Balancing in relative canonical resolutions and a unirational moduli space of K3 surfaces

Transcription

1 Balancing in relative canonical resolutions and a unirational moduli space of K3 surfaces Frank-Olaf Schreyer Universität des Saarlandes schreyer@math.uni-sb.de. The Prospects for Commutative Algebra Osaka, 10 July 2017

2 In this talk I report on recent work of two of my students Christian Bopp Michael Hoff

3 Relative canonical embedding Throughout this talk I denote by π : C D P 1 a smooth non-hyperelliptic curve of genus g together with a base point free complete pencil of divisors of degree d g 1. The canonical embedding of C factors C ι P g 1 π P 1 π X = P(E) through a P d 2 -bundle over P 1 associated to π ω C = ωp 1 E, where E is a vector bundle of rank d 1 and degree f = g d + 1 on P 1, hence slope g d+1 d 1.

4 Maroni invariant The splitting type E = O(e 1 )... O(e d 1 ) is called the Maroni invariant of (C, D ). Theorem (Ballico) For (C, D ) W 1 g,d general, E is balanced, i.e. e i e j 1.

5 Maroni invariant The splitting type E = O(e 1 )... O(e d 1 ) is called the Maroni invariant of (C, D ). Theorem (Ballico) For (C, D ) W 1 g,d general, E is balanced, i.e. e i e j 1. Pic X = ZH ZR of X = P(E) is generated by the hyperplane class H and the ruling R with intersection products d 1 H d 1 = e i = f, H d 2 R = 1 and R 2 = 0. i=1 The canonical class of X is ω X = OX ( (d 1)H + (f 2)R.

6 Relative canonical resolution Theorem (Schreyer, 1986) C P 1 a degree d cover by a curve C of genus g as above. Then as an O = O X -module O C has a locally free resolution of shape 0 O C O O( 2H) π N 1 O( 3H) π N O( (d 2)H) π N d 3 O( dh + (f 2)R) 0, where the N i are vector bundles on P 1 of rank N i = d(d 2 i) (i+1) ( d 2 i 1 )

7 Relative canonical resolution Theorem (Schreyer, 1986) C P 1 a degree d cover by a curve C of genus g as above. Then as an O = O X -module O C has a locally free resolution of shape 0 O C O O( 2H) π N 1 O( 3H) π N O( (d 2)H) π N d 3 O( dh + (f 2)R) 0, where the N i are vector bundles on P 1 of rank N i = d(d 2 i) (i+1) ( d 2 i 1 and slope µ i = (i+1)(g d 1) d (Bopp-Hoff, 2015). )

8 Relative canonical resolution Theorem (Schreyer, 1986) C P 1 a degree d cover by a curve C of genus g as above. Then as an O = O X -module O C has a locally free resolution of shape 0 O C O O( 2H) π N 1 O( 3H) π N O( (d 2)H) π N d 3 O( dh + (f 2)R) 0, where the N i are vector bundles on P 1 of rank N i = d(d 2 i) (i+1) ( d 2 i 1 and slope µ i = (i+1)(g d 1) d (Bopp-Hoff, 2015). Question Are the vector bundles N i balanced for (C, D ) W 1 g,d general? Yes, if d g 1 (Bujokas, Patel, 2015). )

9 Relative quadrics and N 1 Theorem (Bopp-Hoff, 2015) (C, D ) W 1 g,d M g general with Brill-Noether number ρ = ρ(g, d, 1) = g 2(g d + 1) > 0. Then N 1 is unbalanced iff (d ρ 7 2 )2 > d.

10 Relative quadrics and N 1 Theorem (Bopp-Hoff, 2015) (C, D ) W 1 g,d M g general with Brill-Noether number ρ = ρ(g, d, 1) = g 2(g d + 1) > 0. Then N 1 is unbalanced iff (d ρ 7 2 )2 > d. Idea: The image of C P(H 0 (ω C ( D)) = P f 1 lies on a quadric iff C X is contained in divisor of class 2H 2R.

11 Relative quadrics and N 1 Theorem (Bopp-Hoff, 2015) (C, D ) W 1 g,d M g general with Brill-Noether number ρ = ρ(g, d, 1) = g 2(g d + 1) > 0. Then N 1 is unbalanced iff (d ρ 7 2 )2 > d. Idea: The image of C P(H 0 (ω C ( D)) = P f 1 lies on a quadric iff C X is contained in divisor of class 2H 2R. The maximal rank conjecture holds for these curves P f 1.

12 Relative quadrics and N 1 Theorem (Bopp-Hoff, 2015) (C, D ) W 1 g,d M g general with Brill-Noether number ρ = ρ(g, d, 1) = g 2(g d + 1) > 0. Then N 1 is unbalanced iff (d ρ 7 2 )2 > d. Idea: The image of C P(H 0 (ω C ( D)) = P f 1 lies on a quadric iff C X is contained in divisor of class 2H 2R. The maximal rank conjecture holds for these curves P f 1. The numerical condition gives the existence of a quadric.

13 Relative quadrics and N 1 Theorem (Bopp-Hoff, 2015) (C, D ) W 1 g,d M g general with Brill-Noether number ρ = ρ(g, d, 1) = g 2(g d + 1) > 0. Then N 1 is unbalanced iff (d ρ 7 2 )2 > d. Idea: The image of C P(H 0 (ω C ( D)) = P f 1 lies on a quadric iff C X is contained in divisor of class 2H 2R. The maximal rank conjecture holds for these curves P f 1. The numerical condition gives the existence of a quadric. Can compute h 0 (X, J C/X (2H 2R). Remains a summand N 1 of N 1 with slope < 1. So also a summand O P 1 occurs.

14 Relative quadrics and N 1 Theorem (Bopp-Hoff, 2015) (C, D ) W 1 g,d M g general with Brill-Noether number ρ = ρ(g, d, 1) = g 2(g d + 1) > 0. Then N 1 is unbalanced iff (d ρ 7 2 )2 > d. Idea: The image of C P(H 0 (ω C ( D)) = P f 1 lies on a quadric iff C X is contained in divisor of class 2H 2R. The maximal rank conjecture holds for these curves P f 1. The numerical condition gives the existence of a quadric. Can compute h 0 (X, J C/X (2H 2R). Remains a summand N 1 of N 1 with slope < 1. So also a summand O P 1 occurs. Conjecture (Bopp-Hoff, 2015) For general (C, D ) Wg,d 1 and ρ = ρ(g, d, 1) 0 the bundle N 1 is balanced.

15 Cubic Resolvent Let C X P 1 be a tetragonal defined over K. Then C X is a complete intersection of two quadric bundles and N 1 = O(b 1 ) O(b 2 ) with b 1 + b 2 = g 5. The cubic resolvent of the field extension of function fields of degree [K (C) : K (t)] = 4 can be identified with the rank 2 quadrics of the pencil in each fiber P 2 t R. Proposition (Casnati, 1998) The cubic resolvent corresponding to the normal subgroup D 4 S 4 of index 3 is a trigonal curve C of genus g + 1 (if smooth) and Maroni invariant (e 1, e 2 ) = (b 1 + 2, b 2 + 2).

16 Resolvents Conjecture (Castryck-Zhou, 2017) C X P 1 be a 5-gonal of genus g defined over K. N 2 = O(b 1 )... O(b 5 ) = Hom(N 1, O(f 2)). The Cayley resolvent corresponding to the subgroup F 20 S 5 defines a 6-gonal curve of genus 3g + 7 (if smooth) with Maroni invariant (b 1 + 4,..., b 5 + 4). Here, F 20 = AffineAut(Z/5Z).

17 Resolvents Conjecture (Castryck-Zhou, 2017) C X P 1 be a 5-gonal of genus g defined over K. N 2 = O(b 1 )... O(b 5 ) = Hom(N 1, O(f 2)). The Cayley resolvent corresponding to the subgroup F 20 S 5 defines a 6-gonal curve of genus 3g + 7 (if smooth) with Maroni invariant (b 1 + 4,..., b 5 + 4). Here, F 20 = AffineAut(Z/5Z). A quintic equation f (x) = 0 for f K [x] is solvable by radicals iff its Cayley resolvent has a K -rational root.

18 Resolvents Conjecture (Castryck-Zhou, 2017) C X P 1 be a 5-gonal of genus g defined over K. N 2 = O(b 1 )... O(b 5 ) = Hom(N 1, O(f 2)). The Cayley resolvent corresponding to the subgroup F 20 S 5 defines a 6-gonal curve of genus 3g + 7 (if smooth) with Maroni invariant (b 1 + 4,..., b 5 + 4). Here, F 20 = AffineAut(Z/5Z). A quintic equation f (x) = 0 for f K [x] is solvable by radicals iff its Cayley resolvent has a K -rational root. Castryk and Zhou have further experimental findings for various other resolvents associated to subgroups G S d.

19 A unbalanced case Theorem (Bopp-Hoff, 2017) Let (C, D ) W9,6 1 be a general curve of genus 9 together with a general pencil of degree 6. Then X = P(O P 1(1) 4 O P 1) and the relative canonical resolution of O C as O = O X -module has shape O( 2H + R) 6 O( 3H + 2R) 2 O O( 2H) 3 O( 3H + R) 12 O( 4H + 2R) 3 O( 6H + 2R) 0 O( 3H) 2 O( 4H + R) 6 In particular, N 2 = O P 1(2) 2 O P 1(1) 12 O 2 P 1 is unbalanced.

20 A unbalanced case Theorem (Bopp-Hoff, 2017) Let (C, D ) W9,6 1 be a general curve of genus 9 together with a general pencil of degree 6. Then X = P(O P 1(1) 4 O P 1) and the relative canonical resolution of O C as O = O X -module has shape O( 2H + R) 6 O( 3H + 2R) 2 O O( 2H) 3 O( 3H + R) 12 O( 4H + 2R) 3 O( 6H + 2R) 0 O( 3H) 2 O( 4H + R) 6 In particular, N 2 = O P 1(2) 2 O P 1(1) 12 O 2 is unbalanced. P 1 Unlike a balanced case, it is not possible to prove this by exhibiting an example and a semi-continuity argument.

21 A unbalanced case Theorem (Bopp-Hoff, 2017) Let (C, D ) W9,6 1 be a general curve of genus 9 together with a general pencil of degree 6. Then X = P(O P 1(1) 4 O P 1) and the relative canonical resolution of O C as O = O X -module has shape O( 2H + R) 6 O( 3H + 2R) 2 O O( 2H) 3 O( 3H + R) 12 O( 4H + 2R) 3 O( 6H + 2R) 0 O( 3H) 2 O( 4H + R) 6 In particular, N 2 = O P 1(2) 2 O P 1(1) 12 O 2 is unbalanced. P 1 Unlike a balanced case, it is not possible to prove this by exhibiting an example and a semi-continuity argument. But we can study an example!

22 Idea of proof: Syzygy schemes Consider a syzygy s: O( 2H + R) 6 O( 3H + 2R) O( 2H) 3 h 0 (O X (H R)) = 4 and h 0 (O X (H 2R)) = 0. So this syzygy is really only a relation among 4 quadrics in H 0 ((J C/X (2H R)), and by [S. 1991] there exists a skew symmetric 5 5 matrix ψ = (ψ ij ) with entries as indicate H R H R H R H R H R H H H H R H H H H R H H H H R H H H such that 4 of the 5 pfaffians are the given quadrics. All five of them define a codimension 3 Gorenstein subscheme which turns out to be a K3 surface Y X.

23 The K3 surfaces Indeed, by the Buchsbaum-Eisenbud complex, ω Y = Ext 3 X (O Y, ω X ) = O Y and h 1 (O Y ) = 0 O(H) embeds Y into P 8 = PH 0 (O X (1)). Hence H cuts out on Y a family of curves of genus 8. The pencil R gives a pencil of elliptic curves on Y of degree 5 and we also have C Y. Thus the intersection products of these curves on Y are H C R H C = M. R 5 6 0

24 The K3 surfaces Indeed, by the Buchsbaum-Eisenbud complex, ω Y = Ext 3 X (O Y, ω X ) = O Y and h 1 (O Y ) = 0 O(H) embeds Y into P 8 = PH 0 (O X (1)). Hence H cuts out on Y a family of curves of genus 8. The pencil R gives a pencil of elliptic curves on Y of degree 5 and we also have C Y. Thus the intersection products of these curves on Y are H C R H C = M. R Since we have pencil of syzygies N 2 = O P 1(2) 2 O P 1(1) 12 O 2 P 1 in our (randomly choosen) example, we expect that each each pair (C, D ) W9,6 1 give rise to a pencil of such K3 surfaces.

25 A dimension count Let F M denote the moduli space of with M lattice polarized K 3 surfaces, so Y F M is a tuple Y = (Y, O Y (H), O Y (C), O Y (R)) P =? ϕ P 9 S P 1 F M W 1 9,6 M 9 (Y, C) syz. scheme (C, D, s) Y (C, D ) W 1 6 (C) C with ϕ(y, C) = (C, O C (R) )

26 A dimension count Let F M denote the moduli space of with M lattice polarized K 3 surfaces, so Y F M is a tuple Y = (Y, O Y (H), O Y (C), O Y (R)) P =? ϕ P 9 S P 1 F M W 1 9,6 M 9 (Y, C) syz. scheme (C, D, s) Y (C, D ) W 1 6 (C) C with ϕ(y, C) = (C, O C (R) ) and dimensions 26 =?? = g 3 = 24

27 Key Lemma P =? ϕ P 9 S P 1 F M W 1 9,6 M 9 26 =??

28 Key Lemma P P 9 =? ϕ S P 1 F M W 1 9,6 M 9 26 =?? Lemma (Bopp-Hoff) (Y, C) P general, (C, D ) W9,6 1 its image under ϕ. Then 1. Y X and its O X -resolution is an Buchsbaum-Eisenbud complex with skew matrix as in the example above. In particular N 2 (C, D ) is unbalanced: N 2 (C, D ) = O P 1(2) a O P 1(1) 16 2a O a P 1 with a 1 2. dim ϕ 1 (C, D ) a 1.

29 Key Lemma P P 9 =? ϕ S P 1 F M W 1 9,6 M 9 26 =?? Lemma (Bopp-Hoff) (Y, C) P general, (C, D ) W9,6 1 its image under ϕ. Then 1. Y X and its O X -resolution is an Buchsbaum-Eisenbud complex with skew matrix as in the example above. In particular N 2 (C, D ) is unbalanced: N 2 (C, D ) = O P 1(2) a O P 1(1) 16 2a O a P 1 with a 1 2. dim ϕ 1 (C, D ) a 1. = a 2

30 Key Lemma P P 9 =? ϕ S P 1 F M W 1 9,6 M 9 26 =?? Lemma (Bopp-Hoff) (Y, C) P general, (C, D ) W9,6 1 its image under ϕ. Then 1. Y X and its O X -resolution is an Buchsbaum-Eisenbud complex with skew matrix as in the example above. In particular N 2 (C, D ) is unbalanced: N 2 (C, D ) = O P 1(2) a O P 1(1) 16 2a O a P 1 with a 1 2. dim ϕ 1 (C, D ) a 1. = a 2 Existence of an example defined over Q with a = 2 = ϕ is dominant and a = 2 holds generically.

31 Key Lemma P P 9 birational ϕ S P 1 F M W 1 9,6 M 9 26 =! Lemma (Bopp-Hoff) (Y, C) P general, (C, D ) W9,6 1 its image under ϕ. Then 1. Y X and its O X -resolution is an Buchsbaum-Eisenbud complex with skew matrix as in the example above. In particular N 2 (C, D ) is unbalanced: N 2 (C, D ) = O P 1(2) a O P 1(1) 16 2a O a P 1 with a 1 2. dim ϕ 1 (C, D ) a 1. = a 2 Existence of an example defined over Q with a = 2 = ϕ is dominant and a = 2 holds generically.

32 Unirationality P P 9 birational ϕ S P 1 Corollary P and F M are unirational. F M W9,6 1 M 9 Proof: W9,6 1 is unirational by [Segre, 1926]: A general curve of genus g = 9 has a plane model of degree 9 with one triple and 16 double points. The projection from the triple points gives the the pencil of degree 6. Indeed, ( ) ( ) = 9 and = Thus 17 = general points in P 2 specifies a pair (C, D ) uniquely.

33 The plane curve Take the triple point defined by the ideal (x 1, x 2 ), the sixteen points by their Hilbert-Burch matrix which we choose randomly with 1 digit coefficients. This leads to a plane curve C defined by the form x0 6 x x0 5 x x0 4 x x 1 x x2 9 and (logarithmic) height = , i.e. binary Information about 2534 bits.

34 The family of K3 containing C It is no surprise that C is contained in a K3 surface. Actually the linear system ω C ( D) embeds C P 3 and C P 3 is contained in a net (a P 2 ) of quartics. The general such quartic has Picard rank 2 generated by the hyperplane class and the curve C. (Reality check: = 18)

35 The family of K3 containing C It is no surprise that C is contained in a K3 surface. Actually the linear system ω C ( D) embeds C P 3 and C P 3 is contained in a net (a P 2 ) of quartics. The general such quartic has Picard rank 2 generated by the hyperplane class and the curve C. (Reality check: = 18) Inside this P 2 the rational curve of our Y s form a Noether-Lefschetz component, which turns out to be a nodal cubic curve.

36 Equation of the NL-cubic: x 3 2 which has (logarithmic) height =

37 Equation of the NL-cubic: x 3 2 which has (logarithmic) height = pts plane mod. adj. syst. rel. can. eqs. 5 5 K3 surf

38 The nodal point For (C, D ) general, the node of the cubic corresponds to a K3 surface of Picard rank 4 and lattice h C L 1 L 2 h C L = N L where h H i R i and H i C L i, hence R i C L i h. Theorem (Bopp-Hoff) W 1 9,6 is birational to the universal family PN of genus 9 curves over the lattice polarized moduli space of K 3 surfaces F N.

39 The nodal point For (C, D ) general, the node of the cubic corresponds to a K3 surface of Picard rank 4 and lattice h C L 1 L 2 h C L = N L where h H i R i and H i C L i, hence R i C L i h. Theorem (Bopp-Hoff) W 1 9,6 is birational to the universal family PN of genus 9 curves over the lattice polarized moduli space of K 3 surfaces F N. Thank you!