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algebraic equivalence of divisors
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(Definition)
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Let $X$ be a surface (a two-dimensional algebraic variety).
Definition 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 $T$ .
- 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.
- Two (Weil) divisors $D,D'$ on $X$ are algebraically equivalent if there is a finite sequence $D=D_0, D_1, \ldots, D_n=D'$ with $D_i$ and $D_{i+1}$ prealgebraically equivalent for all $0\leq i < n$ .
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"algebraic equivalence of divisors" is owned by alozano.
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Cross-references: finite sequence, equivalent, divisors, closed points, flat, Cartier divisor, effective, curve, non-singular, effective divisors, variety, algebraic, surface
There is 1 reference to this entry.
This is version 1 of algebraic equivalence of divisors, born on 2005-11-09.
Object id is 7474, canonical name is AlgebraicEquivalenceOfDivisors.
Accessed 1696 times total.
Classification:
| AMS MSC: | 14C20 (Algebraic geometry :: Cycles and subschemes :: Divisors, linear systems, invertible sheaves) |
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Pending Errata and Addenda
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