Abel maps of Gorenstein curves

Abstract  For a Gorenstein curve X and a nonsingular point P∈ X, we construct Abel maps and , where J_Xⁱ is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A_P are shown to have the same arithmetic genus of X. Also, A and A_P are shown to be embeddings away from rational subcurves L⊂ X meeting in separating nodes. Finally we establish a connection with Seshadri’s moduli scheme U_X(1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U_X(1). Keywords Abel map, Torsion-free rank-1 sheaf, Compactified Jacobian, Gorenstein singularity Mathematics Subject Classification (2000) 14H40, 14H60     

Primitive holomorphic maps of curves

Let t: XY be a proper morphism of non-singular irreducible affine curves over . This paper shows that there is seldom a holomorphic function f such that (t,f): X Y × is a holomorphic embedding.     

Invariant curves for birational surface maps

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and number of irreducible components of the curve. In the case of an invariant curve with genus equal to one, we show that there is an associated invariant meromorphic two-form.

     

On residue maps for affine curves

We establish several compatibility results between residue maps in étale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and in fact have already been used, for the study of finiteness properties of the unramified cohomology of function fields of affine curves over number fields.     

Mond's conjecture for maps between curves

A theorem by D. Mond shows that if is finite and has has degree one onto its image (Y , 0), then the ‐codimension is less than or equal to the image Milnor number , with equality if and only if (Y , 0) is weighted homogeneous. Here we generalize this result to the case of a map germ , where (X , 0) is a plane curve singularity.     

Abel maps for curves of compact type

Recently, the first Abel map for a stable curve of genus g≥2 has been constructed. Fix an integer d≥1 and let C be a stable curve of compact type of genus g≥2. We construct two d-th Abel maps for C, having different targets, and we compare the fibers of the two maps. As an application, we get a characterization of hyperelliptic stable curves of compact type with two components via the second Abel map.     

Topological entropy of maps on regular curves

In [G.T. Seidler, The topological entropy of homeomorphisms on one-dimensional continua, Proc. Amer. Math. Soc. 108 (1990) 1025-1030], G.T. Seidler proved that the topological entropy of every homeomorphism on a regular curve is zero. Also, in [H. Kato, Topological entropy of monotone maps and confluent maps on regular curves, Topology Proc. 28 (2) (2004) 587-593] the topological entropy of confluent maps on regular curves was investigated. In particular, it was proved that the topological entropy of every monotone map on any regular curve is zero. In this paper, furthermore we investigate the topological entropy of more general maps on regular curves. We evaluate the topological entropy of maps f on regular curves X in terms of the growth of the number of components of f−n(y) (y∈X).     

Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps f:S ² → S ² that are post-critically finite. In addition, we assume that f is expanding in a suitable sense. It is shown that each sufficiently high iterate F = f ⁿ of f is semi-conjugate to z ^d :S ¹ → S ¹, where d = deg F. More precisely, for such an F we construct a Peano curve γ:S ¹ → S ² (onto), such that F°γ(z) = γ(z ^d ) (for all z∈S ¹).     

Clique-width of graphs defined by one-vertex extensions

Let G be a graph and u be a vertex of G. We consider the following operation: add a new vertex v such that v does not distinguish any two vertices which are not distinguished by u. We call this operation a one-vertex extension. Adding a true twin, a false twin or a pendant vertex are cases of one-vertex extensions. Here we are interested in graph classes defined by a subset of allowed one-vertex extension. Examples are trees, cographs and distance-hereditary graphs. We give a complete classification of theses classes with respect to their clique-width. We also introduce a new graph parameter called the modular-width, and we give a relation with the clique-width.     

Invariant curves for planar competitive and cooperative maps

In this paper we present results on the existence of invariant curves for planar maps that are monotone with respect to either the south-east or north-east ordering. Some of these curves are the stable or unstable manifolds of hyperbolic fixed points (saddle points) or non-hyperbolic fixed points, and are also the boundary of basins of attraction of such points.     

Plane Cremona maps, exceptional curves and roots

We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection and , respectively) on a smooth complex projective surface which admits a birational morphism to . The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of is in the -orbit of the class of the total transform of some point blown up by if is exceptional, or in the -orbit of a simple root if is root, where is the Weyl group acting on ; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor is a necessary and sufficient condition in order that could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.

     

Equivariant harmonic maps into the sphere via isoparametric maps

By using concrete isoparametric maps we obtain some new equivariant harmonic maps between spheres and solve equivariant boundary value problems for harmonic maps from unit open ballB ^m+1 intoS ⁿ. Research partially supported by NNSFC, SFECC and ICTP.     

Triquotient maps via ultrafilter convergence

In this paper we characterize triquotient maps as those that are surjective on chains of convergent ultrafilters, extending the result known for triquotient maps between finite topological spaces.

     

Gaussian maps of plane curves with nine singular points

We consider nonsingular curves which are the normalization of plane curves with nine ordinary singular points, viewing them as embedded in the blow-up X of the projective plane along their singular points. For a large class of such curves we show that the gaussian map relative to the canonical line bundle has corank one. The proof makes essential use of the geometry of X.     

Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps

Mathematische Annalen -     

On maps between modular Jacobians and Jacobians of Shimura curves

Fix a squarefree integer N, divisible by an even number of primes, and let Γ′ be a congruence subgroup of level M, where M is prime to N. For each D dividing N and divisible by an even number of primes, the Shimura curve X ^D (Γ₀(N/D) ∩ Γ′) associated to the indefinite quaternion algebra of discriminant D and Γ₀(N/D) ∩ Γ′-level structure is well defined, and we can consider its Jacobian J ^D (Γ₀(N/D) ∩ Γ′). Let J ^D denote the N/D-new subvariety of this Jacobian. By the Jacquet-Langlands correspondence [J-L] and Faltings’ isogeny theorem [Fa], there are Hecke-equivariant isogenies among the various varieties J ^D defined above. However, since the isomorphism of Jacquet-Langlands is noncanonical, this perspective gives no information about the isogenies so obtained beyond their existence. In this paper, we study maps between the varieties J ^D in terms of the maps they induce on the character groups of the tori corresponding to the mod p reductions of these varieties for p dividing N. Our characterization of such maps in these terms allows us to classify the possible kernels of maps from J ^D to J ^D′, for D dividing D′, up to support on a small finite set of maximal ideals of the Hecke algebra. This allows us to compute the Tate modules J ^D of J ^D at all non-Eisenstein of residue characteristic l > 3. These computations have implications for the multiplicities of irreducible Galois representations in the torsion of Jacobians of Shimura curves; one such consequence is a “multiplicity one” result for Jacobians of Shimura curves.