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On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in 3 . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, B and E , we give a necessary and sufficient condition for B to be the branch curve of a surface X in N and E to be the image of the double curve of a 3 -model of X . In the classical Segre theory, a plane curve...

On the Difference of 4-Gonal Linear Systems on some Curves

Ohbuchi, Akira (1997)

Serdica Mathematical Journal

Let C = (C, g^1/4 ) be a tetragonal curve. We consider the scrollar invariants e1 , e2 , e3 of g^1/4 . We prove that if W^1/4 (C) is a non-singular variety, then every g^1/4 ∈ W^1/4 (C) has the same scrollar invariants.

On the variety of linear series on a singular curve

E. Ballico, C. Fontanari (2002)

Bollettino dell'Unione Matematica Italiana

Let Y be an integral projective curve with g := p a Y 2 . For all positive integers d , r let W d r Y * A * be the set of all L Pic d Y with h 0 Y , L r + 1 and L spanned. Here we prove that if d g - 2 , then dim W d r Y * A * d - 3 r except in a few cases (essentially if Y is a double covering).

On the variety of quadrics of rank four containing a projective curve

Alexis G. Zamora (1999)

Bollettino dell'Unione Matematica Italiana

Sia X P H 0 X , L * una curva proeittiva e lissa, generali nel senso di Brill-Noether, indichiamo con R 4 X l'insieme algebrico di quadrici di rango 4 contenendo a X . In questo lavoro noi descriviamo birazionalmente i componenti irriducibile di R 4 X .

Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

Yuji Kodama, Shigeki Matsutani, Emma Previato (2013)

Annales de l’institut Fourier

A lattice model with exponential interaction, was proposed and integrated by M. Toda in the 1960s; it was then extensively studied as one of the completely integrable (differential-difference) equations by algebro-geometric methods, which produced both quasi-periodic solutions in terms of theta functions of hyperelliptic curves and periodic solutions defined on suitable Jacobians by the Lax-pair method. In this work, we revisit Toda’s original approach to give solutions of the Toda lattice in terms...

Singularities of 2 Θ -divisors in the jacobian

Christian Pauly, Emma Previato (2001)

Bulletin de la Société Mathématique de France

We consider the linear system | 2 Θ 0 | of second order theta functions over the Jacobian J C of a non-hyperelliptic curve C . A result by J.Fay says that a divisor D | 2 Θ 0 | contains the origin 𝒪 J C with multiplicity 4 if and only if D contains the surface C - C = { 𝒪 ( p - q ) p , q C } J C . In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing 𝒪 with multiplicity 6 , divisors containing the fourfold C 2 - C 2 = { 𝒪 ( p + q - r - s ) p , q , r , s C } , and divisors singular along C - C , using the third exterior...

Torelli theorem for stable curves

Lucia Caporaso, Filippo Viviani (2011)

Journal of the European Mathematical Society

We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of finite fibers. We also bound the dimension of infinite fibers.

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