# 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 list, a list $\{a,i,j,k,\ldots\}$ of nonnegative integers
• Optional inputs:
• Outputs:
• , 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