algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, Reider's theorem gives conditions for a

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

on a projective surface to be

very ample In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...

Statement

Let ''D'' be a nef

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

on a smooth projective surface ''X''. Denote by ''K''_''X'' the

canonical divisor In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...

of X. * If ''D''² > 4, then the

linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction o ...

, ''K''_''X''''+D'', has no base points unless there exists a nonzero effective divisor ''E'' such that **

DE = 0, E^2 = -1

, or **

DE = 1, E^2 =0

; * If ''D''² > 8, then the linear system , ''K''_''X''''+D'', is very ample unless there exists a nonzero effective divisor ''E'' satisfying one of the following: **

DE = 0, E^2 = -1

-2

; **

DE = 1, E^2 = 0

-1

; **

DE = 2, E^2 = 0

; **

DE = 3, D = 3E, E^2 = 1

Applications

Reider's theorem implies the surface case of the

Fujita conjecture In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved . It is named after Takao Fujita, who formulated it in 1985. Statement In complex geometry, the conjecture states that for a posi ...

. Let ''L'' be an ample line bundle on a smooth projective surface ''X''. If ''m'' > 2, then for ''D''=''mL'' we have * ''D''² = ''m''² ''L''² ≥ ''m''² > 4; * for any effective divisor ''E'' the ampleness of ''L'' implies ''D · E'' = ''m(L · E)'' ≥ m > 2. Thus by the first part of Reider's theorem , ''K''_''X''''+mL'', is base-point-free. Similarly, for any ''m'' > 3 the linear system , ''K''_''X''''+mL'', is very ample.

References

* Algebraic surfaces Theorems in algebraic geometry {{algebraic-geometry-stub