What can computer Algebraic Geometry do today? Gregory G. Smith Wolfram Decker Mike Stillman 14 July 2015 Essential Questions ̭ What can be computed? ̭ What software is currently available? ̭ What would you like to compute? ̭ How should software advance your research? Basic Mathematical Types ̭ Polynomial Rings, Ideals, Modules, ̭ Varieties (affine, projective, toric, abstract), ̭ Sheaves, Divisors, Intersection Rings, ̭ Maps, Chain Complexes, Homology, ̭ Polyhedra, Graphs, Matroids, ̯ Established Geometric Tools ̭ Elimination, Blowups, Normalization, ̭ Rational maps, Working with divisors, ̭ Components, Parametrizing curves, ̭ Sheaf Cohomology, ঠ-modules, ̯ Emerging Geometric Tools ̭ Classification of singularities, ̭ Numerical algebraic geometry, ̭ ैक़௴Ь, Derived equivalences, ̭ Deformation theory,Positivity, ̯ Some Geometric Successes ̭ GEOGRAPHY OF SURFACES: exhibiting surfaces with given invariants ̭ BOIJ-SÖDERBERG: examples lead to new conjectures and theorems ̭ MODULI SPACES: computer aided proofs of unirationality Some Existing Software ̭ GAP,Macaulay2, SINGULAR, ̭ CoCoA, Magma, Sage, PARI, RISA/ASIR, ̭ Gfan, Polymake, Normaliz, 4ti2, ̭ Bertini, PHCpack, Schubert, Bergman, an idiosyncratic and incomplete list Effective Software ̭ USEABLE: documented examples ̭ MAINTAINABLE: includes tests, part of a larger distribution ̭ PUBLISHABLE: Journal of Software for Algebra and Geometry; www.j-sag.org ̭ CITATIONS: reference software Recent Developments in Singular Wolfram Decker Janko B¨ohm, Hans Sch¨onemann, Mathias Schulze Mohamed Barakat TU Kaiserslautern July 14, 2015 Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 1 / 24 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? A computer algebra system for polynomial computations, with special emphasis on algebraic geometry, Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 2 / 24 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? A computer algebra system for polynomial computations, with special emphasis on algebraic geometry, commutative and non-commutative algebra, Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 2 / 24 and with packages for convex and tropical geometry. It is free and open-source under the GNU General Public Licence. What is Singular? A computer algebra system for polynomial computations, with special emphasis on algebraic geometry, commutative and non-commutative algebra, singularity theory, Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 2 / 24 It is free and open-source under the GNU General Public Licence. What is Singular? A computer algebra system for polynomial computations, with special emphasis on algebraic geometry, commutative and non-commutative algebra, singularity theory, and with packages for convex and tropical geometry. Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 2 / 24 What is Singular? A computer algebra system for polynomial computations, with special emphasis 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. Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 2 / 24 Open development model Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 3 / 24 Open development model Singular issue tracker Interaction with user base, bug & feature tracking Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 4 / 24 What is Singular? Over 30 development teams worldwide, over 130 libraries for advanced topics. Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 5 / 24 What is Singular? Over 30 development teams worldwide, over 130 libraries for advanced topics. Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 5 / 24 libraries, written in the Singular language which provides a convenient way of user interaction and adding new mathematical features, and a comprehensive online manual and help function. What is Singular? Singular consists of akernel,writteninC /C ++, and containing the core algorithms, Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 6 / 24 a comprehensive online manual and help function. What is Singular? Singular consists of akernel,writteninC /C ++, and containing the core algorithms, libraries, written in the Singular language which provides a convenient way of user interaction and adding new mathematical features, and Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 6 / 24 What is Singular? Singular consists of akernel,writteninC /C ++, and containing the core algorithms, libraries, written in the Singular language which provides a convenient way of user interaction and adding new mathematical features, and a comprehensive online manual and help function. Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 6 / 24 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); Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 7 / 24 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); Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 8 / 24 Example: The Parametrization Algorithm 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) Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 9 / 24 Example: The Parametrization Algorithm 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) Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 10 / 24 Singular Libraries Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 11 / 24 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)); Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 12 / 24 Example: Intersection Theory Example (continued) 27 Some keywords Schubert calculus, double point formulas, excess intersection formula, equivariant intersection theory using Bott’s formula, Gromov-Witten invariants. Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 13 / 24 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. Key Algorithms in Singular Basic stu↵ Gr¨obner and standard Bases; Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 14 / 24 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. Key Algorithms in Singular Basic stu↵ Gr¨obner and standard Bases; free resolutions; Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 14 / 24 More advanced stu↵ Primary decomposition, algorithms of Gianni-Trager-Zacharias, Shimoyama-Yokoyama, Eisenbud-Huneke-Vasconcelos: primdec.lib.; normalization: normal.lib, locnormal.lib, modnormal.lib. Key Algorithms in Singular Basic stu↵ Gr¨obner and standard Bases; free resolutions; polynomial factorization: Factory. Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 14 / 24 normalization: normal.lib, locnormal.lib, modnormal.lib. Key Algorithms in Singular Basic stu↵ Gr¨obner 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.; Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 14 / 24 Key Algorithms in Singular Basic stu↵ Gr¨obner 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. Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 14 / 24 > 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: 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); Wolfram Decker (TU-KL) Recent Developments in Singular July 14, 2015 15 / 24 Example: Coarse Grained Parallelism in Singular Example > LIB("parallel.lib","random.lib"); >