94 research outputs found
Generalized Lazarsfeld-Mukai bundles and a conjecture of Donagi and Morrison
Let S be a K3 surface and assume for simplicity that it does not contain any
(-2)-curve. Using coherent systems, we express every non-simple
Lazarsfeld-Mukai bundle on S as an extension of two sheaves of some special
type, that we refer to as generalized Lazarsfeld-Mukai bundles. This has
interesting consequences concerning the Brill-Noether theory of curves C lying
on S. From now on, let g denote the genus of C and A be a complete linear
series of type g^r_d on C such that d<= g-1 and the corresponding Brill-Noether
number is negative. First, we focus on the cases where A computes the Clifford
index; if r>1 and with only some completely classified exceptions, we show that
A coincides with the restriction to C of a line bundle on S. This is a
refinement of Green and Lazarsfeld's result on the constancy of the Clifford
index of curves moving in the same linear system. Then, we study a conjecture
of Donagi and Morrison predicting that, under no hypothesis on its Clifford
index, A is contained in a g^s_e which is cut out from a line bundle on S and
satisfies e<= g-1. We provide counterexamples to the last inequality already
for r=2. A slight modification of the conjecture, which holds for r=1,2, is
proved under some hypotheses on the pair (C,A) and its deformations. We show
that the result is optimal (in the sense that our hypotheses cannot be avoided)
by exhibiting, in the Appendix, some counterexamples obtained jointly with
Andreas Leopold Knutsen.Comment: 28 pages, final version, to appear in Adv. Math. with an Appendix
joint with Andreas Leopold Knutse
Stability of rank-3 Lazarsfeld-Mukai bundles on K3 surfaces
Given an ample line bundle L on a K3 surface S, we study the slope stability
with respect to L of rank-3 Lazarsfeld-Mukai bundles associated with complete,
base point free nets of type g^2_d on curves C in the linear system |L|. When d
is large enough and C is general, we obtain a dimensional statement for the
variety W^2_d(C). If the Brill-Noether number is negative, we prove that any
g^2_d on any smooth, irreducible curve in |L| is contained in a g^r_e which is
induced from a line bundle on S, thus answering a conjecture of Donagi and
Morrison. Applications towards transversality of Brill-Noether loci and higher
rank Brill-Noether theory are then discussed.Comment: 29 pages, final version, to appear in Proc. Lon. Math. So
A codimension 2 component of the Gieseker-Petri locus
We show that the Brill-Noether locus M^3_{18,16} is an irreducible component
of the Gieseker-Petri locus in genus 18 having codimension 2 in the moduli
space of curves. This result disproves a conjecture predicting that the
Gieseker-Petri locus is always divisorial.Comment: Final version, to appear in Journal of Algebraic Geometr
Green's Conjecture for curves on rational surfaces with an anticanonical pencil
Green's conjecture is proved for smooth curves C lying on a rational surface
S with an anticanonical pencil, under some mild hypotheses on the line bundle L
defined by C. Constancy of Clifford dimension, Clifford index and gonality of
curves in the linear system |L| is also obtained.Comment: Final version, to appear in Math. Zei
Severi Varieties and Brill-Noether theory of curves on abelian surfaces
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 with polarization of type , we prove
nonemptiness and regularity of the Severi variety parametrizing -nodal
curves in the linear system for (here is
the arithmetic genus of any curve in ). We also show that a general genus
curve having as nodal model a hyperplane section of some -polarized
abelian surface admits only finitely many such models up to translation;
moreover, any such model lies on finitely many -polarized abelian
surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is
proved concerning the possibility of deforming a genus curve in
equigenerically to a nodal curve. The rest of the paper deals with the
Brill-Noether theory of curves in . It turns out that a general curve in
is Brill-Noether general. However, as soon as the Brill-Noether number is
negative and some other inequalities are satisfied, the locus of
smooth curves in possessing a 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 having the expected
codimension in the moduli space of curves . For , the
results are generalized to nodal curves.Comment: 29 pages, 3 figures. Comments are welcome. 2nd version: added some
references in Rem. 7.1
Genus two curves on abelian surfaces
This paper deals with singularities of genus 2 curves on a general (d1, d2)- polarized abelian surface (S, L). In analogy with Chen’s results concerning rational curves on K3 surfaces [Ch1, Ch2], it is natural to ask whether all such curves are nodal. We prove that this holds true if and only if d2 is not divisible by 4. In the cases where d2 is a multiple of 4, we exhibit genus 2 curves in |L| that have a triple, 4-tuple or 6-tuple point. We show that these are the only possible types of unnodal singularities of a genus 2 curve in |L|. Furthermore, with no assumption on d1 and d2, we prove the existence of at least one nodal genus 2 curve in |L|. As a corollary, we obtain nonemptiness of all Severi varieties on general abelian surfaces and hence generalize [KLM, Thm. 1.1] to nonprimitive polarizations.acceptedVersio
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