rationalMap(PolynomialRing,List) -- rational map defined by the linear system of hypersurfaces passing through random points with multiplicity

Synopsis

Function: rationalMap
Usage:

rationalMap(R,{a,i,j,k,...})
Inputs:
- R, a polynomial ring
- a list, a list $\{a,i,j,k,\ldots\}$ of nonnegative integers
Optional inputs:
- Dominant => ..., default value null,
Outputs:
- a rational map, the rational map defined by the linear system of hypersurfaces of degree $a$ in $Proj(R)$ having $i$ random base points of multiplicity 1, $j$ random base points of multiplicity 2, $k$ random base points of multiplicity 3, and so on until the last integer in the given list.

Description

In the example below, we take the rational map defined by the linear system of septic plane curves with 3 random simple base points and 9 random double points.

i1 : ringP2 = ZZ/65521[vars(0..2)];

i2 : phi = rationalMap(ringP2,{7,3,9})

o2 = -- rational map --
                    ZZ
     source: Proj(-----[a, b, c])
                  65521
                    ZZ
     target: Proj(-----[t , t , t , t , t , t ])
                  65521  0   1   2   3   4   5
     defining forms: {
                       2 5           6         7         6          5            4 2         3 3          2 4           5          6          5 2         4   2         3 2 2         2 3 2          4 2         5 2         4 3        3   3         2 2 3           3 3         4 3        3 4         2   4           2 4        3 4         2 5             5         2 5           6          6         7
                      a b  - 17617a*b  + 12900b  + 19307a c - 18229a b*c - 30969a b c + 1575a b c + 24969a b c + 2454a*b c - 20297b c - 23940a c  + 19340a b*c  - 31968a b c  + 30464a b c  + 8682a*b c  - 16059b c  - 27897a c  + 9414a b*c  - 17051a b c  + 16858a*b c  + 16891b c  + 6542a c  + 11200a b*c  + 18611a*b c  + 4864b c  + 25805a c  + 17308a*b*c  - 26633b c  - 18043a*c  + 4571b*c  - 25269c ,
                      
                       3 4          6         7         6          5           4 2          3 3          2 4           5        6          5 2         4   2         3 2 2         2 3 2           4 2         5 2         4 3         3   3         2 2 3           3 3        4 3         3 4         2   4           2 4         3 4         2 5            5         2 5           6           6        7
                      a b  - 6417a*b  - 31396b  - 28886a c - 21440a b*c + 6917a b c - 10431a b c - 23992a b c - 7580a*b c - 788b c - 27190a c  + 20014a b*c  - 20537a b c  + 27021a b c  - 23283a*b c  + 17852b c  - 24937a c  - 16723a b*c  + 24564a b c  + 26115a*b c  - 8339b c  - 18678a c  - 15728a b*c  + 20581a*b c  - 14384b c  + 25509a c  + 2926a*b*c  - 17705b c  - 28136a*c  - 21430b*c  + 8233c ,
                      
                       4 3           6         7        6          5            4 2          3 3        2 4            5          6          5 2        4   2        3 2 2         2 3 2         4 2         5 2         4 3         3   3         2 2 3           3 3        4 3       3 4         2   4           2 4         3 4         2 5            5         2 5           6           6         7
                      a b  + 20866a*b  + 31138b  - 7183a c + 17095a b*c - 23521a b c - 13812a b c - 778a b c + 29807a*b c + 15984b c + 29820a c  - 7941a b*c  + 2117a b c  + 13574a b c  + 736a*b c  - 25606b c  - 31365a c  - 18421a b*c  - 10600a b c  - 29664a*b c  - 6435b c  - 630a c  + 20506a b*c  - 28923a*b c  - 21257b c  + 16629a c  + 2906a*b*c  - 16104b c  + 21098a*c  - 19604b*c  - 15373c ,
                      
                       5 2          6        7         6         5           4 2         3 3          2 4            5          6          5 2         4   2        3 2 2         2 3 2           4 2         5 2        4 3         3   3         2 2 3           3 3         4 3         3 4         2   4          2 4         3 4         2 5            5        2 5         6           6         7
                      a b  + 4714a*b  + 3006b  + 25391a c - 1317a b*c - 6878a b c - 2149a b c - 14185a b c - 11763a*b c + 20233b c + 11143a c  + 13264a b*c  - 3964a b c  + 28248a b c  + 31891a*b c  - 23380b c  - 1149a c  - 13443a b*c  + 21881a b c  + 32594a*b c  - 21744b c  - 13857a c  + 16583a b*c  - 5648a*b c  + 22435b c  + 23399a c  + 6718a*b*c  - 6761b c  - 371a*c  + 30515b*c  + 19313c ,
                      
                       6            6        7        6         5            4 2          3 3          2 4            5          6         5 2         4   2        3 2 2        2 3 2           4 2         5 2        4 3        3   3         2 2 3           3 3         4 3         3 4         2   4          2 4        3 4         2 5             5        2 5           6           6         7
                      a b - 19717a*b  + 2972b  + 5457a c + 2562a b*c - 30501a b c - 20360a b c + 25738a b c + 18600a*b c - 22423b c - 3993a c  - 10791a b*c  - 5480a b c  - 6946a b c  + 21025a*b c  - 10758b c  - 3912a c  - 6515a b*c  - 11351a b c  + 14417a*b c  + 17851b c  + 14014a c  - 31942a b*c  + 2029a*b c  - 2448b c  - 13209a c  - 31099a*b*c  + 8072b c  + 11291a*c  - 12950b*c  + 10256c ,
                      
                       7           6        7         6          5           4 2         3 3          2 4           5          6          5 2         4   2        3 2 2         2 3 2           4 2         5 2         4 3         3   3         2 2 3           3 3         4 3         3 4         2   4           2 4        3 4        2 5             5        2 5           6          6         7
                      a  - 28386a*b  + 6699b  - 13796a c + 17142a b*c - 3532a b c + 7955a b c - 12957a b c + 7222a*b c - 32443b c + 23975a c  - 28394a b*c  + 1532a b c  - 17058a b c  + 20899a*b c  + 16537b c  + 25830a c  - 26584a b*c  - 10084a b c  + 32513a*b c  - 28825b c  - 16911a c  - 18959a b*c  - 20721a*b c  - 4891b c  - 5083a c  - 21548a*b*c  + 1713b c  + 29336a*c  + 3763b*c  - 30117c
                     }

o2 : RationalMap (rational map from PP^2 to PP^5)

i3 : describe phi!

o3 = rational map defined by forms of degree 7
     source variety: PP^2
     target variety: PP^5
     image: surface of degree 10 and sectional genus 6 in PP^5 cut out by 10 hypersurfaces of degree 3
     dominance: false
     birationality: false
     degree of map: 1
     projective degrees: {1, 7, 10}
     number of minimal representatives: 1
     dimension base locus: 0
     degree base locus: 30
     coefficient ring: ZZ/65521

Ways to use this method: