The canonical ring of a stacky curve

Transcription

1 The canonical ring of a stacky curve David Zureick-Brown (Emory University) John Voight (Dartmouth) Automorphic Forms and Related Topics Fall Southeastern Sectional Meeting University of North Carolina at Greensboro, Greensboro, NC Nov 8, 2014

2 Modular forms Let Γ be a Fuchsian group (e.g. Γ = Γ 0 (N) SL 2 (Z)). Definition A modular form for Γ of weight k Z 0 is a holomorphic function f : H C such that f (γz) = (cz + d) k f (z) for all γ Γ and such that the limit lim z f (z) exists for all cusps. Definition Let M k (Γ) be the C-vector space of modular forms for Γ of weight k. David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

3 Ring of Modular forms Definition (Ring of Modular forms) M(Γ) := M k (Γ) k 2Z 0 Example M(SL 2 (Z)) = C[E 4, E 6 ] Theorem (Wagreich) M(Γ) is generated by two elements if and only if Γ = SL 2 (Z), Γ 0 (2), or Γ(2). David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

4 Ring of Modular forms Definition (Ring of Modular forms) M(Γ) := M k (Γ) k 2Z 0 Example (LMFDB) M(Γ 0 (11)) = C[E 2, f E, g 4 ]/(g4 2 F (E 2, f E )) Example (Ji, 1998) M(Γ 2,3,7 ) = C[ 12, 16, 30 ]/f ( 12, 16, 30 ) David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

5 Rustom s conjectures (2012) Conjecture (Rustom) The C-algebra M(Γ 0 (N)) is generated in weight at most 6 with relations in weight at most 12. This was proved by Wagreich in 1980/81. Conjecture (Rustom) The Z [1/6N]-algebra M(Γ 0 (N), Z [1/6N]) is generated in weight at most 6 with relations in weight at most 12. M k (Γ 0 (N), R) consists of forms with q-expansion in R q. David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

6 Main Theorem Conjecture (Rustom) The Z [1/6N]-algebra M(Γ 0 (N), Z [1/6N]) is generated in weight at most 6 with relations in weight at most 12. Theorem (Voight, ZB) Rustom s conjecture is true. David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

7 Translation to Geometry (Kodaira Spencer) Modular curves 1 Y = [H/Γ] 2 X = Y = [H/Γ] Kodaira-Spencer M k (Γ) = H 0 (X, Ω 1 ( ) k/2 ) f (z) f (z) dz k/2 Log canonical ring M(Γ) = R X, := k H 0 (X, Ω 1 ( ) k ) David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

8 Example: X 0 (11) (fundamental domain) David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

9 Example: X 0 (11), = P + Q Example (LMFDB) k 2Z 0 M k (Γ 0 (11)) = C[E 2, f E, g 4 ]/(g 2 4 F (E 2, f E )) Remark (Via Kodaira Spencer) k 2Z 0 M k (Γ 0 (11)) = Remark (Riemann Roch) k Z 0 H 0 (X 0 (11), k(p + Q)) dim H 0 (X 0 (11), k(p + Q)) = max{1, 2k} ( ) dim im H 0 (X 0 (11), P + Q) 2 H 0 (X 0 (11), 2(P + Q)) = 3 David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

10 Log canonical map/ring Definition The canonical map φ K : C P g 1 is given by P [ω 1 (P) :... : ω g (P)]. (An embedding iff C is not hyperelliptic.) Facts C = Proj R X, = Proj H 0 (X, Ω 1 ( ) k ) k Facts The relations among R X,1 are the defining equations of φ K (C). David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

11 Petri s theorem Let C be non-hyperelliptic, non-trigonal, not a plane quintic. Theorem (Enriques-Noether-Baggage-Petri) The canonical ring R C is generated in degree 1 with relations in degree 2. Remark 1 For C trigonal or a plane quintic R C is generated in degree 1 with relations in degrees 2 and 3 2 (unless g(c) = 3, which has a single relation in degree 4) 3 For C hyperelliptic, there are generators in degrees 1,2, relations in degrees up to 4. David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

12 Log Petri s theorem Let C be a curve and a log divisor. Theorem (Voight, ZB) The log canonical ring R C is generated in degree at most 3 with relations in degree at most 6. Remark Lots of exceptional cases if 0 < deg 2. Remark (Things stabilize) 1 Generators in degree 1 with relations in degree 2,3 if = 3 2 (Mumford.) Generators in degree 1 with relations in degree 2 if 4 David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

13 Log Petri s theorem Let C be a curve and a log divisor. Theorem (Voight, ZB) The log canonical ring R C is generated in degree at most 3 with relations in degree at most 6. Corollary Rustom s conjecture is true if Γ acts without stabilizers. David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

14 Translation to Geometry (Kodaira Spencer) Modular curves 1 Y = [H/Γ] 2 X = Y = [H/Γ] Kodaira-Spencer M k (Γ) = H 0 (X, Ω 1 ( ) k/2 ) f (z) f (z) dz k/2 Log canonical ring M(Γ) = R X, := k H 0 (X, Ω 1 ( ) k ) David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

15 Fundamental Domain for X (1) David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

16 Fundamental Domain for X (1) D = K + = d dd dim H 0 (X, dd ) dim M 2d (SL 2 (Z)) David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

17 Fractional divisors µ a1 µ a1 µ a1 Remark 1 Divisors are now fractional. 2 D = D 0 + n 1 a! P 1 + n 2 a 2 P 2 + n 3 a 3 P 3 Fact K X = K X + e P 1 P e P David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

18 Floors Definition The floor D of a Weil divisor D = i a ip i on X is the divisor on X given by D = ai π(p i ). #G Pi i Fact H 0 (X, D) = H 0 (X, D ) David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

19 Example: X (1) D = K + = 1 2 P Q d dd deg dd dim H 0 (X, dd ) M 2d (SL 2 (Z)) P + Q E 4 3 P + 2Q E 6 4 2P + 2Q E P + 3Q E 4 E 6 6 3P + 4Q E4 3, E 6 2 David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

20 Main theorem Theorem (Voight,ZB) Let (X, ) be a tame log stacky curve with signature (g; e 1,..., e r ; δ) over a field k, and let e = max(1, e 1,..., e r ). Then the canonical ring R(X, ) = H 0 (X, Ω( ) d ) d=0 is generated as a k-algebra by elements of degree at most 3e with relations of degree at most 6e. Remark Moreover, if 2g 2 + δ 0, then R(X, ) is generated in degree at most max(3, e) with relations in degree at most 2 max(3, e). David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21

21 Final comments Remark 1 We generalize to the relative and spin cases. 2 We give (relative) Gröbner bases, generic initial ideals. 3 Exact formulations of theorems are amenable to computation. David Zureick-Brown and John Voight The canonical ring of a stacky curve Nov 8, / 21