Toric divisors

Toric divisors: Examples

Divisors on a toric variety <#Include Label="Divisors">

Polytope of toric divisors <#Include Label="PolytopeOfToricDivisor">
The GAP category true or false The GAP category of torus invariant Weil divisors.

Properties true or false Checks if the torus invariant Weil divisor divi is Cartier i.e. if it is locally principal. true or false Checks if the torus invariant Weil divisor divi is principal which in the toric invariant case means that it is the divisor of a character. true or false Checks if the Weil divisor divi represents a prime divisor, i.e. if it is a standard generator of the divisor group. true or false Checks if the divisor divi is basepoint free. true or false Checks if the divisor divi is ample, i.e. if it is colored red, yellow and green. true or false Checks if the divisor divi is very ample. true or false Checks if the divisor divi is nef.
Attributes a list Returns the Cartier data of the divisor divi, if it is Cartier, and fails otherwise. a homalg module element Returns the character corresponding to the principal divisor divi. a homalg module element Returns the class group element corresponding to the divisor divi. a polytope Returns the polytope corresponding to the divisor divi. a list Returns a basis of the global section module of the quasi-coherent sheaf of the divisor divi. an integer Returns an integer n such that n \cdot divi is very ample. a variety Returns the toric variety which contains the prime divisors of the divisor divi. a homalg module element Returns an element which represents the divisor divi in the Weil group. a variety Returns the closure of the torus orbit corresponding to the prime divisor divi. Not implemented for other divisors. Maybe we should add the support here. Is this even a toric variety? Exercise left to the reader. an integer Returns the degree of the divisor divi. This is not to be confused with the (divisor) class of divi! a variety Returns the variety corresponding to the polytope of the divisor divi. a list Returns the monoms in the Cox ring of degree equal to the (divisor) class of the divisor divi. a ring A basepoint free divisor divi defines a map from its ambient variety in a projective space. This method returns the Cox ring of such a projective space. a ring map A basepoint free divisor divi defines a map from its ambient variety in a projective space. This method returns the morphism between the cox ring of this projective space to the cox ring of the ambient variety of divi.
Methods a divisor Returns a very ample multiple of the ample divisor divi. The method will fail if divisor is not ample. a list Returns characters for closed embedding defined via the ample divisor divi. The method fails if the divisor divi is not ample. a divisor Returns the sum of the divisors divi1 and divi2. a divisor Returns the divisor divi1 minus divi2. a divisor Returns k times the divisor divi. a list Returns the monoms of the Cox ring of the variety vari with degree equal to the class group element elem. The variable elem can also be a list. a divisor Computes a divisor of the variety divi which is member of the divisor class presented by elem. The variable elem can be a homalg element or a list presenting an element. Adds the divisor divi to the Weil divisor list of its ambient variety. a polytope Returns the polytope of the divisor divi. Another name for PolytopeOfDivisor for compatibility and shortness. a ring Given a toric divisor divi, it induces a toric morphism. The target of this morphism is a toric variety. This method returns the Cox ring of this target. The variables are named according to string.
Constructors a divisor Returns the divisor of the toric variety vari which corresponds to the character elem. a divisor Returns the divisor of the toric variety vari which corresponds to the character which is created by the list lis. a divisor Returns the divisor of the toric variety vari which corresponds to the Weil group element elem. by the list lis. a divisor Returns the divisor of the toric variety vari which corresponds to the Weil group element which is created by the list lis.