algebraic equivalence of divisors

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

Definition 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.
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 $0\\leq i<n$ .

单词	AlgebraicEquivalenceOfDivisors
释义	algebraic equivalence of divisors Let $X$ be a surface (a two-dimensional algebraic variety). Definition 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. 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 $0\\leq i<n$ .
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