algebraic equivalence of divisors
Let $X$ be a surface (a twodimensional algebraic variety).
Definition 1.

1.
An algebraic family of effective divisors on $X$ parametrized by a nonsingular^{} 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 nonsingular 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^{} $D={D}_{0},{D}_{1},\mathrm{\dots},{D}_{n}={D}^{\prime}$ with ${D}_{i}$ and ${D}_{i+1}$ prealgebraically equivalent for all $$.
Title  algebraic equivalence of divisors 

Canonical name  AlgebraicEquivalenceOfDivisors 
Date of creation  20130322 15:34:10 
Last modified on  20130322 15:34:10 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  4 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 14C20 