Syzygies of curves via K3 surfaces Michael Kemeny - - PowerPoint PPT Presentation

Syzygies of curves via K3 surfaces

Michael Kemeny

Humboldt-Universit¨ at zu Berlin

August 28, 2015. Haeundae, Busan, Korea

Notation

◮ L denotes a globally generated line bundle on a smooth, projective

curve C.

◮ For M any line bundle Ki,j(C, M; L) denotes the middle cohomology

in the sequence

i+1

• H0(C, L) ⊗ H0(C, (j − 1)L + M) →

i

• H0(C, L) ⊗ H0(C, jL + M)

→

i−1

• H0(C, L) ⊗ H0(C, (j − 1)L + M)

◮ Ki,j(C, L) is short for Ki,j(C, OC; L) ◮ The line bundle L is said to satisfy property (Np) if Ki,j(C, L) = 0

for i ≤ p, j ≥ 2.

p-very ampleness

A line bundle L is called p-very ample iff for any effective divisor D of degree p + 1 ev : H0(C, L) → H0(D, L|D) is surjective. Equivalently, L is not p v. a. iff φL : C → Pr admits a (p + 1) secant p − 1 plane. Thus L is 0-very ample ⇔ L is globally generated. Thus L is 1-very ample ⇔ L is very ample.

Secant Conjecture

Conjecture (Secant Conjecture, Green–Lazarsfeld, 86)

Let L be a globally generated line bundle of degree d on a curve C of genus g such that d ≥ 2g + p + 1 − 2h1(C, L) − Cliff(C). Then L fails property Np if and only if L is not p + 1-very ample. The line bundle L is not p + 1-very ample iff φL admits a (p + 2)-secant p plane.

Special case: p = 0

Firstly the direction “L not (p + 1) v.a. = ⇒ L fails Np” is straightforward and was shown by GL when they formulated the

• conjecture. So the conjecture concerns the other direction.

For L very ample, condition N0 is equivalent to φL : C → Pr being projectively normal. In this case the Secant Conjecture becomes:

Theorem (Green–Lazarsfeld ’86)

Let L be a very ample line bundle on C with deg(L) ≥ 2g + 1 − 2h1(C, L) − Cliff(C). Then φL : C → Pr is projectively normal.

Special case L = ωC

We next consider the special case L = ωC. Assume p < Cliff(C). Then ωC is p + 1-very ample. Indeed, otherwise ∃D ∈ Cp+2 such that H0(ωC) → H0(D, ωC |D) is non-surjective. Equivalently H0(D, ωC |D) → H1(C, ωC(−D)) is nonzero, so the surjection H1(C, ωC(−D)) ։ H1(C, ωC) is non-injective. Thus h0(C, O(D)) = h1(C, ωC(−D)) ≥ 2 so Cliff(C) ≤ p which is a contradiction.

Green’s conjecture

Thus the Secant Conjecture becomes, in case L = ωC:

Conjecture (Green’s Conjecture, 1984)

The canonical bundle ωC satisfies (Np) for p < Cliff(C).

K3 Surfaces

A K3 surface is a smooth, projective surface with h1(X, OX) = 0 and ωX ≃ OX. By the adjunction formula, if C ⊆ X is smooth curve then O(C)|C ≃ ωC. K3 surfaces enter the picture via the following proposition:

Theorem (Lefschetz Thm. Green 1984, Farkas, K.- 2015)

Let X be a K3 surface and let L, H be line bundles on X. Assume H is effective and base point free and either:

• 1. L the trivial line bundle OX, or
• 2. (L · H) > 0.

In addition assume H1(X, qH − L) = 0 for all q ≥ 0. Then, for any smooth D ∈ |H|, restriction gives an isomorphism Kp,q(X, −L; H) ≃ Kp,q(D, −L; ωD).

Proof of Lefschetz Thm

Proof.

By the assumptions we have, for each q ∈ Z, a short exact sequence 0 → H0(X, (q − 1)H − L)

s

→ H0(X, qH − L) → H0(D, qωD − L|D) → 0, where the first map is multiplication by the section s ∈ H0(X, H) defining D. Thus we have a s.e.s of graded SymH0(X, H) modules 0 →

• q

H0(X, (q−1)H−L) →

• q

H0(X, qH−L) →

• q

H0(D, qωD−L|D) → 0. Taking Koszul cohomology leads to the l.e.s → Kp,q−1(X, −L, H)

s

→ Kp,q(X, −L, H) → Kp,q(B, H0(X, H)) → Kp−1,q(X, −L, H)

s

→ where B is the graded SymH0(X, H) module

q H0(D, qωD − L|D).

Proof Cont.

Proof (Cont.)

But multiplication by s induces the zero map on cohomology so we have Kp,q(B, H0(X, H)) ≃ Kp,q(X, −L, H) ⊕ Kp−1,q(X, −L, H). Next we have 0 → H0(X, OX)

s

→ H0(X, H) → H0(D, KD) → 0 so choose a splitting H0(X, H) ≃ H0(D, KD) ⊕ C{s}. This gives

p

• H0(X, H) ≃

p

• H0(D, KD) ⊕

p−1

• H0(D, KD).

A diagram chase then gives Kp,q(B, H0(X, H)) ≃ Kp,q(D, −L, ωD) ⊕ Kp−1,q(D, −L, ωD). Induction

• n p now gives the claim.

Consequences

The Lefschetz Theorem has some interesting consequences for sections of K3 surfaces. For example, if H is very ample then it implies Kp,q(D1, ωD1) ≃ Kp,q(X, H) ≃ Kp,q(D2, ωD1) for any two smooth sections D1, D2 ∈ |H|. Thus, Green’s conjecture would imply that Cliff(D) is constant for all D ∈ |H|. This was verified by Green and Lazarfseld in 87. In fact, one can always compute the Clifford index of sections of a very ample line bundle in terms of the Picard lattice of a K3 surface.

Lazarsfeld’s conjecture

For our purposes, we will only need one special case:

Theorem (Lazarsfeld, 86)

Let H be a line bundle on a K3 surface, and assume that H is base point free and (H)2 > 0. Assume there is no decomposition H = L1 ⊗ L2 for line bundles Li with h0(Li) ≥ 2 for i = 1, 2. Then a general smooth curve C ∈ |H| is Brill–Noether–Petri general. In particular, Cliff(C) = ⌊ g(C)−1

2

⌋ in the conclusion of the theorem above.

Green’s conjecture for curves on K3

The results on the previous slides suggest that it might be a good idea to try and prove Green’s conjecture for a curve C on a K3 surface. This was done, by C. Voisin in 2002 and 2005, in the case Pic(X) ≃ Z[C] and by Aprodu–Farkas (2011) in general:

Theorem (Voisin, Aprodu–Farkas)

Green’s Conjecture holds for a smooth curve C ⊆ X lying on a K3 surface X. In particular we deduce by semicontinuity the fact that Green’s conjecture holds for general curves. But we also get even more, e.g. by applying the Thm to K3s with Picard rank two generated by classes C and E with (E)2 = 0, (C · E) = k one sees that Green’s conjecture holds for general curves of each gonality (warning: this is not how this fact was first proven!).

Kernel Bundles

We will need the kernel bundle description of Koszul cohomology. Let L be a globally generated line bundle on a variety. Set ML := Ker(H0(X, L) ⊗ OX ։ X), where the morphism is evaluation of sections. For any second line bundle F (or coherent sheaf), we have Kp,q(X, F; L) ≃ coker(

p+1

• H0(L) ⊗ H0(X, F ⊗ Lq−1) → H0(

p

• ML ⊗ F ⊗ Lq))

≃ ker(H1(

p+1

• ML ⊗ F ⊗ Lq−1)) →

p+1

• H0(L) ⊗ H1(F ⊗ Lq−1))

Kernel Bundles: Hyperelliptic curves

In particular, if either H0(p ML ⊗ F ⊗ Lq)) = 0 or H1(p+1 ML ⊗ F ⊗ Lq−1)) = 0 then Kp,q(X, F; L) = 0. One more fact will be useful. If C is a hyperelliptic curve and E ∈ Pic2(C) denotes the g 1

2 then

ωC ≃ (g − 1)E. Further, the canonical morphism φωC : C → Pg−1 is the composite of φE : C → P1 with the Veronese v : P1 ֒ → Pg−1.

Kernel Bundles: Hyperelliptic curves

The Euler sequence can be written 0 → ΩPg−1(1) → H0(O(1)) ⊗ OPg−1 → O(1) → 0. Thus MωC ≃ φ∗

E(ΩPg−1(1)).

Further, v ∗(ΩPg−1(1)) ≃ OP1(−1)⊕g−1. So we have MωC ≃ OC(−E)⊕g−1.

Generic Secant Conjecture

We can now prove our first main result.

Theorem (Farkas, K.-)

The Secant Conjecture holds for general curve C of genus g with a general line bundle L of degree d on C. The word general means that the property holds on a dense open subset

• f the appropriate moduli space.

Proof of Generic Secant Conj.

Proof.

We first state some simple reductions. In the case h1(C, L) ≥ 1, the claim reduces to generic Green’s Conj, by an straightforward argument of Koh–Stillman from 1989. So we may assume h1(C, L) = 0. Assume L is p + 1 very ample with h1(L) = 0 and d ≥ 2g + p + 1 − Cliff(C). We need to show L satisfies (Np). In fact, L is projectively normal under this bound on d (and h1(L) = 0), so by a result of GL we only need to prove Kp,2(C, L) = 0. As h1(C, L) = 0, then L fails to be p + 1 very ample iff there exists D ∈ Cp+2 such that h0(C, L(−D)) ≥ d − g − p. By Riemann–Roch this is equivalent to h1(C, L(−D)) ≥ 1 or −(L − D − KC) effective.

Proof of Secant Conj.

Proof cont.

In other words, if L is p + 1 very ample then L − KC / ∈ Cp+2 − C2g−d+p. It is an easy exercise to prove that if a + b ≥ g then the difference map Ca × Cb → Pica−b(C) is surjective. Thus we must have (p + 2) + (2g − d + p) ≤ g − 1 or d ≥ g + 2p + 3 (1) in addition to d ≥ 2g + p + 1 − Cliff(C). (2)

Proof of Secant Conj.

Proof cont.

For simplicity, assume g = 2i + 1 is odd, the even case is similar. It is easy to see h1(C, L) = 0 and Kp,2(C, L) = 0 implies Kp−1,2(C, L(−x)) = 0 for any x ∈ C with L(−x) globally generated. Using this we reduce to the case p ≥ i − 1. In this case inequalities (1),(2) reduce to d ≥ 2p + 2i + 4. Using the projective normality result (from two slides back) we see that it suffices to assume d = 2p + 2i + 4.

Proof of Secant Conj.

Proof cont.

Thus we need to show that, for any p ≥ i − 1, there exists a line bundle L of degree 2p + 2i + 4 on a curve C of genus 2i + 1 satisfying Kp,2(C, L) = 0. To this end, consider a K3 surface X with Picard group generated by two classes H, η giving the intersection matrix 4p + 4 2p − 2i 2p − 2i −4

• .

We let C be a smooth curve in the linear system H − η, and will show Kp,2(C, H|C ) = 0. Following Green, from the sequence of graded Sym(H0(X, H)) modules 0 →

• q

H0(X, qH − C) →

• q

H0(X, qH) →

• q

H0(C, qH) → 0 we deduce the long exact sequence → Kp,2(X, H) → Kp,2(C, H) → Kp−1,3(X, −C, H) →

Proof of Secant Conj. cont.

Proof cont.

Using Lazarsfeld’s theorem, one shows D is BNP general for D ∈ |H| general and thus Cliff(D) = p + 1 for any D ∈ |H| smooth. By the Lefschetz hyperplane theorem and Green’s Conj for K3 surfaces Kp,2(X, H) = 0. Thus we need to show Kp−1,3(X, −C, H) = 0 or equivalently, Kp−1,3(D, OD(−C), ωD) = 0, by the Lefschetz theorem. For this, using Lazarsfeld’s kernel bundle description of Koszul cohomology it is in turn sufficient to show H1(D,

p

• MKD ⊗ KD ⊗ η|D) = 0

where MKD is the kernel of ev : H0(D, KD) ⊗ OD ։ KD.

Proof of Secant Conj. cont.

Proof cont.

For the last vanishing H1(D, p MKD ⊗ KD ⊗ η|D) = 0, we further degenerate so that D becomes hyperelliptic, by adding an elliptic curve E with (E · C) = 2, (E · η) = 0 into the Picard group of the K3. As D is hyperelliptic KD = (g(D) − 1)E. Further, MKD ≃ OD(−E)⊕g(D) so

j

• MKD ≃ OD(−jE)⊕(

g(D)−1 j

). This reduces the statement to the easy verification H1(D, (p + 2)E + η) = 0. Note that ((p + 2)E + η)|D has degree 2(g(D) − 1) − 2i, so this vanishing is expected so long as 2i ≤ g(D) − 1. Since g(D) = 2p + 3, this is just the equality p ≥ i − 1 we started with.

The specific Secant Conj.

In some special cases we can prove effective versions of the secant

• conjecture. Consider the ‘extremal case’, where g = 2i + 1 is odd and

d = 2g = 4i + 2, p = i − 1. This is the case where the two inequalities (1), (2) in the proof become equalities. In this case we have an optimal result.

Theorem (Farkas–K.)

The GL Secant Conjecture holds for every smooth curve C of odd genus g and every line bundle L ∈ Pic2g(C), that is, one has the equivalence K g−3

2 ,2(C, L) = 0 ⇔ Cliff(C) < g − 1

2

• r

L − KC ∈ C g+1

2 − C g−3 2 .

(the condition L − KC ∈ C g+1

2 − C g−3 2

is the same as saying L is not g−1

2

very ample)

The specific Secant Conj.

The extremal case of the Secant conjecture is proven by a divisor calculation on Mg,2g. Let π : Mg,2g → Mg be the forgetful morphism. Define three divisors:

• 1. The Hurwitz divisor M1

g,i+1 := {[C] ∈ Mg : W 1 i+1(C) = ∅} of

curves with non-maximal Clifford index and its pull-back Hur := π∗(M1

g,i+1).

• 2. The syzygy divisor Syz :=
• [C, x1, . . . , x2g] ∈ Mg,2g :

Ki−1,2

• C, OC(x1 + · · · + x2g)
• = 0
• .
• 3. The secant divisor Sec :=
• [C, x1, . . . , x2g] ∈ Mg,2g :

OC(x1 + · · · + x2g) ∈ KC + Ci+1 − Ci−1

• .

The specific Secant Conj.

One now calculates the class of Syz in CH1(Mg,2g). To do this, one first writes it as a degeneracy locus and then uses Grothendieck–Riemann–Roch. On the other hand, the classes of Sec and Hur have already been computed in previous work of Farkas and Harris–Mumford. Comparing the formulae, one sees [Syz] = [Sec] + i · [Hur] ∈ CH1(Mg,2g) where g = 2i + 1. This gives the claim.

The specific Secant Conj.

Using the extremal case we can deduce some other statements, such as the extremal case for even genus , i.e. where deg(L) = 2g + 1, we prove the expected statement for BNP general curves and arbitrary line bundles

• n them:

Theorem

The Green-Lazarsfeld Conjecture holds for a Brill–Noether–Petri general curve C of even genus and every line bundle L ∈ Pic2g+1(C), that is, K g

2 −1,2(C, L) = 0 ⇔ L − KC ∈ C g 2 +1 − C g 2 −2.

The specific Secant Conj.

Another result which follows from the extremal cases is:

Theorem

Let C be a BNP general curve of genus g . For p ≥ 0, let d = 2g + p + 1 − Cliff(C). If L ∈ Picd(C) is a non-special line bundle such that the secant variety V g−p−4

g−p−3 (2KC − L) has the expected dimension d − g − 2p − 4, then

Kp,2(C, L) = 0. Here V g−p−4

g−p−3 (2KC − L) denotes the locus of g − p − 3 secant g − p − 5

planes.

Natural resolutions

Let L be a globally generated line bundle on a curve C of genus g. We say that the resolution of the graded SymH0(C, L) module ΓC(L) := ⊕qH0(C, qL) is natural if, for every j at most one of the spaces Kp,j−p(C, L) is non-zero for all p. Equivalently, at most one term in each diagonal of the Betti table is non-zero. If h1(C, L) = 0 and d := deg(L), r := h0(C, L) − 1 then dim Kp,1(C, L) − dim Kp−1,2(C, L) = p d − g p

• (d + 1 − g

p + 1 − d d − g ). So in fact one can determine the whole Betti table for ΓC(L) if one knows that it has a natural resolution (by Green’s theorem the spaces Kp,q(C, L) = 0 for q ≥ 3 and by the Vanishing theorem Kp,q(C, L) = 0 for q ≤ 0 unless p = q = 0).

Prym–Green conjecture

Note that the generic Green’s conjecture implies that the resolution of ΓC(ωC) is natural for the general curve C. The Prym–Green conjecture predicts that this generalises to the general paracanonical curve, i.e. it states that for the general element [C, η] in the moduli space Rg,l parametrising curves with a torsion line bundle η ∈ Pic0(C) of order l, the graded module ΓC(ωC ⊗ η) has a natural resolution. The Prym–Green conjecture has been shown been tested probabilistically via Macaulay and to very high likelihood fails for g = 8 and g = 16 and l = 2. This suggests one should amend the conjecture appropriately, e.g. by insisting g = 2k for l = 2. Nevertheless, we have . . .

Prym–Green conjecture

Theorem (Farkas, K.-)

The Prym–Green conjecture holds for g odd and l = 2 or g ≥ 11, l ≥

• g+2

2 . I.e. for such values ΓC(ωC ⊗ η) has a natural resolution for

the general [C, η] ∈ Rg,l.

Proof in case l = 2

Proof.

Write g = 2i + 1. Then naturality of ΓC(ωC ⊗ η) reduces to two vanishings, namely Ki−1,1(C, ωC ⊗ η) = 0 and Ki−3,2(C, ωC ⊗ η). We will proceed by finding a K3 surface X together with two primitive classes C, η ∈ Pic(X) with C integral of genus g and with at worst nodes, with η2 = −4, (η · C) = 0 and η|C torsion of order l.

Proof of PG

Proof Cont.

Assume such a K3 exists. Let H = C + η. If we can check that H0(X, H − C) = 0 and H1(X, qH − C) = 0 for all q (by setting q = 1 and using RR for surfaces, this forces η2 = −4), then we get → Kp,q(X, H) → Kp,q(C, H) → Kp−1,q+1(X, −C, H) → The proof now follows that of the GL Secant conjecture. Namely, one first shows that, on the given K3s, that D ∈ |H| is BNP general for D general, using Lazarsfeld’s theorem. This gives Ki−1,1(X, H) = 0 and Ki−3,2(X, H) = 0, via Green’s conjecture for K3s. The vanishing Ki−2,2(X, −C, H) = 0 and Ki−4,3(X, −C, H) is then dealt with via a hyperelliptic degeneration.

Proof of PG

Proof Cont.

It remains to construct such K3s. For l = 2, use a classical construction of Nikulin. Namely, one takes a K3 surface with symplectic involution i together with a big and nef line bundle L which is invariant under i and (L)2 = 2g − 2. There exist such K3s of Picard rank 9 and containing a class η with the desired properties, where is C can be any smooth element of |L|. The construction in the case l ≥

• g+2

2

is given by the resolution of the following conjecture due to Barth–Verra.

Case l ≥

• g+2

2 Let Υg be the rank two lattice generated by elements L, η with (L)2 = 2g − 2, η2 = −4, (L · η) = 0.

Theorem (Barth–Verra 88, Farkas–K.)

Fix and l ≥ 2 and assume 2l2 − 2 ≥ g. Let Xg be a general Υg-polarised K3 surface. Then there exist precisely 2l2 − 2 g

• integral divisors C ∈ |L|

such that η|C ∈ Pic0(C) is nontrivial and ηl

|C ≃ OC. All such divisors C

have at worst nodal singularities.

Proof of Barth–Verra Conj

The count 2l2 − 2 g

• from the previous slide does not give the number
• f hyperplane sections C such that η|C has order exactly l unless l is

prime; nevertheless this number is positive and is given by an explicit formula via subtracting the counts of sections with order dividing l. The key step to proving the Barth–Verra conjecture is to degenerate to a K3 with rank three Picard lattice described in terms of the ordered basis {E, Γ, η} with intersection matrix:   1 1 −2 −4  

Proof of Barth–Verra Conj cont.

We degenerate L to Γ + gE. The proof now runs by observing the following two facts. Firstly, each divisor in |Γ + gF| has the form Γ + Fi + . . . + Fg, for Fi ∈ |E| elliptic fibres. Secondly, using the theory of elliptic surfaces one can show that there exists 2l2 − 2 fibres F ∈ |E| such that ηl

|F ≃ OF. This leads to the

desired formula (modulo a transversality statement which requires additional work).