On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety

In this work we shall propose definitions for the tangent spaces TZ(X) and TZ(X) to the groups Z(X) and Z(X) of 0-cycles and divisors, respectively, on a smooth n-dimensional algebraic variety. Although the definitions are algebraic and formal, the motivation behind them is quite geometric and much of the text is devoted to this point. It is noteworthy that both the regular differential forms of all degrees and the field of definition enter significantly into the definition. An interesting and subtle algebraic point centers around the construction of the map THilb(X) → TZ(X). Another interesting algebraic/geometric point is the necessary appearance of spreads and absolute differentials in higher codimension. For an algebraic surface X we shall also define the subspace TZ rat(X) ⊂ TZ(X) of tangents to rational equivalences, and we shall show that there is a natural isomorphism TfCH (X) ∼= TZ(X)/TZ rat(X) where the left hand side is the formal tangent space to the Chow groups defined by Bloch. This result gives a geometric existence theorem, albeit at the infinitesimal level. The “integration” of the infinitesimal results raises very interesting geometric and arithmetic issues that are discussed at various places in the text. PUTangSp March 1, 2004 PUTangSp March 1, 2004