Numerical equivalence

In algebraic geometry , numerical equivalence is an equivalence relation between algebraic cycles of a variety .

definition

Two cycles of the same dimension in a variety are called numerically equivalent if the equation for all cycles with ${\displaystyle Z_{1},Z_{2}}$ ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle dim(T)=kodim(Z_{i})}$

${\displaystyle deg(Z_{1}\cap T)=deg(Z_{2}\cap T)}$

applies. Here the degree of the sub-varieties denotes , i.e. the number of intersections with a set of hyperplanes in a general position. ${\displaystyle deg}$

properties

1. Linearity: Numerical equivalence is compatible with the addition of cycles.
2. Chow's Moving Lemma : For cycles in a variety there is a numerically equivalent cycle that is too in general position. ${\displaystyle A,B}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle A^{\prime }}$ ${\displaystyle B}$
3. Push-Forwards: Be a cycle in and a cycle in that is too in general. If numeric is zero-equivalent, then the projection of on is numeric zero-equivalent. ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle B}$ ${\displaystyle X\times Y}$ ${\displaystyle A\times Y}$ ${\displaystyle A}$ ${\displaystyle B\cdot (A\times Y)}$ ${\displaystyle Y}$

The same properties also have the equivalence relations rational equivalence , algebraic equivalence and homological equivalence , among which numerical equivalence is the weakest equivalence relation.

literature

• Uwe Jannsen : "Equivalence relations on algebraic cycles", The Arithmetic and Geometry of Algebraic Cycles, pp. 225-260, Kluwer Ac. Publ. Co. (2000)