Non-hyperelliptic Curves Of Genus Research Articles

Slope equality of non-hyperelliptic Eisenbud–Harris special fibrations of genus 4

AbstractThe Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus 4 with unique trigonal structure.

Maximality of Ciani curves over finite fields

In this paper, we will study Ciani curves in characteristic p≥3, in particular their standard forms C:x4+y4+z4+rx2y2+sy2z2+tz2x2=0. It is well-known that any Ciani curve is a non-hyperelliptic curve of genus 3, and its Jacobian variety is isogenous to the product of three elliptic curves. As a main result, we will show that if C is superspecial, then r,s,t belong to Fp2 and C is maximal or minimal over Fp2. Moreover, in this case we will provide a simple criterion in terms of r,s,t,p that tells whether C is maximal (resp. minimal) over Fp2.

Thomae-Weber Formula: Algebraic Computations of Theta Constants

Abstract We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated with a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic curves of genus 4, in particular, such curves lying on a singular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, we obtain a complete level 2 structure of the curves by studying their theta characteristic divisors via exceptional divisors of the del Pezzo surfaces as the structure is required for the method.

ORBIFOLD ASPECTS OF CERTAIN OCCULT PERIOD MAPS

We first characterize the automorphism groups of Hodge structures of cubic threefolds and cubic fourfolds. Then we determine for some complex projective manifolds of small dimension (cubic surfaces, cubic threefolds, and nonhyperelliptic curves of genus 3 or 4), the action of their automorphism groups on Hodge structures of associated cyclic covers, and thus confirm conjectures made by Kudla and Rapoport in (Pacific J. Math.260(2) (2012), 565–581).

Algebraic rank on hyperelliptic graphs and graphs of genus 3

Let G¯=(G,ω) be a vertex-weighted graph, and let δ be a divisor class on G. Let rG¯(δ) denote the (combinatorial) rank of δ. Caporaso has introduced the algebraic rank rG¯alg(δ) of δ by using nodal curves with dual graph G¯. In this paper, when G¯ is hyperelliptic or of genus 3, we show that rG¯alg(δ)≥rG¯(δ) holds, generalizing our previous result. We also show that, with respect to the specialization map from a nonhyperelliptic curve of genus 3 to its reduction graph, any divisor on the graph lifts to a divisor on the curve of the same rank.

UNIVERSAL WEIERTSRASS n-SEMIGROUPS

We study Weierstrass n-semigroups, which are realized on every non-hyperelliptic curve of genus g and called them universal n-semigroups. If g = 3 we classify all the ones with n = 1,2,3. When n = 1 there is only the non-special one, but for n = 2,3 there are several special ones.

On the moduli space of smooth plane quartic curves with a sextactic point

Let Mg be the moduli space of smooth algebraic curves of genus g over C: In this paper, we prove that the set Sr M 3 of moduli points of smooth plane quartic curves (nonhyperelliptic curves of genus 3) having at least one sextactic point of sextact multiplicity r; where r 2 f1;2;3g; is an irreducible, closed and rational subvariety of codimensional r 1 of M3 H 3 (where H3 M 3 is the hyperelliptic locus ).

Moduli of stable sheaves on a smooth quadric and a Brill–Noether locus

We prove that the moduli space of stable sheaves of rank 2 with the Chern classes c 1 = O Q ( 1 , 1 ) and c 2 = 2 on a smooth quadric Q in P 3 is isomorphic to P 3 . Using this identification, we give a new proof that a Brill–Noether locus, defined as the closure of the stable bundles with at least three linearly independent sections, on a non-hyperelliptic curve of genus 4, is isomorphic to the Donagi–Izadi cubic threefold.

Focal varieties of curves of genus 6 and 8

In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill-Noether varieties. Among other results, it is shown that the focal variety of a general, canonical and nonhyperelliptic curve of genus 6 is a hypersurface.

A note on a Brill-Noether locus over a non-hyperelliptic curve of genus 4

AbstractWe prove that a certain Brill-Noether locus over a non-hyperelliptic curve C of genus 4, is isomorphic to the Donagi-Izadi cubic threefold in the case when the pencils of the two trigonal line bundles of C coincide.

Non-hyperelliptic curves of genus three over finite fields of characteristic two

Let k be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non-singular quartic plane curves defined over k. We find explicit rational models and closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non-hyperelliptic locus M 3 nh of the moduli space M 3 of curves of genus 3. By adding to these computations the results of Oort [Moduli of abelian varieties and Newton polygons, C.R. Acad. Sci. Paris 312 (1991) 385–389] and Nart and Sadornil [Hyperelliptic curves of genus three over finite fields of characteristic two, Finite Fields Appl. 10 (2004) 198–200] on the hyperelliptic locus we obtain a complete picture of these strata for M 3 .

Some Properties of Second Order Theta Functions o Prym Varieties

Let P ∪ P′ be the two component Prym variety associated to an etale double cover C C of a non-hyperelliptic curve of genus g ≥6 and let |2Ξ0| and |2Ξ′0| be the linear systems of second order theta divisors on P and P′ respectively. The component P′ contains canonically the Prym curve C. We show that the base locus of the subseries of divisors containing C ⊂ P′ is exactlythe curve C. We also prove canonical isomorphisms between some subseries of |2Ξ0| and |2Ξ′0| and some subseries of second order theta divisors on the Jacobian of C.