Algebraic and Analytic Geometry WIT A Structured and Comprehensive Package for Computations in Intersection Theory

Transcription

1 AMS SPECIAL SESSION Computational Algebraic and Analytic Geometry for Low dimensional Varieties WIT A Structured and Comprehensive Package for Computations in Intersection Theory SEBASTIAN XAMBÓ DESCAMPS FACULTAT DE MATEMÀTIQUES I ESTADÍSTICA UNIVERSITAT POLITÈCNICA DE CATALUNYA BARCELONA (SPAIN)

2 2 CREDITS JOINT WORK WITH: Josep M Miret for the geometrical aspects Francesc Massanés for the Java interfaces to the WIT system THANKS TO THE ORGANIZERS M. Seppälä, T. Shaska, E. Volcheck

3 3 MAIN POINTS The WIT approach to computational IT Enumerative geometry problems Algebraic geometry foundations (IT) Intersection rings Chern classes Vanishing locus of a vector bundle section The method of degenerations Enumerative significance Archetypal examples Some new Schubert tables Results Methods Concluding remarks

4 4 The Wit approach to computational IT Java interface ( ) Help page and User s guide Select, load and run file example Save a session Note. There is a similar interface for error correcting codes ( ) and for the plain mathematical engine ( )

5 5 Examples # Bernoulli polynomials -- after Shafarevich { bernoulli_polynomial(k,x) with k in 1..3 }# k=100; bernoulli_polynomial(k,x) - bernoulli_polynomial(k,x-1)# # should be x^k

6 6 # Bernoulli polynomials -- after Shafarevich u> { bernoulli_polynomial(k,x) with k in 1..3 }# > {1/2*x2+1/2*x, 1/3*x3+1/2*x2+1/6*x, 1/4*x4+1/2*x3+1/4*x2} u> k=100; u> bernoulli_polynomial(k,x) - bernoulli_polynomial(k,x-1)# > x^100 # should be x^k

7 # Examples: Intersection multiplicity of two plane # curves at a point obtained by Fulton's algorithm 7 # 0. Conventions # For simplicity, here a plane curve is represented # by a polynomial in the variables x and y. # 1. Reduction to the point {0,0} mult(f,g,{a,b}):= mult(eval(f,{x+a,y+b}),eval(g,{x+a,y+b})); # 2. The function mult(f,g) mult(f,g):= begin local f, g, r, s, a, b # it is assumed that the variables are {x,y} if not(subset?(variables(f),{x,y}) & subset?(variables(g),{x,y}) ) then return 0 end

8 8 # mult is 0 if one of the curves does not pass # through the origin if ( eval(f,{0,0})!=0 eval(g,{0,0})!= 0 ) then return 0 end #.. and it is if one of the equations is 0 if F==0 G==0 then return Infinity end # Otherwise continue with Fulton's algorithm f=collect(f,y).0 g=collect(g,y).0 r=degree(f) s=degree(g)

9 if f==0 then if g==0 then Infinity # y is a common factor else trailing_degree(g)+mult(f/y,g) end else # f!=0 if g==0 then trailing_degree(f)+mult(f,g/y) else # g!=0 a=leading_coefficient(f) b=leading_coefficient(g) if r<=s then mult(f,a*g-b*x^(s-r)*f) else mult(b*f-a*x^(r-s)*g,g) end end end end; 9

10 # Examples F=(x^2+y^2)^3-4*x^2*y^2; G=(x^2+y^2)^2+3*x^2*y-y^3; mult(f,g)# # should be 14 f=collect(f,y); g=collect(g,y); resultant(f,g)# 80*x^ *x^ *x^18 10 K=Zmod(5); A=K[x,y]; F=F:A; G=G:A; mult(f,g)# # should be 18

11 11 Structure wit h wit power wit types wit sheaves wit varieties wit chow wit morphism Credits. The WIT system is inspired on the Schubert package (programmed in MAPLE) of S. Katz and S. A. Str mme. A twofold recognition of SCHUBERT as the soul of WIT.

12 12 Types (a sample) Variety. This class, also called Manifold, or VAR, is intended to capture the structural details of the algebraic varieties appearing in intersection theory, and it is defined as follows: Variety ={ # Name of field dim : Integer # dimension kind : Identifier gcs : Vector # generating classes degs : Vector # degrees of gcs monomials : List monomial_values : Relation pt : Polynomial # point relations : Vector basis : List dual_basis : List tan_bundle : Sheaf # tangent bundle todd_class : Vector # Todd class }

13 Grassmannian. This class, also called GRASS, is an extension of Variety that allows us to handle Grassmannian varieties. It is defined as follows: Grassmannian = Variety { tautological_bundle : Sheaf tautological_quotient : Sheaf } Thus Grassmannian is a Variety with two additional fields, both of type Sheaf. As suggested by their identifiers, these fields will hold the tautological bundle and the tautological quotient of a given instance of the class. 13

14 14 Morphism. This class, also called MOR, is defined follows: Morphism = { source : Variety target : Variety dim : Integer upperstardata : Rule kind : Identifier }

15 Projective_bundle. This class, also called PROJ_BUNDLE, is defined as follows: Projective_Bundle = Variety { base_variety : Variety vector_bundle : Sheaf fiber_dim : Integer base_dim : Integer lowerdata : Rule section : Alg } Thus we see that a Projective_bundle is a Variety with six additional fields. 15

16 16 Blowup. This class, also called BLOWUP, is defined follows: Blowup = Variety { exceptional_class : Alg blowup_map : Morphism blowup_locus : Variety locus_inclusion : Morphism } Thus we see that a Blowup is an extension of the class Variety with four additional fields.

17 17 Constructors (a sample) projective_space(n,h) constructs a Variety that implements the notion of projective space of dimension with hypeplane class. Its fields have, letting c = [binomial(n+1,j) h j with j in 1..n], the following values: dim = n gcs = [h:polynomial] degs = [1] relations = [h^(n+1)] monomials = { {h^j} with j in 1..n} monomial_values = {h^n->1} pt = h^n tan_bundle = sheaf(n,c) todd_class = todd_vector(sheaf(n,c)) grassmannian(k, n, c) constructs a Variety that implements the notion of the Grassmannian variety of planes in the dimensional projective space and with = [c1, c2,... ] = vector(c,k+1) the vector of Chern classes of the tautological bundle on G. Its fields have, letting

18 S = bundle(k+1,c), T = trivial_bundle(n+1), M = { monomial_list ( C, [1..(k+1)], j) with j in 1..(k+1) (n k) }, and Q = T / dual(s), the following values: dim = (k+1) (n-k) kind = _grassmannian_ gcs = C degs = [1..(k+1)] relations = take(invert(c,n+1),-k-1) pt = (C.(k+1))^(n-k) monomials = M monomial_values = find_monomial_values( (k+1) (n-k), C, [1..(k+1)], R, pt ) tautological_bundle = S tautological_quotient = dual(s) tan_bundle = hom(q,dual(s)) todd_class = todd_vector(hom(q,dual(s))) 18

19 blowup(i,e,f) constructs, if is a Morphism, and we let =target( ) and =source( ), the blowup, say, of along in such a way that is the exceptional class and : the projection map. 19

20 20 Enumerative geometry problems lines meeting 4 lines in lines meeting 6 planes in lines contained in a cubic in conics in meeting 8 lines conics tangent to 5 conics in the plane lines contained in 5ic in conics contained in a 5ic in

21 21 Nodal cubics in whose plane passes through a point, whose node lies on a plane, and which meets 6 lines and is tangent to 3 planes. Symbolically: to find. b

22 22 Why are these problems interesting? They have stimulated the development of modern IT. Verifying and completing Schubert s (an others) computations. Understanding why many numbers could not be computed by Schubert s symbolic calculus. They lead to interesting geometry (parameter spaces of the figures, and of their degenerate forms). It is an excellent ground for testing mathematical engines.

23 23 Algebraic geometry foundations Intersections. Let be a smooth algebraic variety of dimension and let denote the group of cycles on ( 0,1,, ). If,, be positive integers such that, then we have a map given by the intersection product. If is complete, and / 0, then there is induced map. Moreover, the degree map deg: induces a map.

24 24 deg deg,

25 25 Algebraic geometry foundations Chern classes. Given a rank vector bundle on, there are classes ( 0,1,, ) that satisfy the following properties, where we write : Normalization. If is the line bundle associated to a divisor, then 1. Functoriality. If : is a map of smooth varieties, and a vector bundle on,. Additivity. If 0 0 is an exact sequence of vector bundles on, then.

26 26 Algebraic geometry foundations Vanishing locus of a vector bundle section. If has rank, its top Chern class is. This is justified by the fact that 0 for. This class has a direct geometric interpretation: if is a section of such that has codimension, then Thus. if. deg

27 27 Algebraic geometry foundations The method of degenerations. Let be the space parameterizing the objects we are interested in (say nodal cubics in ). This space is smooth, but rarely complete. In order to be able to apply the machinery of intersection theory to solve the enumerative problem deg, suppose we are able to construct a smooth complete variey containing such that, a hypersurface of, and with the property that, where we let be the closure of in. Then we clearly have ( ). deg deg

28 28 Algebraic geometry foundations Enumerative significance. When the cycles are defined by specifying relations to other objects (data), such as meeting a line, or being tangent to a plane, Kleiman s transversality theorem guarantees that if the data are in general position then the intersection is proper and with (in characteristic 0) multiplicities 1. Therefore, in such cases the method of degenerations provides numbers with enumerative significance, in the sense that indeed are the number of (non degenerate) figures that satisfy the conditions. Terminology. The irreducible components of are usually called (first order) degenerations (of the figures parameterized by ).

29 29 Archetypal examples Chasles space of complete conics. The space is formed by the smooth conics (an open set in ). There is a compactificacion of such that the degenerations are as indicated in the picture. If is the hypersurface in of conics E tangent to a given smooth conic, then it turns out that D 6 2, where, a hyperplane in, and. Therefore, 6 2 is the number of conics tangent to 5 smooth conics in general position. Remark: can be obtained as the blow up on along the surface of double lines.

30 30 Lines meeting 4 lines in or 6 planes in # Number of lines in P^n meeting # 2*n-2 (n-2)-planes. # It is known to be binomial(2n-2,n)/(n-1) L(n):=binomial(2n-2,n)/(n-1); {L(n) with n in 3..10} # # Lines in P^3 meeting 4 lines n=3; d=2*(n-1); G=grassmannian(1,n,s); integral(g,s1^d) # Wit> 4

31 31 # Lines in P^4 meeting 6 planes n=4; d=2*(n-1); G=grassmannian(1,n,s); integral(g,s1^d) # Wit> 5

32 32 Lines contained in a cubic in # Examples: integral # 27 lines on a cubic surface in P^3 G=grassmannian(1,3,c); S=tautological_bundle\G; F=symm(3,dual(S)); c4=chern(rk\f,f) # integral(g,c4) # Wit> 27

33 33 Conics meeting 8 lines in # Planes in P^3 and the rank 3 tautological # bundle. The class c1 represents that the # plane goes through a point G=grassmannian(2,3,c); S=tautological_bundle\G; # Rank 6 bundle on G of conic equations and the # the space of conics in P^3, with tautological # class e E=symm(2,S); P=projective_bundle(G, E, e); # As 2*c1+e represents that the conic meets a # line, the number sought is given by integral(g,lowerstar(p,(2*c1+e)^8)) : ZZ # Wit> 92

34 34 Conics tangent to 5 conics in # Examples: blowup # Chasles space of complete conics P5=projective_space(5,H); S =projective_space(2,h); i=morphism(s,p5,[2*h]); X=blowup(i,e,f); integral(x,(6*h-2e)^5) # Wit> 3264

35 35 Lines contained in a 5ic in G=grassmannian(G,1,4,c); S=tautological_bundle\G; F=symm(5,dual(S)); c6=chern(rk\f,f) # integral(g,c6) # Wit> 2875

36 36 Conics contained in a 5ic in G=grassmannian(2,4,c); # Planes in P^4 S=tautological_bundle\G; # has rank 3 Q=symm(2,dual(S)); # 2ic forms, rk=6 # The projective bundle of conics P=projective_bundle(G,Q,z); S5=symm(5,dual(S)); # 5ic forms, rk=21 S3=symm(3,dual(S)); # 3ic forms, rk=10 # The rank 11 bundle of quintics restricted # to the universal conic F= S5 / ( S3 * o_(-z,p) ) ; C=chern(rk\F,F); # Top Chern class of F c=lowerstar(p,c) # pushforward of C to G # The number of conics on a quintic integral(g,c) : ZZ # Wit>

37 37 Some new Schubert tables Results for nodal cubics in Numbers Example: 6080 Numbers Example: 53772

38 38 Numbers Example: Numbers Example: 4556 Remark. The numbers in blue were determined by Schubert. Those in red read are new.w.

39 39 Some new Schubert tables Degenerations used u p x b π π x b u p Degenerations and

40 40 x b π u p Degeneration

41 41 x b u z π u p Degeneration

42 42 x b u z π u s u p Degeneration

43 43 x b x v u p = u z = u s π Degeneration

44 44 x b x v π u p = u z = u s Degeneration Remark. In this degeneration the point cubic is a triple line and there are 8 point on it: the node, ; the 3 inflections, ; and four foci that determine the dual curve. These points are not free (dimension count) and understanding the relation of each to the other 7 is a crucial step for the determination of the intersection ring, which in turn is needed for finding the numbers about nodal cubics.

45 45 References Fulton, W. Intersection Theory, 2 nd ed. Ergebnisse der Math., 3. Folge, Band 2, Springer Verlag, Hernández, X., Miret, J., Xambó, S. Computing the characteristic numbers of the variety of nodal cubics in. Journal of Symbolic Computation, 42, no. 1 2, , Kleiman, S., The transversality of a general translate. Compositio Math. 38, , Kleiman, S., Str mme, S. A. and Xambó, S. Sketch of a verification of Schubert's number of twisted cubics. Springer LNM 1266, , Miret, J., Xamb\'o, S. On the geometry of nodal cubics: the condition. Contemporary Mathematics, 123, AMS, 1991.

46 46 Miret, J., Xambó, S., Hernández, X. Completing H. Schubert's work on the enumerative geometry of cuspidal cubics in. Comm. in Algebra, 31(8), , Pandharipande, R. Intersections on divisors on Kontsevich's moduli space,, and enumerative geometry. Transactions of the American Mathematical Society, 351 (4), , Schubert, H. Kalkül der abzählenden Geometrie. Teubner, Reprint by~springer, 1879 (with a preface by S. Kleiman). Xambó Descamps, S. Using Intersection Theory. Aportaciones matemáticas, serie textos, 7. Sociedad Matemática Mexicana, 1996.

47 47 Concluding remarks Computations in intersection theory are a wonderful ground for testing the expressiveness and power of computational mathematics systems. WIT is a structured package of functions (WiP) that allows the user to express intersection theory computations in a rather compact way and with a syntax that is quite close to the mathematics involved. We have seen illustrations of its working in some archetypal examples and also in completing Schubert s tables for nodal cubics in (work to appear in the JSC: Computing some fundamental numbers of the variety of nodal cubics in. WIT is intended to be freely accessed with any one of the familiar navigators at The plain engine can be accessed at The companion system for computations in block error correction codes at

48 48 Brevity is the soul of wit (Shakespeare)