Families of Algebraic Cycles

  • William Fulton


If T is a non-singular curve, and p: S → is a morphism, any (k+ 1)-cycle \( \alpha = \sum {n_i}\left[ {{\mathfrak{F}_i}} \right] \) On S determines an algebraic family of k-cycles αt, on the fibres Y t =P -1 (t):
$$ {\alpha _t} = \begin{array}{*{20}{c}} \sum \\ {{\gamma _i}} \end{array}{n_i}\left[ {{{\left( {{V_i}} \right)}_t}} \right] $$
Rationally equivalent (k + 1)-cycles on S determine rationally equivalent k-cycles in each fibre. The basic operations of intersection theory preserve algebraic families. For example, if S is smooth over T, and {αt} and {βt} are algebraic families of cycles, then the intersection products α t · β t , also vary in an algebraic family. These facts are consequences of the general theorems of Chap. 6, and the recognition of α t , as the image of α by the refined Gysin homomorphism constructed from the diagram
$$ \begin{array}{*{20}{c}} {{Y_t}}& \to &\gamma \\ \downarrow &{}&{{ \downarrow ^p}} \\ {\left\{ t \right\}}& \to &T \end{array} $$

In this formulation, T may be replaced by any variety of arbitrary dimension, with t a regular point of T. This provides a simple method for studying algebraic equivalence.

The principle of continuity, or conservation of number, has two parts. First, in an algebraic family of zero-cycles, on a scheme which is proper over the parameter space, all the cycles have the same degree. Second, as mentioned above, the operations of intersection theory preserve algebraic families.

Refined intersection theory yields an improvement over classical formulations of this principle. For example, the ambient variety need not be complete; all that is necessary is that the locus of intersections is proper over the parameter space. This is useful for applications to enumerative geometry, when the ambient space is a space of non-degenerate geometric figures. In the last section an example of this kind is worked out: the formula for the number of curves in an r-dimensional family of plane curves which are tangent to r given plane curves in general position, in terms of the characteristics of the family, and the degrees and classes of the given curves.

where n i , is the degree of the i th surface, mi its first class (i.e. the number of points in a general plane section at which the tangent plane passes through a fixed general point), and the characteristic \( {v^i}{\rho ^{r - i}} \) is the number of curves in the family tangent to i general lines and r - i general planes. For the family of all (plane) conics in \( {\mathbb{P}^3} \) these characteristics were found by Chasles, cf. Schubert (1)§ 20:
$$ {v^8} = 92,{v^7}\rho = 116,{v^6}{\rho ^2} = 128,{v^5}{\rho ^3} = 104,{v^4}{\rho ^4} = 64,{v^3}{\rho ^5} = 32,{v^2}{\rho ^6} = 16,v{\rho ^7} = 8,{\rho ^8} = 4 $$

Similarly, there are 666,841,088 quadric surfaces in \( {\mathbb{P}^3} \) tangent to 9 given quadrics in general position.

A method for calculating the characteristics for the family of all quadrics of dimension m in \( {\mathbb{P}^n} \) was given by Schubert (4), based on the beautiful geometry of complete quadrics. This was reconsidered by Semple (1) and Tyrell (1), and recently by Demazure, Vainsencher, De Concini and Procesi.




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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • William Fulton
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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