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An effective Cartier divisor on X is an ideal sheaf I which is invertible and such that for every point x in X, the stalk I x is principal. It is equivalent to require that around each x , there exists an open affine subset U = Spec A such that U ∩ D = Spec A / ( f ) , where f is a non-zero divisor in A .
In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, [1] or equivalently if the map from R to R that sends x to ax is not injective. [a] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0.
If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take x = 0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that ax = 0. [2]
Let X be a scheme which is finite type over a field k. An algebraic r-cycle on X is a formal linear combination [] of r-dimensional closed integral k-subschemes of X. The coefficient n i is the multiplicity of V i. The set of all r-cycles is the free abelian group
For example, using single-precision IEEE arithmetic, if x = −2 −149, then x/2 underflows to −0, and dividing 1 by this result produces 1/(x/2) = −∞. The exact result −2 150 is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.
A linear system of divisors algebraicizes the classic geometric notion of a family of curves, as in the Apollonian circles.. In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.
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A divisor on a Riemann surface C is a formal sum = of points P on C with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of C, defining () as the vector space of functions having poles only at points of D with positive coefficient, at most as bad as the coefficient indicates, and having zeros at points of D with negative ...