka Ishida

Keyword(s):

Rational Surfaces

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On the first Steenrod square for Chow groups

American Journal of Mathematics ◽

10.1353/ajm.2013.0003 ◽

2013 ◽

Vol 135 (1) ◽

pp. 53-63 ◽

Cited By ~ 4

Author(s):

Olivier Haution

Keyword(s):

Chow Groups ◽

Steenrod Square

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The surfaces whose prime-sections contain a

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100016583 ◽

1934 ◽

Vol 30 (2) ◽

pp. 170-177 ◽

Cited By ~ 2

Author(s):

J. Bronowski

Keyword(s):

Hyperelliptic Curves ◽

Ruled Surfaces ◽

Minimum Order ◽

Rational Surfaces ◽

Pencil Of Conics

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

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