On the GIT moduli of semistable pairs consisting of a plane cubic curve and a line

Transcription

1 On the GIT moduli of semistable pairs consisting of a plane cubic curve and a line Masamichi Kuroda at Tambara Institute of Mathematical Sciences The University of Tokyo August 26, 2015 Masamichi Kuroda The GIT moduli of pairs August / 35

2 Contents 1 Introduction 2 The GIT moduli P 1,3 of semistable pairs The classification of stability Nontrivial identifications of semistable pairs 3 A comparison between AP 1,3 and P 1,3 4 Maps from blowing-ups of SQ 1,3 P 2 to AP 1,3 and P 1,3 5 A recent study and problem Masamichi Kuroda The GIT moduli of pairs August / 35

3 1 Introduction 2 The GIT moduli P 1,3 of semistable pairs 3 A comparison between AP 1,3 and P 1,3 4 Maps from blowing-ups of SQ 1,3 P 2 to AP 1,3 and P 1,3 5 A recent study and problem Masamichi Kuroda The GIT moduli of pairs August / 35

4 Introduction Purpose To study in some detail the GIT moduli P 1,3 of semistable pairs (C, L) consisting of a plane cubic curve C and a line L Main results The classification of unstable, semistable and stable pairs, and nontrivial identifications of semistable pairs in P 1,3, A comparison between AP 1,3 and P 1,3, Maps from blowing-ups of SQ 1,3 P 2 to AP 1,3 and P 1,3, where AP 1,3 : the first nontrivial case of Alexeev s moduli SQ 1,3 ( P 1 ) : the moduli of Hesse cubics defined by I Nakamura Masamichi Kuroda The GIT moduli of pairs August / 35

5 1 Introduction 2 The GIT moduli P 1,3 of semistable pairs 3 A comparison between AP 1,3 and P 1,3 4 Maps from blowing-ups of SQ 1,3 P 2 to AP 1,3 and P 1,3 5 A recent study and problem Masamichi Kuroda The GIT moduli of pairs August / 35

6 A quick review of the Geometric Invariant Theory X : a variety, G : algebraic group acting on X For the construction of a quotient of X by G, we desire the quotient to have a natural structure of variety However, in general the orbit space does not have it = We need to modify the definition of the quotient (the GIT quotient) We denote by X//G the GIT quotient of X by G, when it exists Proposition If X is an affine variety X = Spec R, then X//G = Spec R G, where R G is the subring of G-invariant functions But, for a projective variety X, the GIT quotient does not exists in general = We can take an open G-inv subset of X which has a GIT quotient This open subset is the set X ss of semistable points Proposition Ịf X is a projective variety X = Proj R, then X ss //G = Proj R G Masamichi Kuroda The GIT moduli of pairs August / 35

7 The definition of P 1,3 Notations k : an algebraic closed field of characteristic 2, 3 V : the dual space of k[x 0, x 1, x 2 ] 1 P(V ) := Proj(Sym(V )) : the space of lines on P 2 k = Proj(k[x 0, x 1, x 2 ]) P(S 3 V ) := Proj(Sym(S 3 V )) : the space of cubic curves on P 2 k The natural action of PGL(3) on P 2 k induces an action of PGL(3) on P(S 3 V ) P(V ): g (C, L) = g (V (F ), V (S)) := (V (F (g 1 x)), V (S(g 1 x))), where g PGL(3), F (S 3 V ), S V Masamichi Kuroda The GIT moduli of pairs August / 35

8 Definition 21 Let ( P(S 3 V ) P(V ) ) ss be the subset of all semistable pairs We denote by P 1,3 the GIT quotient of ( P(S 3 V ) P(V ) ) ss by PGL(3): p : ( P(S 3 V ) P(V ) ) ss P 1,3 := ( P(S 3 V ) P(V ) ) ss //PGL(3) Remark 22 We do not give the definition of GIT moduli in detail However by the definition, we have the following: For any z 1, z 2 ( P(S 3 V ) P(V ) ) ss, p(z 1 ) = p(z 2 ) PGL(3) z 1 PGL(3) z 2 ( P(S 3 V ) P(V ) ) ss Thus P 1,3 is the set of closures of orbits of semistable pairs Masamichi Kuroda The GIT moduli of pairs August / 35

9 The numerical criterion of stability Definition 23 (i) λ : an one parameter subgroup (1-PS) of PGL(3) def λ : G m PGL(3) is a nontrivial hom of algebraic groups (ii) An 1-PS λ is called normalized def r i Z, r 0 r 1 r 2 and r 0 + r 1 + r 2 = 0 st λ(t) = Diag(t r 0, t r 1, t r 2 ) For any z = (C, L) = (V (F ), V (S)) P(S 3 V ) P(V ), F (x) = a ij x 3 i j 0 x i 1x j 2, S(x) = b k x k, and any normalized 1-PS λ, we define H(x, y) := F (x)s(y) = a ij b k x 3 i j 0 x i 1x j 2 y k, µ(z, λ) := max {R ijk := (3 i j)r 0 + ir 1 + jr 2 + r k a ij b k 0} Masamichi Kuroda The GIT moduli of pairs August / 35

10 How to classify stability of pairs Theorem 24 (The numerical criterion of stability) For any z P(S 3 V ) P(V ), z : semistable µ(g z, λ) 0, λ : normalized 1-PS, g PGL(3), z : stable µ(g z, λ) > 0, λ : normalized 1-PS, g PGL(3) An unstable pair is a non-semistable pair (see Mumford, [GIT]) Step 1 Determine the geometric conditions of pairs z st µ(z, λ) < 0 for some λ = We obtain unstable conditions Step 2 Determine the geom conditions which do not satisfy any of unstable conditions = We obtain semistable conditions Step 3 Determine the geom cond s of semistable pairs z with µ(z, λ) = 0 for some λ = We obtain semistable but not stable conditions, and stable conditions Masamichi Kuroda The GIT moduli of pairs August / 35

11 Known Example The GIT moduli of plane cubics We consider the GIT moduli of plane cubics: p : ( P(S 3 (V )) ) ss ( P(S 3 (V )) ) ss //PGL(3) =: P Table 1: The semistable and stable cubic C (1) (2) (3) (4) C stability stable semistable semistable semistable ( P(S 3 V ) ) ss Oi : the subset of semistable cubics in the row (i) in Table 1 Then p(o 2 ) = p(o 3 ) = p(o 4 ) is a single point in P, and hence P P 1 k In fact, we have O i = PGL(3) z i (i = 2, 3, 4), where z 2 = V (x x 3 2 3x 0 x 1 x 2 ), z 3 = V (x 0 (x 1 x 2 + x 2 0)), z 4 = V (x 0 x 1 x 2 ) g b := Diag(b, 1, 1) PGL(3) (b 0), g b z 2 = V (b(x x 3 2) 3x 0 x 1 x 2 ), g 1 b z 3 = V (x 0 (x 1 x 2 + b 2 x 2 0)), and hence lim b 0 g b z 2 = lim b 0 g 1 b z 3 = z 4 p(z 2 ) = p(z 3 ) = p(z 4 ) P Masamichi Kuroda The GIT moduli of pairs August / 35

12 The classification of unstable, semistable and stable pairs We study the GIT quotient p : ( P(S 3 V ) P(V ) ) ss P 1,3 = ( P(S 3 V ) P(V ) ) ss //PGL(3) Step 1 (Unstable conditions) The pair (C, L) is unstable iff one of the following is true: L is a triple tangent to C, L C red, L passes through a double point of C, C has a triple point, C is nonreduced Step 2 (Semistable conditions) The pair (C, L) is semistable iff any of the following is true: C is reduced and free from triple points, L is not contained in C, L does not pass through any double point of C, L is not a triple tangent to C Masamichi Kuroda The GIT moduli of pairs August / 35

13 Proposition 25 (Step 3, Semistable and stable pairs) k (C, L) stability stable semistable stable stable semistable semistable semistable stable semistable stable semistable Masamichi Kuroda The GIT moduli of pairs August / 35

14 Nontrivial identifications of semistable pairs in P 1,3 S k : the subset of semistable or stable pairs in the row k in Prop 25 Then we have S 5 = PGL(3) z 5 + PGL(3) z 7, z 5 := (V (x 0 (x 0 x 2 + x 1 (x 1 + x 2 ))), V (x 2 )), S 6 = PGL(3) z 6 + PGL(3) z 7, z 6 := (V (x 0 (x 0 x 2 + x 1 (x 0 + x 1 ))), V (x 2 )), S 7 = PGL(3) z 7, z 7 := ( V (x 0 (x 0 x 2 + x 2 1)), V (x 2 ) ), S 11 = PGL(3) z 11, z 11 := ( V (x 2 0x 2 + x 2 1(x 0 + x 1 )), V (x 2 ) ) g a := Diag(a, 1, 1/a), h a := Diag(1, 1/a, 1/a 2 ) PGL(3) (a 0), g a z 5 = (V (x 0 (x 0 x 2 + x 1 (x 1 + ax 2 ))), V (x 2 )) z 7 (as a 0), g 1 a z 6 = (V (x 0 (x 0 x 2 + x 1 (ax 0 + x 1 ))), V (x 2 )) z 7 (as a 0), h a z 11 = ( V (x 2 0x 2 + x 2 1(x 0 + ax 1 )), V (x 2 ) ) z 7 (as a 0) Thus p(z 5 ) = p(z 6 ) = p(z 7 ) = p(z 11 ), and hence we have Proposition 26 p(s 5 ) = p(s 6 ) = p(s 7 ) = p(s 11 ) is a single point in P 1,3 Masamichi Kuroda The GIT moduli of pairs August / 35

15 1 Introduction 2 The GIT moduli P 1,3 of semistable pairs 3 A comparison between AP 1,3 and P 1,3 4 Maps from blowing-ups of SQ 1,3 P 2 to AP 1,3 and P 1,3 5 A recent study and problem Masamichi Kuroda The GIT moduli of pairs August / 35

16 The first nontrivial case of Alexeev s moduli AP 1,3 AP 1,3 : the moduli of pairs (C, L) st C : a cubic curve with at worst nodal singularities L : a line which does not path through singularities of C (C, L) Table 2: The pairs (C, L) in AP 1,3 (1) (2) (3) (4) (5) (6) (7) (8) (9) W T := {(C, L) AP 1,3 L is a triple tangent to C} = {(C, L) AP 1,3 (C, L) satisfies (3) or (9)} Masamichi Kuroda The GIT moduli of pairs August / 35

17 A birational map from P 1,3 to AP 1,3 W Cusp := p(s 10 ) + p(s 11 ) = {(C, L) P 1,3 C is a cusp cubic}, Y := P 1,3 \ W Cusp, Z := AP 1,3 \ W T The identity map f : Y Z, (C, L) (C, L) defines a birational map f : P 1,3 AP 1,3 The base loci of f, f 1 are W Cusp, W T, respectively G(f) := Im ((id, f) : Y Y Z) P 1,3 AP 1,3 : the graph of f Theorem 31 f : P 1,3 AP 1,3 is a birational map and G(f) \ G(f) = pr 1 1 (W Cusp ) pr 1 2 (W T ) = W Cusp W T P 1 P 1, where pr 1, pr 2 are the canonical projections from P 1,3 AP 1,3 to P 1,3, AP 1,3, respectively Masamichi Kuroda The GIT moduli of pairs August / 35

18 1 Introduction 2 The GIT moduli P 1,3 of semistable pairs 3 A comparison between AP 1,3 and P 1,3 4 Maps from blowing-ups of SQ 1,3 P 2 to AP 1,3 and P 1,3 5 A recent study and problem Masamichi Kuroda The GIT moduli of pairs August / 35

19 Notations SQ 1,3 ( P 1 k ) : the moduli of Hesse cubics over k defined by I Nakamura P(V )( P 2 k ) : the space of lines on P2 k We have rational maps φ (resp ψ) : X := SQ 1,3 P(V ) AP 1,3 (resp P 1,3 ), (µ 0 : µ 1 ) (b 0 : b 1 : b 2 ) (C, L), C : µ 0 (x x x 3 2) = 3µ 1 x 0 x 1 x 2, L : b 0 x 0 + b 1 x 1 + b 2 x 2 = 0 To see a rough structure of AP 1,3 (resp P 1,3 ), we construct a scheme X (resp ˆX) such that φ (resp ψ) can be extended on it We give a relation between the birational in Section 3 and these two extended maps Masamichi Kuroda The GIT moduli of pairs August / 35

20 The base loci of birational maps φ and ψ Recall that φ and ψ are defined by φ (resp ψ) : X := SQ 1,3 P(V ) AP 1,3 (resp P 1,3 ), (µ 0 : µ 1 ) (b 0 : b 1 : b 2 ) (C, L), C : µ 0 (x x x 3 2) = 3µ 1 x 0 x 1 x 2, L : b 0 x 0 + b 1 x 1 + b 2 x 2 = 0 φ : X AP 1,3 is not well-defined on four 3-gons {(0 : 1) (b 0 : b 1 : b 2 ) X b 0 b 1 b 2 = 0}, {(β : 1) (b 0 : b 1 : b 2 ) X (b 0 + b 1 + βb 2 )(b 0 + ζ 3 b 1 + βζ 2 3b 2 )(b 0 + ζ 2 3b 1 + βζ 3 b 2 ) = 0 }, and ψ : X P 1,3 is not well-defined on the above 3-gons and nine lines { } A (j) i := (µ 0 : µ 1 ) (b 0 : b 1 : b 2 ) X b i+1 = b i+2 ζ 2j 3, b i+2µ 1 = b i ζ 2j 3 µ 0 where β 3 = 1 We are going to blow up X along the above base loci Masamichi Kuroda The GIT moduli of pairs August / 35

21 How to construct blowing-ups X and ˆX of X We first construct X by blowing up X along the above four 3-gons and define an extended map φ : X AP 1,3 Since the other cases are similar, it suffices to blow up an affine subset ( [ ]) 1 U := Spec k s, t, u, u 3, s := b 0, t := b 1, u := µ 0 1 b 2 b 2 µ 1 along two lines L 0 := (s = u = 0) and L 1 := (t = u = 0) Masamichi Kuroda The GIT moduli of pairs August / 35

22 Sketch of construction The rational map φ on U is given by { C : u(x 3 φ : U AP 1,3, (s, t, u) 0 + x x 3 2) = 3x 0 x 1 x 2, L : sx 0 + tx 1 + x 2 = 0, and its base loci are L 0 = (s = u = 0) and L 1 = (t = u = 0) We first blow up U at O = L 0 L 1 = (0, 0, 0) We denote it by π 1 : U U On{ U 1, let x 2 = uy 2, x i = y i (i = 0, 1) C : y y1 3 + u 3 y2 3 = 3y 0 y 1 y 2, L : (s/u)y 0 + (t/u)y 1 + y 2 = 0 =: φ(s/u, t/u, u) This pair is contained in AP 1,3 On{ U 2, let x 2 = sy 2, x i = y i (i = 0, 1) C : (u/s)(y y1 3 + s 3 y2) 3 = 3y 0 y 1 y 2, =: φ(s, t/s, u/s) L : y 0 + (t/s)y 1 + y 2 = 0 This pair is not contained in AP 1,3 if and only if u/s = t/s = 0 = We blow up U along two strict transforms L i of L i (i = 0, 1) Masamichi Kuroda The GIT moduli of pairs August / 35

23 We blow up U three times as follows: U π 1 U π 2 U π 3 Ũ, where π 1 : the blowing-up of U at the origin O = (0, 0, 0) of U π 2 : the blowing-up of U along two strict transforms L i of L i (i = 0, 1) π 3 : the blowing-up of U along two lines L i in exceptional sets of π 2 Masamichi Kuroda The GIT moduli of pairs August / 35

24 Ũ i := (i) : affine subsets of Ũ (1 i 7), and Ũ = 1 i 7 Ũi Ẽ 0 : the strict transform of (u = 0) under π := π 1 π 2 π 3 Ẽ l : the strict transforms of the exceptional sets E l of π l under π Masamichi Kuroda The GIT moduli of pairs August / 35

25 The definition of φ : X AP 1,3 Definition 41 φ : X AP 1,3 is defined on Ũ by { C : y 3 Ũ 1 (s/u, t/u, u) 0 + y1 3 + u 3 y2 3 = 3y 0 y 1 y 2, L : (s/u)y 0 + (t/u)y 1 + y 2 = 0, { C : y0 (y0 Ũ 2 (s, t/u, u/s) 2 3y 1 y 2 ) = (u/s) 3/2 (y1 3 + s 3 y2), 3 L : (u/s) 1/2 (t/u)y 0 + y 1 + y 2 = 0, { Ũ 3 (s, t 2 C : y0 (y0 /su, u/t) 2 3y 1 y 2 ) = (u/t) 3 (t 2 /su) 3/2 (y1 3 + s 3 y2), 3 L : (t 2 /su) 1/2 y 0 + y 1 + y 2 = 0, { Ũ 4 (s, t/s, su/t 2 C : 3y0 y ) 1 y 2 = (su/t 2 )(y0 3 + (t/s) 3 (y1 3 + s 3 y2)), 3 L : y 0 + y 1 + y 2 = 0 On Ũ5 (resp Ũ 6, Ũ7), φ is defined by y 0 y 1 and s t in Ũ4 (resp Ũ 3, Ũ2) Def 41 is independent of choices of square roots of u/s and t 2 /su We can glue these maps on Ũi, and hence we obtain φ on Ũ Masamichi Kuroda The GIT moduli of pairs August / 35

26 Remark 42 For any v Ũ, we put φ(v) = (C, L) AP 1,3 Then we have C : a 3-gon v Ẽ0, C : an irreducible conic plus a line v (Ẽ2 Ẽ3) \ Ẽ0, C : a nodal v Ẽ1 \ (Ẽ0 Ẽ2 Ẽ3), C : a smooth v Ũ \ (Ẽ0 Ẽ1 Ẽ2 Ẽ3) In summary, X is given by the blowing-up of X three times: X π 1 X π 2 X π 3 X Then we can define the extended map φ : X AP 1,3 explicitly The exceptional sets E π of π := π 1 π 2 π 3 correspond to the pairs (C, L) st C is a nodal curve or a line + a conic X \ E π corresponds to the pair (C, L) st C is a smooth curve or 3-gon Masamichi Kuroda The GIT moduli of pairs August / 35

27 We can define a rational map ψ : X P 1,3 which is same as above φ The base loci of ψ are the strict transforms Ã(j) i of A (j) i Similarly to above, by blowing up X along nine lines Ã(j) i, we can construct ˆX such that ψ can be extended on it In fact, ˆX is given by a blowing-up of X three times: X p 1 X p 2 X p 3 ˆX We can define a extended map ψ : ˆX P 1,3 Let p := p 1 p 2 p 3 Remark 43 The exceptional sets Ê of p correspond to the pairs (C, L) st C is a cusp cubic ˆX \ Ê corresponds to the pairs (C, L) st C is a smooth curve, a 3-gon, a nodal cubic, or a line + a conic (same as Rem 42) Masamichi Kuroda The GIT moduli of pairs August / 35

28 Proposition 44 We have a commutative diagram ψ ˆX P 1,3 f p π X X φ AP 1,3 where X = SQ 1,3 P 2 f : the birational map defined in Section 3 π = π 1 π 2 π 3 and p = p 1 p 2 p 3 : compositions of blowing-ups = φ 1 (W T ) is the center à (j) i of p, ψ 1 (W Cusp ) is the exceptional set Ê of p Masamichi Kuroda The GIT moduli of pairs August / 35

29 1 Introduction 2 The GIT moduli P 1,3 of semistable pairs 3 A comparison between AP 1,3 and P 1,3 4 Maps from blowing-ups of SQ 1,3 P 2 to AP 1,3 and P 1,3 5 A recent study and problem Masamichi Kuroda The GIT moduli of pairs August / 35

30 A recent study and problem Let k = C Let P 1,3 := p(s 1 ) = {(C, L) P 1,3 C : smooth, L : transv to C} By associating each pair (C, L) P 1,3 with the mixed Hodge structure (H 1 (C \ (C L)), W, F ), we can define the period map ϕ : P 1,3 Γ\D, where D : the classifying space of mixed Hodge str s of some type Φ 0, Γ : the monodromy group Problems What is the compactification Γ\D Σ of Γ\D? Can we extend the period map to a map ϕ : P 1,3 Γ\D Σ? where D Σ : the set of all nilpotent orbits, Γ\D Σ : the toroidal partial compactification of Γ\D constructed by K Kato, C Nakayama and S Usui Masamichi Kuroda The GIT moduli of pairs August / 35

31 The Hodge date Φ 0 Φ 0 is a quadruple (H 0, W, (, w ) w Z, (h p,q ) p,q Z ), where H 0 = Ze 1 + Ze 2 + Ze 3 + Ze 4 W 0 = 0 W 1 = Re 1 + Re 2 W 2 = H 0,R := R Z H 0 Let e 1, e 2 (resp e 3, e 4 ) be images of e 1, e 2 (resp e 3, e 4 ) in gr W 1 := W 1 /W 0 (resp gr W 2 := W 2/W 1 ), respectively, w is the nondegenerate { R-bilinear form gr W w gr W w R st symmetric if w is even,, w is and skew-symmetric if w is odd, e 2, e 1 1 = 0, e 3, e 3 2 = e 4, e 4 2 = 1, e 3, e 4 2 = (p, q) = (1, 1), h p,q = 1 (p, q) = (0, 1), (1, 0) 0 otherwise Masamichi Kuroda The GIT moduli of pairs August / 35

32 The classifying space D and the monodromy group Γ By a simple calculation, we have D = {F (τ, z 1, z 2 ) τ H, z 1, z 2 C} H C 2, where H is the upper half-plane and F = F (τ, z 1, z 2 ) is defined by 0 = F 2 F 1 = C(τe 1 + e 2 ) + C(z 1 e 1 + e 3 ) + C(z 2 e 1 + e 4 ) F 0 = H 0,C, and we have Γ = g = A 1 O A 2 [ ( ) where G Z gr W = 1 1 a 1 a 3 a 2 a 4 ], [ a 1, a 2, a 3, a 4 Z A 1 SL 2 (Z), A 2 G Z ( gr W 2 ) ] [ 1 0, 0 1 ], Then Γ acts on D by (g, F (τ, z 1, z 2 )) g F (τ, z 1, z 2 ) = F (τ, z 1, z 2), where τ = A 1 τ = aτ + b cτ + d, [ ] [z 1 z 2] z1 = cτ + d + a 1 a 2 τ z 2 cτ + d + a 3 a 4 τ A 1 2 Masamichi Kuroda The GIT moduli of pairs August / 35

33 The period map ϕ : P 1,3 Γ\D The period map ϕ : P 1,3 Γ\D is given by the following: For any pair (C, L) P 1,3, there exists τ H st C E τ := C/(Z + Zτ), C L {p 1, p 2, p 3 } E τ Let [z 1 ] = p 1 p 2, [z 2 ] = p 2 p 3 E τ (z 1, z 2 C) Then we have ϕ : P 1,3 Γ\D Γ\ ( H C 2), (C, L) [F (τ, z 1, z 2 )] [(τ, z 1, z 2 )] Aim To construct the compactification Γ\D Σ of Γ\D explicitly To extend the above period map to a map ϕ : P 1,3 Γ\D Σ Masamichi Kuroda The GIT moduli of pairs August / 35

34 Appendix The definition of the GIT quotient Definition Let G be an algebraic group acting on a variety X Then a G-inv mor p : X Y (ie p(g x) = p(x) for g G, x X) is called a GIT quotient, if p satisfies the following properties: (i) For all open U Y, p : O Y (U) O X (p 1 (U)) G O X (p 1 (U)) (ii) If W X is closed and G-inv (ie g W = W for g G), then p(w ) Y is closed (iii) If V 1, V 2 X are closed, G-inv, and V 1 V 2 =, then p(v 1 ) p(v 2 ) = Proposition Let p : X Y be a GIT quotient Then for any x 1, x 2 X, p(x 1 ) = p(x 2 ) G x 1 G x 2 Masamichi Kuroda The GIT moduli of pairs August / 35

35 Appendix The definition of stability Definition Let G be a reductive group acting on a projective variety X which has an embedding φ : X P n Then a point x X is called (i) semistable, if there exists some G-invariant homogeneous polynomial f of positive degree such that f(x) 0, (ii) stable, if there exists f as in (i) and additionally G x := {g G g x = x} is finite and all orbits of G in X f := {y X f(y) 0} are closed, (iii) unstable, if it is not semistable Proposition Let R be the coordinate ring of X Then there exists a GIT quotient p : X ss X ss //G = Proj R G Masamichi Kuroda The GIT moduli of pairs August / 35