algebraic equivalence of divisors

Let $X$ be a surface (a two-dimensional algebraic variety).

1.

An algebraic family of effective divisors on $X$ parametrized by a non-singular curve $T$ is defined to be an effective Cartier divisor $\mathcal{D}$ on $X\times T$ which is flat over $T$ .
2.

If $\mathcal{F}$ is an algebraic family of effective divisors on $X$ , parametrized by a non-singular curve $T$ , and $P,Q\in T$ are any two closed points on $T$ , then we say that the corresponding divisors in $\mathcal{F}$ , $D_{P},D_{Q}$ , are prealgebraically equivalent.
3.

Two (Weil) divisors $D,D^{\prime}$ on $X$ are algebraically equivalent if there is a finite sequence $D=D_{0},D_{1},\ldots,D_{n}=D^{\prime}$ with $D_{i}$ and $D_{i+1}$ prealgebraically equivalent for all $0\leq i<n$ .

Title	algebraic equivalence of divisors
Canonical name	AlgebraicEquivalenceOfDivisors
Date of creation	2013-03-22 15:34:10
Last modified on	2013-03-22 15:34:10
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Definition
Classification	msc 14C20