# algebraic equivalence of divisors

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

###### Definition 1.
1. 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. 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. 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.

Title algebraic equivalence of divisors AlgebraicEquivalenceOfDivisors 2013-03-22 15:34:10 2013-03-22 15:34:10 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 14C20