Severi varieties and Brill–Noether theory of curves on abelian surfaces

Andreas Leopold Knutsen; Margherita Lelli-Chiesa; Giovanni Mongardi

doi:10.1515/crelle-2016-0029

Published by De Gruyter August 11, 2016

Severi varieties and Brill–Noether theory of curves on abelian surfaces

Andreas Leopold Knutsen , Margherita Lelli-Chiesa and Giovanni Mongardi

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2016-0029

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Abstract

Severi varieties and Brill–Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type (1,n), we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system |L| for 0≤δ≤n-1=p-2 (here p is the arithmetic genus of any curve in |L|). We also show that a general genus g curve having as nodal model a hyperplane section of some (1,n)-polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many (1,n)-polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus g curve in S equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in |L|. It turns out that a general curve in |L| is Brill–Noether general. However, as soon as the Brill–Noether number is negative and some other inequalities are satisfied, the locus |L|dr of smooth curves in |L| possessing a gdr is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill–Noether locus ℳp,dr having the expected codimension in the moduli space of curves ℳp. For r=1, the results are generalized to nodal curves.

Funding statement: The second named author was supported by the Centro di Ricerca Matematica Ennio De Giorgi in Pisa and the third named author by “Firb 2012, Spazi di moduli ed applicazioni”.

A Appendix: A stronger version of Theorem 5.7

We will prove a strengthening of Theorem 5.7 for r≥p-1-d. In this range the bundles ℰC,𝒜 have nonvanishing H1 and the existence of stable extensions yields a stronger bound than (5.8). Also note the slightly stronger assumption on [C], which is no longer an open condition in the moduli space of polarized abelian surfaces, but holds off a countable union of proper closed subsets.

Theorem A.1.

Let C be a reduced and irreducible curve of arithmetic genus p=pa⁢(C) on an abelian surface S such that [C] generates NS⁡(S). Assume that C possesses a globally generated torsion free rank one sheaf A such that deg⁡A=d≤p-1, h0⁢(A)=r+1 and r≥p-1-d. Set γ:=⌊rp-1-d⌋ if d<p-1. Then one has

(A.1)ρ⁢(p,r,d)+r⁢(r+2)

≥{12⁢r⁢(r+1),if ⁢d=p-1,γ⁢(p-1-d)⁢(r+1-12⁢(p-1-d)⁢(γ+1)),if ⁢r≥p-1-d>0.

The proof needs the following technical result.

Proposition A.2.

Let E be a vector bundle on an abelian surface S satisfying:

[c1⁢(ℰ)] generates NS⁡(S),
ℰ is generically globally generated,
H2⁢(ℰ)=0.

Let N≥0 be an integer such that, for i=1,…,N, there exists a sequence of “universal extensions”

(A.2)0→𝒪S⊕hi→ℰi→ℰi-1→0,

where E0:=E, hi:=h1⁢(Ei-1) and the coboundary map H1⁢(Ei-1)→H2⁢(OS⊕hi)≃Chi is an isomorphism (this condition is empty for N=0).

Then each Ei is stable with respect to any polarization and satisfies H2⁢(Ei)=0.

Proof.

We proceed by induction on i.

Let i=0. Then H2⁢(ℰ0)=H2⁢(ℰ)=0 by assumption (iii). Assume that ℰ0=ℰ is not stable. Consider any destabilizing sequence

(A.3)0→ℳ→ℰ→𝒬→0,

where ℳ and 𝒬 are torsion free sheaves of positive rank; this gives c1⁢(ℰ)=c1⁢(ℳ)+c1⁢(𝒬). For any ample line bundle H on S, we have that c1⁢(ℳ).H>0 because ℳ destabilizes ℰ, and c1⁢(𝒬).H>0 by Lemma 5.8, because H2⁢(𝒬)=0 and 𝒬 is globally generated off a codimension-one set by (A.3) and assumptions (ii)-(iii). This contradicts assumption (i).

Now assume that i>0 and ℰi-1 is H-stable with H2⁢(ℰi-1)=0. The fact that H2⁢(ℰi)=0 is an immediate consequence of (A.2) and the coboundary map being an isomorphism. If ℰi is not H-stable, then we have a destabilizing sequence

(A.4)0→ℳ→ℰi→𝒬→0,

with ℳ and 𝒬 torsion free sheaves of positive rank. Let ℳ′ denote the image of the composition ℳ→ℰi→ℰi-1 of maps from (A.4) and (A.2). Then we have a commutative diagram with exact rows and columns:

defining 𝒦, 𝒦′ and 𝒬′. Since 𝒦′ is globally generated, we have c1⁢(𝒦).H=-c1⁢(𝒦′).H≤0. Hence c1⁢(ℳ′).H=c1⁢(ℳ).H-c1⁢(𝒦).H≥c1⁢(ℳ).H>0, as ℳ destabilizes ℰi. In particular, as ℳ′ is torsion free, we have rk⁡ℳ′>0.

If rk⁡𝒬′>0, then rk⁡ℳ′<rk⁡ℰi-1. As ℰi-1 is H-stable, we must have

c1⁢(ℳ′).Hrk⁡ℳ′<c1⁢(ℰi-1).Hrk⁡ℰi-1.

In particular, 0<c1⁢(ℳ′).H<c1⁢(ℰi-1).H, so that c1⁢(𝒬′).H>0. But then

c1⁢(ℰ)=c1⁢(ℰi-1)=c1⁢(ℳ′)+c1⁢(𝒬′)

with both c1⁢(ℳ′).H>0 and c1⁢(𝒬′).H>0, contradicting hypothesis (i).

Hence we have rk⁡𝒬′=0 and c1⁢(𝒬′), if nonzero, is represented by the effective cycle of the one-dimensional support of 𝒬′. Then c1⁢(𝒬).H=c1⁢(𝒦′).H+c1⁢(𝒬′).H≥0 and strict inequality follows from Lemma 5.8 because h2⁢(𝒬)=h2⁢(ℰi)=0 and 𝒬 is globally generated off the one-dimensional support of 𝒬′, as 𝒦′ is globally generated. As ℳ destabilizes ℰi, we get that c1⁢(ℰ)=c1⁢(ℰi)=c1⁢(ℳ)+c1⁢(𝒬) with both c1⁢(ℳ).H>0 and c1⁢(𝒬).H>0, again contradicting hypothesis (i). ∎

Proof of Theorem A.1.

Consider the vector bundle ℰ0:=ℰC,𝒜 from the proof of Theorem 5.7, which satisfies conditions (i)–(iii) of Proposition A.2. Moreover, (5.10) and (5.11) imply

h1:=h1⁢(ℰC,𝒜)=h0⁢(ℰC,𝒜)-χ⁢(ℰC,𝒜)≥r+2+d-p>0.

Therefore, we have a “universal extension”

0→𝒪S⊕h1→ℰ1→ℰ0→0,

where the coboundary map H1⁢(ℰ0)→H2⁢(𝒪S⊕h1)≃ℂh1 is an isomorphism.

If h2:=h1⁢(ℰ1)>0, we can iterate the construction. Hence, there is an integer N>0 and a sequence of universal extensions as in (A.2), where the coboundary maps

H1⁢(ℰi-1)→H2⁢(𝒪S⊕hi)≃ℂhi

are isomorphisms and, by Proposition A.2, all ℰi are stable with H2⁢(ℰi)=0 for i=0,…,N. (We do not claim that there is a maximal such N; indeed, it may happen that the process does not terminate, i.e. all hi>0, in which case any N>0 fulfills the criteria.)

By (A.2) and properties (5.10) of ℰ0, we have

(A.5)rk⁡ℰN=r+1+h1+⋯+hN,

c1⁢(ℰN)=[C],

χ(ℰN)=χ(ℰ0)=p-1-d=:χ≥0.

Since ℰN is stable, it is simple, whence (5.13) and (A.5) yield

(A.6)p-1≥χ⁢rk⁡ℰN.

The sequence (A.2) and coboundary maps H1⁢(ℰi-1)→H2⁢(𝒪S⊕hi) being isomorphisms yield

hi+1=h1⁢(ℰi)≥2⁢hi-h0⁢(ℰi-1)=2⁢h1⁢(ℰi-1)-h0⁢(ℰi-1)=hi-χ,i≥1.

In particular, we obtain that

(A.7)hi≥h1-(i-1)⁢χ≥r+1-i⁢χ,i=1,…,N.

Hence, the procedure of taking extensions goes on at least until i=N, for any N≥1 satisfying

(A.8)N⁢χ≤r.

Note that if N satisfies (A.8), then inequalities (A.6) increase in strength as N increases.

By (A.5) and (A.7) we have

rk⁡ℰN=∑i=1Nhi+r+1≥∑i=0N(r+1-i⁢χ)=(N+1)⁢(r+1)-χ⁢∑i=0Nhj

=(N+1)⁢(r+1)-N⁢(N+1)⁢χ2=(N+1)⁢(r+1-N⁢χ2).

Hence, we obtain from (A.6) with i=N that

p-1≥χ⁢(N+1)⁢(r+1-N⁢χ2)

or, equivalently,

ρ⁢(p,r,d)+r⁢(r+2)≥χ⁢N⁢(r+1-χ⁢(N+1)2).

If χ>0, the strongest inequality is obtained by using the largest N satisfying (A.8), which is N=γ:=⌊rχ⌋. This proves (A.1) if χ>0.

If χ=0, the left-hand side of (A.1) is simply d=p-1, so we may assume that r≥2. The torsion free sheaf 𝒜′:=𝒜⊗𝒪C⁢(-P) for a general P∈C is still globally generated and satisfies deg𝒜′=d-1=:d′ and r′:=h0⁢(𝒜′)-1=r. Since 1=p-1-d′<r′, this falls into the lower line of (A.1), which easily rewrites as the desired inequality

ρ⁢(p,r,d)+r⁢(r+2)≥12⁢r⁢(r+1).∎

Remark A.3.

We do not know if the stronger condition (A.1) is optimal, although it gets rid of the cases occurring in Examples 5.13 and 5.14. One can easily verify that the inequality d≥r⁢(r+1) is stronger than (A.1) in this range. We must however recall that our Theorem 1.4 yields the existence of birational linear series on smooth curves; it is plausible that nonbirational linear systems, as well as torsion free sheaves on singular curves (cf. Example 5.15), may cover a wider value range of p, r and d. We also remark that the torsion free sheaves in the quoted example satisfy equality in (A.1).

Remark A.4.

The fact that bundles ℰC,𝒜 with h1⁢(ℰC,𝒜)>0 do indeed exist when 0≤p-1-d≤r follows from Theorem 1.4 (ii). This shows that additional complexity arises for abelian surfaces in comparison with K⁢3 surfaces, where the analogous bundles always have vanishing H1.

Acknowledgements

We thank the Max Planck Institute for mathematics and the Hausdorff Center for Mathematics in Bonn, the University of Bonn and the Universities of Roma La Sapienza, Roma Tor Vergata and Roma Tre, for hosting one or more of the authors at different times enabling this collaboration. In particular, part of this work was carried out during the Junior Hausdorff Trimester Program “Algebraic Geometry”. We also thank the referee for a very careful reading and for suggestions that improved the readability of the paper.

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Received: 2015-04-14

Revised: 2016-02-27

Published Online: 2016-08-11

Published in Print: 2019-04-01

Severi varieties and Brill–Noether theory of curves on abelian surfaces

Abstract

A Appendix: A stronger version of Theorem 5.7

Theorem A.1.

Proposition A.2.

Proof.

Proof of Theorem A.1.

Remark A.3.

Remark A.4.

Acknowledgements

References

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