Ampler divisor

In mathematics, the ample divisor is a term from algebraic geometry . Algebraic geometry links the equations of abstract algebra with geometry . Divisors describe the zeros of algebraic curves , and somewhat simplified they are traffic lights if their basic functions have no common zeros.

The amplified divisors show whether the algebraic curves described by polynomials can be mapped onto a projective space .

definition

Is a divisor on a algebraic curve and the vector space that of rational functions on whose principal divisor the inequality satisfied. ${\displaystyle D}$ ${\displaystyle C}$ ${\displaystyle H^{0}(C,{\mathcal {O}}(D))}$ ${\displaystyle C}$ ${\displaystyle (f)}$ ${\displaystyle (f)\geq -D}$

${\displaystyle D}$is called very traffic light when there is a base of so that the functions have no common zero on and the mapping ${\displaystyle f_{0},\ldots ,f_{n-1}}$ ${\displaystyle H^{0}(C,{\mathcal {O}}(D))}$ ${\displaystyle f_{0},\ldots ,f_{n-1}}$ ${\displaystyle C}$

${\displaystyle \phi _{D}\colon C\to P^{n-1}}$

${\displaystyle \phi _{d}(P)=\left[f_{0}(P)\colon \ldots \colon f_{n-1}(P)\right]}$

is an embedding in the projective space .

${\displaystyle D}$is called traffic light if there is a natural number so that is very traffic light. ${\displaystyle l\in \mathbb {N} }$ ${\displaystyle lD}$

Examples

• Let be the projective line and . A rational function with must therefore only have one pole and there at most of degree 2. This is of the form for a homogeneous polynomial of degree 2. A basis of the vector space of these functions is, for example . The mapping defined with this base is the embedding of the projective straight line as a closed parabola in . So it's very traffic light. ${\displaystyle C=P^{1}}$ ${\displaystyle D=2\left[0:1\right]}$ ${\displaystyle f}$ ${\displaystyle (f)\geq -D}$ ${\displaystyle \left[0:1\right]}$ ${\displaystyle f}$ ${\displaystyle f={\tfrac {p}{x_{0}^{2}}}}$ ${\displaystyle p}$ ${\displaystyle f_{0}=1,f_{1}={\tfrac {x_{1}}{x_{0}}},f_{2}={\tfrac {x_{1}^{2}}{x_{0}^{2}}}}$ ${\displaystyle \phi _{d}}$ ${\displaystyle P^{2}}$ ${\displaystyle D}$
• An elliptic curve in intersects a projective straight line at three points . Then it is very traffic light. ${\displaystyle P^{2}}$ ${\displaystyle P_{1},P_{2},P_{3}}$ ${\displaystyle D:=P_{1}+P_{2}+P_{3}}$
• The canonical divisor of an algebraic curve by gender is very traffic light if the curve is not hyperelliptical . ${\displaystyle g\geq 2}$

literature

• E. Arbarello, M. Cornalba, PA Griffiths, J. Harris: Geometry of algebraic curves. Vol. I (Fundamental Principles of Mathematical Sciences, 267). Springer-Verlag, New York 1985, ISBN 0-387-90997-4

Web links

• Christina Birkenhake : Algebraic Geometry - An Insight (PDF) 2008

Individual evidence

1. ↑ Birkenhake, op, cit., Examples 4 & 7
2. ↑ Birkenhake, op, cit., Example 9
3. ↑ Birkenhake, Algebraic Geometry - An Insight, 2008 (see web links), example 8