What can computer Algebraic Geometry do today? · What can computer Algebraic Geometry do today? GregoryG.Smith WolframDecker MikeStillman 14July2015

WhatcancomputerAlgebraicGeometrydotoday?

Gregory G. SmithWolfram DeckerMike Stillman

14 July 2015

EssentialQuestionsWhat can be computed?What software is currently available?What would you like to compute?How should software advance yourresearch?

BasicMathematicalTypesPolynomial Rings, Ideals, Modules,Varieties (affine, projective, toric, abstract),Sheaves, Divisors, Intersection Rings,Maps, Chain Complexes, Homology,Polyhedra, Graphs, Matroids,

EstablishedGeometricToolsElimination, Blowups, Normalization,Rational maps, Working with divisors,Components, Parametrizing curves,Sheaf Cohomology, -modules,

EmergingGeometricToolsClassification of singularities,Numerical algebraic geometry,

, Derived equivalences,Deformation theory, Positivity,

SomeGeometricSuccessesGEOGRAPHY OF SURFACES: exhibitingsurfaces with given invariantsBOIJ-SÖDERBERG: examples lead tonew conjectures and theoremsMODULI SPACES: computer aidedproofs of unirationality

SomeExistingSoftwareGAP, Macaulay2, SINGULAR,CoCoA, Magma, Sage, PARI, RISA/ASIR,Gfan, Polymake, Normaliz, 4ti2,Bertini, PHCpack, Schubert, Bergman,

an idiosyncratic and incomplete list

EffectiveSoftwareUSEABLE: documented examplesMAINTAINABLE: includes tests, part ofa larger distributionPUBLISHABLE: Journal of Software forAlgebra and Geometry; www.j-sag.orgCITATIONS: reference software

Recent Developments in Singular

Wolfram DeckerJanko Bohm, Hans Schonemann, Mathias Schulze

Mohamed Barakat

TU Kaiserslautern

July 14, 2015

Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 1 / 24

What is Singular?

A computer algebra system for polynomial computations, with specialemphasis on

algebraic geometry,

commutative and non-commutative algebra,

singularity theory, and with

packages for convex and tropical geometry.

It is free and open-source under the GNU General Public Licence.


What is Singular?


algebraic geometry,






What is Singular?


algebraic geometry,


singularity theory,

and with




What is Singular?


algebraic geometry,






What is Singular?


algebraic geometry,






Open development model


Open development model

Singular issue tracker

Interaction with user base, bug & feature tracking


What is Singular?

Over 30 development teams worldwide, over 130 libraries for advancedtopics.


What is Singular?

Over 30 development teams worldwide, over 130 libraries for advancedtopics.


What is Singular?

Singular consists of

a kernel, written in C/C ++, and containing the core algorithms,

libraries, written in the Singular language which provides aconvenient way of user interaction and adding new mathematicalfeatures, and

a comprehensive online manual and help function.


What is Singular?






What is Singular?






Example: Parametrizing Rational Curves

Example> ring R = 0, (x,y,z), dp;

> poly f = x5+10x4y+20x3y2+130x2y3-20xy4+20y5-2x4z-40x3yz-150x2y2z

-90xy3z-40y4z+x3z2+30x2yz2+110xy2z2+20y3z2;

> LIB "paraplanecurves.lib";

> genus(f);

0

> paraPlaneCurve(f);


Example: The Parametrization Algorithm

Example> ideal AI = adjointIdeal(f); // requires normalization, integral bases

> AI;

[1]=y3-y2z

[2]=xy2-xyz

[3]=x2y-xyz

[4]=x3-x2z

> def Rn = mapToRatNormCurve(f,AI);

> setring(Rn);



Example> RNC;

RNC[1]=y(2)*y(3)-y(1)*y(4)

RNC[2]=20*y(1)*y(2)-20*y(2)^2+130*y(1)*y(4)

+20*y(2)*y(4)+10*y(3)*y(4)+y(4)^2

RNC[3]=20*y(1)^2-20*y(1)*y(2)+130*y(1)*y(3)

+10*y(3)^2+20*y(1)*y(4)+y(3)*y(4)



Example> LIB"sing.lib";

> radical(slocus(RNC));

[1]=y(4)

[2]=y(3)

[2]=y(2)

[1]=y(1)

> rncAntiCanonicalMap(RNC);

[1]=2*y(2)+13*y(4)

[2]=y(4)


Singular Libraries


Example: Intersection Theory

Example> LIB "schubert.lib";

> variety G = Grassmannian(2,4);

> def r = G.baseRing;

> setring r;

> sheaf S = makeSheaf(G,subBundle);

> sheaf B = dualSheaf(S)^3;

> integral(G,topChernClass(B));


Example: Intersection Theory

Example (continued)

27

Some keywords

Schubert calculus, double point formulas, excess intersection formula,equivariant intersection theory using Bott’s formula, Gromov-Witteninvariants.


Key Algorithms in Singular

Basic stu↵Grobner and standard Bases;

free resolutions;

polynomial factorization: Factory.

More advanced stu↵Primary decomposition, algorithms of Gianni-Trager-Zacharias,Shimoyama-Yokoyama, Eisenbud-Huneke-Vasconcelos:primdec.lib.;

normalization: normal.lib, locnormal.lib, modnormal.lib.




free resolutions;







free resolutions;







free resolutions;







free resolutions;





Example: Coarse Grained Parallelism in Singular

Example> LIB("parallel.lib","random.lib");

> ring R = 0,x(1..4),dp;

> ideal I = randomid(maxideal(3),3,100);

> proc sizeStd(ideal I, string monord){def R = basering; list RL = ringlist(R);

RL[3][1][1] = monord; def S = ring(RL); setring(S);

return(size(std(imap(R,I))));}



Example> LIB("parallel.lib","random.lib");

> ring R = 0,x(1..4),dp;

> ideal I = randomid(maxideal(3),3,100);

> proc sizeStd(ideal I, string monord){def R = basering; list RL = ringlist(R);

RL[3][1][1] = monord; def S = ring(RL); setring(S);

return(size(std(imap(R,I))));}



Example> list commands = "sizeStd","sizeStd";

> list args = list(I,"lp"),list(I,"dp");

> parallelWaitFirst(commands, args);

[1] empty list

[2] 11

> parallelWaitAll(commands, args);

[1] 55

[2] 11






[1] empty list

[2] 11


[1] 55

[2] 11






[1] empty list

[2] 11


[1] 55

[2] 11



Singularities

Milnor-, Tjurina-numbers, monodromy and Gauss-Manin systems ofan isolated hypersurface singularity: monodromy.lib, gmssing.lib;

Classification of complex and real isolated hypersurface singularitiesfollowing Arnold: classify.lib and realclassify.lib;

Resolution of singularities: resolve.lib.



Singularities






Singularities





New Libraries in Singular

Framework for hyperplane arrangements: arr.lib;

Riemann-Roch spaces: hess.lib, brillnoether.lib;

New algorithms for computing tropical varieties: gfanlib.so;

De Rham Cohomology: deRham.lib;

GIT-fans in geometric invariant theory: gitfan.lib.






























Connections to other Systems

GAPGroups

SingularAlgebraic Geometry

polymakeConvex Geometry

ANTICNumber Theory

Included subsystems

Factory: Polynomial Factorization;

Flint: arithmetic for Number Theory;

Plural: non-commutative stu↵.



GAPGroups



ANTICNumber Theory

Included subsystems






GAPGroups



ANTICNumber Theory

Included subsystems






GAPGroups



ANTICNumber Theory

Included subsystems






GAPGroups



ANTICNumber Theory

Included subsystems






GAPGroups



ANTICNumber Theory

Included subsystems





Homalg Example: From groups to vector bundles

gap> LoadPackage( "repsn" );;gap> LoadPackage( "GradedModules" );;gap> G := SmallGroup( 1000, 93 );<pc group of size 1000 with 6 generators>gap> Display( StructureDescription( G ) );((C5 x C5) : C5) : Q8



gap> V := Irr( G )[6];; Degree( V );5gap> T0 := Irr( G )[5];; Degree( T0 );2gap> T1 := Irr( G )[8];; Degree( T1 );5gap> mu0 := ConstructTateMap( V, T0, T1, 2 );<A homomorphism of graded left modules>



gap> A := HomalgRing( mu0 );Q{e0,e1,e2,e3,e4}(weights: [ -1, -1, -1, -1, -1 ])gap> M:=GuessModuleOfGlobalSectionsFromATateMap(2, mu0);;gap> ByASmallerPresentation( M );<A graded non-zero module presented by 92relations for 19 generators>



gap> S := HomalgRing( M );Q[x0,x1,x2,x3,x4](weights: [ 1, 1, 1, 1, 1 ])gap> ChernPolynomial( M );( 2 | 1-h+4*h^2 ) -> P^4gap> tate := TateResolution( M, -5, 5 );;



gap> Display( BettiTable( tate ) );

total: 100 37 14 10 5 2 5 10 14 37 100 ? ? ? ?

----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

4: 100 35 4 . . . . . . . . 0 0 0 0

3: * . 2 10 10 5 . . . . . . 0 0 0

2: * * . . . . . 2 . . . . . 0 0

1: * * * . . . . . . 5 10 10 2 . 0

0: * * * * . . . . . . . . 4 35 100

----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---S

twist: -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

-------------------------------------------------------------------

Euler: 100 35 2 -10 -10 -5 0 2 0 -5 -10 -10 2 35 100


Macaulay2

Mike Stillman ([email protected])

Department of Mathematics

Cornell

14 July 2015Summer Institute in Algebraic Geometry

Salt Lake City

Macaulay2 essentials in a nutshell

Written with Dan Grayson since 1990’s, open source system

Web site : www.macaulay2.com

Macaulay2 on the web : web.macaulay2.comNo login or installation required !

Has programming language which allows users to create packagesand new types of (mathematical) objects.

around 100 user written packages distributed with Macaulay2.

Selected Macaulay2 packages

NormalToricVarieties (G.G. Smith), cohomology, divisors, res ofsingularities.

BGG (Eisenbud-Schreyer-et al), Bernstein-Gelfand-Gelfandcorrespondence, sheaf cohomology, Beilinson monads.

Dmodules (Leykin-Tsai), b-functions, restriction, local cohomology,multiplier ideals.

Schubert2 Intersection theory, based on maple package Schubert,by Stromme and Katz, with extensions.

NumericalAlgebraicGeometry (Leykin et al) Numerical pathtracking and decomposition, interfaces to Bertini, PHCpack.

Many others : random curves, deformation theory, intersectiontheory of moduli spaces, Schur functors, . . .

Macaulay2, version 1.8.1with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : printWidth=70

o1 = 70

i2 : needsPackage "BGG";

i3 : kk = ZZ/32003

o3 = kk

o3 : QuotientRing

i4 : S = kk[x_0..x_4]

o4 = S

o4 : PolynomialRing

i5 : -- The next line sets the ideal I load "explicit-surface.m2"

i6 : degrees I

o6 = {{6}, {6}, {6}, {6}, {5}, {5}, {5}, {5}, {5}}

o6 : List

i7 : -- 9 generators: 5 of degree 5, 4 of degree 6 I_0

4 2 3 3 2 4 5 4 o7 = - 2704x x + 9448x x - 14844x x + 5013x x - 13570x x x + 0 1 0 1 0 1 0 1 0 1 2 ----------------------------------------------------------------- 3 2 2 3 4 3 2 2 2 2 15457x x x + 11916x x x - 6767x x x + 7798x x x + 2916x x x 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 ----------------------------------------------------------------- 3 2 2 3 2 3 2 2 2 3 2 + 9106x x x + 6032x x x + 4143x x x + 6548x x x + 5689x x x 0 1 2 0 1 2 0 1 2 0 1 3 0 1 3 ----------------------------------------------------------------- 2 2 2 2 3 2 2 2 2 - 2704x x x x + 5551x x x x + 4314x x x - 1819x x x +

0 1 2 3 0 1 2 3 0 1 4 0 1 4 ----------------------------------------------------------------- 3 2 3 2 2 2 2 2 5909x x x - 2704x x x - 6755x x x x + 10115x x x x - 0 1 4 0 2 4 0 1 2 4 0 1 2 4 ----------------------------------------------------------------- 3 2 2 2 2 2 2 2 2 2 5013x x x + 14821x x x + 15672x x x x + 6767x x x - 1 2 4 0 2 4 0 1 2 4 1 2 4 ----------------------------------------------------------------- 3 2 3 2 4 2 2 2 2 2 7779x x x - 9106x x x - 4143x x - 8003x x x x + 4314x x x x 0 2 4 1 2 4 2 4 0 1 3 4 0 2 3 4 ----------------------------------------------------------------- 2 2 2 2 2 4 4 - 5689x x x x - 5551x x x - 6852x x x + 10076x x x - 1 2 3 4 2 3 4 0 1 4 0 2 4 ----------------------------------------------------------------- 4 2 4 2 4 11623x x x - 15829x x - 15149x x 1 2 4 2 4 3 4

o7 : S

i8 : netList I_*

+------------------------------------------------------------- | 4 2 3 3 2 4 5 4 o8 = |- 2704x x + 9448x x - 14844x x + 5013x x - 13570x x x + ... | 0 1 0 1 0 1 0 1 0 1 2 +------------------------------------------------------------- | 5 4 2 3 3 2 4 5 |2704x x - 9448x x + 14844x x - 5013x x + 13570x x - 1545... | 0 1 0 1 0 1 0 1 0 2 +------------------------------------------------------------- | 3 3 2 4 5 6 3 2 |2704x x - 9448x x + 14844x x - 5013x + 13570x x x - 1545 | 0 1 0 1 0 1 1 0 1 2 +------------------------------------------------------------- | 6 5 4 2 3 3 5 |- 2704x + 9448x x - 14844x x + 5013x x + 14821x x + 5726 | 0 0 1 0 1 0 1 0 2 +------------------------------------------------------------- | 3 2 2 3 4 2 |- 5291x x x + 8060x x x - 15744x x x + 12425x x - 12484x | 0 1 2 0 1 2 0 1 2 1 2 0 +------------------------------------------------------------- | 4 3 2 2 3 3 2 |5291x x - 8060x x x + 15744x x x - 12425x x x + 12484x x | 0 2 0 1 2 0 1 2 0 1 2 0 2

+------------------------------------------------------------- | 4 3 2 2 3 4 |- 13570x x + 636x x x + 6190x x x + 4586x x x + 10884x x | 0 2 0 1 2 0 1 2 0 1 2 1 2 +------------------------------------------------------------- | 4 3 2 2 3 3 |- 2704x x + 9448x x x - 14844x x x + 5013x x x + 14821x x ... | 0 2 0 1 2 0 1 2 0 1 2 0 +------------------------------------------------------------- | 3 2 2 3 4 3 |- 2704x x x + 9448x x x - 14844x x x + 5013x x - 13570x x ... | 0 1 2 0 1 2 0 1 2 1 2 0 +-------------------------------------------------------------

i9 : R = S/I

o9 = R

o9 : QuotientRing

i10 : X = Proj R

o10 = X

o10 : ProjectiveVariety

i11 : -------------------------------- -- Basic numerical invariants -- -------------------------------- dim X -- 2

o11 = 2

i12 : degree X -- 12

o12 = 12

i13 : -- X is a surface in P^4, degree 12. genera X -- pa = 2, sectional genus: 13.

o13 = {2, 13, 11}

o13 : List

i14 : --------------------------------- -- Smoothness ------------------- --------------------------------- -- We check this directly:

ideal singularLocus X

o14 = ideal 1

o14 : Ideal of S

i15 : -- X is a nonsingular surface, degree 12, -- sectional genus 13 in P^4 --------------------------------- -- Basic cohomology ------------- --------------------------------- euler OO_X^1 -- 3

o15 = 3

i16 : HH^1(OO_X) -- 1

1o16 = kk

o16 : kk-module, free

i17 : HH^2(OO_X) -- 3

3o17 = kk

o17 : kk-module, free

i18 : h0 = (euler OO_X^1) + rank HH^1(OO_X) - rank HH^2(OO_X)

o18 = 1

i19 : -- therefore h^0(OO_X) = 1 (takes longer to compute directly). -- h^0(OO_X) - h^1 + h^2 = 3, so hh^0 = 1 -- So X is connected -- Whole cohomology table for OO_X -- How to read it: -- row p, column d is entry for h^p(OO_X(d-p)) -- e.g. h^1(OO_X(1)) = 2 ct = cohomologyTable(sheaf(S^1/I), -2, 6)

-2 -1 0 1 2 3 4 5 6 7 8o19 = 2: 123 75 39 15 3 . . . . . . 1: . . . 1 2 1 . . . . . 0: . . 1 5 16 39 75 123 183 255 339

o19 : CohomologyTally

i20 : -- So: X is a smooth irreducible surface, degree 12. -- X is irregular, and pg = 3. -- Canonical Sheaf wX = Ext^(codim I)(S^1/I, S^{-numgens S})

o20 = cokernel {0} | -3092x_0-2149x_2 ... {0} | 2223x_0-10811x_1+5891x_2 ... {0} | x_0+9850x_1-2696x_2 ... {1} | 0 ...

4o20 : S-module, quotient of S

i21 : cohomologyTable(sheaf wX,-2,6)

-2 -1 0 1 2 3 4 5 6o21 = 2: 75 39 16 5 1 . . . . 1: . 1 2 1 . . . . . 0: . . 3 15 39 75 123 183 255


i22 : ct -- Serre duality holds here!

-2 -1 0 1 2 3 4 5 6 7 8o22 = 2: 123 75 39 15 3 . . . . . . 1: . . . 1 2 1 . . . . . 0: . . 1 5 16 39 75 123 183 255 339


i23 : --------------------------------- -- Intersection numbers --------- --------------------------------- KX = sheaf(wX ** R)


3 1o23 : coherent sheaf on X, quotient of OO ++ OO (-1) X X

i24 : -- Riemann-Roch for surfaces: intersectionNumber = (L,M) ->

euler ring L - euler L - euler M + euler(L**M)

o24 = intersectionNumber

o24 : FunctionClosure

i25 : intersectionNumber(OO_X(1), OO_X(1)) -- H.H = 12, as expected

o25 = 12

i26 : intersectionNumber(KX, KX) -- K.K = 0

o26 = 0

i27 : intersectionNumber(KX, OO_X(1)) -- K.H = 12

o27 = 12

i28 : --------------------------------- -- Hodge diamond and h^(1,1) ---- --------------------------------- -- We need the euler char of the cotangent bundle OmegaX = cotangentSheaf(1,X,Minimize=>false);

i29 : euler image generators module OmegaX - euler image relations module OmegaX

o29 = -30

i30 : -- euler characteristic of Omega_X is -30 -- therefore h^(1,1)(X) = 32 -- Hodge diamond: -- 1 -- 1 1 -- 3 32 3 -- 1 1 -- 1 --------------------------------- -- phi_K : X --> E \subset P^2 -- --------------------------------- -- First: bring KX into the ring R = S/I -- One way: Compute Hom(wX,R), take any non-zero -- map, and let J be its image wXR = wX ** R


4o30 : R-module, quotient of R

i31 : hom = Hom(wXR,R)

o31 = image {0} | -12968x_0^3-4556x_0^2x_1+8642x... {0} | -9226x_0^3-1337x_0^2x_1-13005x... {0} | -1822x_0^3-5878x_0^2x_1-6116x_... {-1} | x_0x_1^2x_2-3859x_1^3x_2-14421...

4o31 : R-module, submodule of R

i32 : J = ideal image homomorphism hom_{0}

3 2 2 3 2 o32 = ideal (- 12968x - 4556x x + 8642x x - 6371x - 13175x x + 0 0 1 0 1 1 0 2 ---------------------------------------------------------------- 2 2 2 3 5650x x x - 5369x x + 6425x x - 14501x x - 15967x - 0 1 2 1 2 0 2 1 2 2 ---------------------------------------------------------------- 2 2 2 2 2 10875x x - 15729x x + 4339x x - 15729x x - 3211x x + 0 3 1 3 2 3 0 4 1 4 ---------------------------------------------------------------- 2 3 2 2 3 2 6118x x , - 9226x - 1337x x - 13005x x - 10041x + 5003x x + 2 4 0 0 1 0 1 1 0 2 ---------------------------------------------------------------- 2 2 2 3 3295x x x - 12368x x - 6467x x + 11335x x - 7793x + 0 1 2 1 2 0 2 1 2 2 ---------------------------------------------------------------- 2 2 2 2 2 14956x x - 10910x x + 309x x - 10910x x + 9354x x + 0 3 1 3 2 3 0 4 1 4 ---------------------------------------------------------------- 2 3 2 2 3 2 7656x x , - 1822x - 5878x x - 6116x x + 2843x - 13909x x + 2 4 0 0 1 0 1 1 0 2 ---------------------------------------------------------------- 2 2 2 3 1234x x x - 10761x x - 12450x x - 9173x x - 9619x + 0 1 2 1 2 0 2 1 2 2 ---------------------------------------------------------------- 2 2 2 2 2 13582x x + 5148x x - 11995x x + 5148x x + 526x x +

0 3 1 3 2 3 0 4 1 4 ---------------------------------------------------------------- 2 2 3 2 2 2 2 2 1430x x , x x x - 3859x x - 14421x x - 9381x x x - 7062x x 2 4 0 1 2 1 2 0 2 0 1 2 1 2 ---------------------------------------------------------------- 3 3 4 2 2 + 4924x x - 9555x x + 11943x - 9823x x x - 9743x x x - 0 2 1 2 2 0 2 3 1 2 3 ---------------------------------------------------------------- 2 2 4 2 2 2 2 164x x + 5341x - 14125x x x + 13772x x x + 12460x x - 2 3 3 0 2 4 1 2 4 2 4 ---------------------------------------------------------------- 2 2 4 5447x x + 8369x ) 3 4 4

o32 : Ideal of R

i33 : -- wXR is isomorphic to a twist of the ideal J basis(0, wXR)

o33 = {0} | 1 0 0 | {0} | 0 1 0 | {0} | 0 0 1 | {1} | 0 0 0 |

o33 : Matrix

i34 : betti J -- 3 generators in degree 3.

0 1o34 = total: 1 4 0: 1 . 1: . . 2: . 3 3: . 1

o34 : BettiTally

i35 : J_0

3 2 2 3 2 o35 = - 12968x - 4556x x + 8642x x - 6371x - 13175x x + 0 0 1 0 1 1 0 2 ---------------------------------------------------------------- 2 2 2 3 5650x x x - 5369x x + 6425x x - 14501x x - 15967x - 0 1 2 1 2 0 2 1 2 2

---------------------------------------------------------------- 2 2 2 2 2 10875x x - 15729x x + 4339x x - 15729x x - 3211x x + 0 3 1 3 2 3 0 4 1 4 ---------------------------------------------------------------- 2 6118x x 2 4

o35 : R

i36 : -- next: compute the image of X under the -- 3 cubics in J -- Image of the map: A = kk[a,b,c]

o36 = A

o36 : PolynomialRing

i37 : phi = map(R,A,{J_0,J_1,J_2})

3 2 2 3 2 o37 = map(R,A,{- 12968x - 4556x x + 8642x x - 6371x - 13175x x + ... 0 0 1 0 1 1 0 2

o37 : RingMap R <--- A

i38 : imF = ker phi -- an elliptic curve. Is it smooth?

3 2 2 3 2 o38 = ideal(a + 15592a b + 9520a*b - 7485b - 8060a c - 1884a*b*c + ---------------------------------------------------------------- 2 2 2 3 53b c - 2746a*c + 2263b*c + 2187c )

o38 : Ideal of A

i39 : codim ideal jacobian imF -- 3, so curve is smooth, therefore of genus 1.

o39 = 3

i40 : -- We can also check the following: -- (1) The general fiber is smooth of genus 1 (and degree 12) -- This follows from: genus(K) = 1 + 1/2 K.(K+K) = 1 -- Therefore: X is an elliptic surface

-- (2) This map X --> E \subset P^2 is base point free -- Can be checked by considering set of (g0,g1,g2) which -- define the same morphism: -- (f0,f1,f2) = toSequence flatten entries phi.matrix

3 2 2 3 2 o40 = (- 12968x - 4556x x + 8642x x - 6371x - 13175x x + 0 0 1 0 1 1 0 2 ---------------------------------------------------------------- 2 2 2 3 5650x x x - 5369x x + 6425x x - 14501x x - 15967x - 0 1 2 1 2 0 2 1 2 2 ---------------------------------------------------------------- 2 2 2 2 2 10875x x - 15729x x + 4339x x - 15729x x - 3211x x + 0 3 1 3 2 3 0 4 1 4 ---------------------------------------------------------------- 2 3 2 2 3 2 6118x x , - 9226x - 1337x x - 13005x x - 10041x + 5003x x + 2 4 0 0 1 0 1 1 0 2 ---------------------------------------------------------------- 2 2 2 3 3295x x x - 12368x x - 6467x x + 11335x x - 7793x + 0 1 2 1 2 0 2 1 2 2 ---------------------------------------------------------------- 2 2 2 2 2 14956x x - 10910x x + 309x x - 10910x x + 9354x x + 0 3 1 3 2 3 0 4 1 4 ---------------------------------------------------------------- 2 3 2 2 3 2 7656x x , - 1822x - 5878x x - 6116x x + 2843x - 13909x x + 2 4 0 0 1 0 1 1 0 2 ---------------------------------------------------------------- 2 2 2 3 1234x x x - 10761x x - 12450x x - 9173x x - 9619x + 0 1 2 1 2 0 2 1 2 2 ---------------------------------------------------------------- 2 2 2 2 2 13582x x + 5148x x - 11995x x + 5148x x + 526x x + 0 3 1 3 2 3 0 4 1 4 ---------------------------------------------------------------- 2 1430x x ) 2 4

o40 : Sequence

i41 : -- The map to P^2 consists, off of a set of codim 1, this

vector. -- Is the map a morphism? SZ = syz matrix{{f1,-f0,0},{f2,0,-f0},{0,f2,-f1}};

3 9o41 : Matrix R <--- R

i42 : -- SZ is the entire map SZ_{2}

o42 = {3} | -10602x_0^3-7528x_0^2x_1+3241x_0x_1^2-4750x_1^3-12938x_0 {3} | 9209x_0^3+3963x_0^2x_1-1974x_0x_1^2-10897x_1^3-12469x_0^ {3} | x_0^3+5224x_1^3+3443x_0^2x_2+10389x_0x_1x_2+9049x_1^2x_2 ---------------------------------------------------------------- ^2x_2+7658x_0x_1x_2+12016x_1^2x_2+10149x_0x_2^2+10791x_1x_ 2x_2-11608x_0x_1x_2-11506x_1^2x_2-929x_0x_2^2-2247x_1x_2^2 -12966x_0x_2^2-8158x_1x_2^2-6389x_2^3+9337x_0x_3^2+10563x_ ---------------------------------------------------------------- 2^2+3082x_2^3-1206x_0x_3^2+13213x_1x_3^2-1291x_2x_3^2+13213x_ +4149x_2^3-8088x_0x_3^2-1983x_1x_3^2+567x_2x_3^2-1983x_0x_4^2 1x_3^2+8963x_2x_3^2+10563x_0x_4^2+15060x_1x_4^2+5226x_2x_4^2 ---------------------------------------------------------------- 0x_4^2-9311x_1x_4^2+5522x_2x_4^2 | -15952x_1x_4^2-5291x_2x_4^2 | |

3 1o42 : Matrix R <--- R

i43 : minors(2,SZ) -- all 0, so morphism is given by any column of SZ

o43 = ideal ()

o43 : Ideal of R

i44 : saturate ideal SZ -- no base points!

o44 = ideal 1

o44 : Ideal of R

i45 : -- (3) Singular fibers? -- This is computationally more intensive -- However, (from Hodge diamond) we compute: e(X) = 36. -- e(X) = 36, so there are potentially 36 -- singular fibers (if all are rational double points) -- exercise: find them! What kinds are they?

What can computer Algebraic Geometry do today? · What can computer Algebraic Geometry do today? GregoryG.Smith WolframDecker MikeStillman 14July2015

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