The moduli space of curves of genus two covering elliptic curves

We consider the moduli spaceS _n of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an unramified cover ofE. By Torelli mapS _n is viewed as a subset of the moduli spaceA ₂ of principally polarized abelian surfaces. On the other hand the Humbert surfaceH _Δ of invariant Δ is defined as a subvariety ofA ₂(C), the set of C-valued points ofA ₂. The purpose of this paper is to releaseS _n withH _Δ.     

Explicit complete curves in the moduli space of curves of genus three

We explicitly describe complete, one-dimensional subvarieties of the moduli space of smooth complex curves of genus 3.Supported by the Netherlands Organization for Scientific Research (N.W.O.).     

Stability of tri-canonical curves of genus two

We completely classify tri-canonically embedded curves of genus two that are Chow semistable, and identify the moduli space of them with the compact moduli space of binary sextics. This moduli space is the log canonical model for the pair for 7/10 9/11 whose log canonical divisor pulls back to on the moduli stack     

Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points

Inventiones mathematicae -     

On certain curves of genus three in characteristic two

We study curves of genus 3 over algebraically closed fields of characteristic 2 with the canonical theta characteristic totally supported in one point. We compute the moduli dimension of such curves and focus on some of them which have two Weierstrass points with Weierstrass directions towards the support of the theta characteristic. We answer questions related to order sequence and Weierstrass weight of Weierstrass points and the existence of other Weierstrass points with similar properties. – Dedicated to the treasured memory of our coauthor, Paulo Henrique Viana     

Examples of torsion points on genus two curves

We describe a method that sometimes determines all the torsion points lying on a curve of genus two defined over a number field and embedded in its Jacobian using a Weierstrass point as base point. We then apply this to the examples , , and .

     

Veech surfaces and complete periodicity in genus two

We present several results pertaining to Veech surfaces and completely periodic translation surfaces in genus two. A translation surface is a pair where is a Riemann surface and is an Abelian differential on . Equivalently, a translation surface is a two-manifold which has transition functions which are translations and a finite number of conical singularities arising from the zeros of .

A direction on a translation surface is completely periodic if any trajectory in the direction is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction . Then, we say that a translation surface is completely periodic if any direction in which there is at least one cylinder of closed trajectories is completely periodic. There is an action of the group on the space of translation surfaces. A surface which has a lattice stabilizer under this action is said to be Veech. Veech proved that any Veech surface is completely periodic, but the converse is false.

In this paper, we use the -invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space of genus two translation surfaces with corresponding Abelian differentials which have a single double zero. Furthermore, we obtain a classification of all completely periodic surfaces in genus two.

     

The genus of space curves

LetC be a curve contained in ℙ _k ³ (k of any characteristic), which is locally Cohen-Macaulay, not contained in a plane and of degreed. We prove thatp _a (C)≤≤((d−2)(d−3))/2. Moreover we show existence of curves with anyd, p _a satisfying this inequality and we characterize those curves for which equality holds.

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Sunto Si dimostra che seC⊃ℙ _k ³ (k di caratteristica qualunque) é una curva, non piana, localmente Cohen-Macaulay, di gradod, allorap _a (C)≤((d−2)(d−3))/2. Si mostra che questa limitazione è ottimale, e si classificano le curve di genere aritmetico massimale.

     

Dyson's lemma for products of two curves of arbitrary genus

Supported by the Miller Institute for Basic Research in Science.     

Examples of genus two CM curves defined over the rationals

We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the well-known example we find 19 non-isomorphic such curves. We believe that these are the only such curves.

     

Singular pencils of quadrics and compactified Jacobians of curves

LetY be an irreducible nodal hyperelliptic curve of arithmetic genusg such that its nodes are also ramification points (char ≠2). To the curveY, we associate a family of quadratic forms which is dual to a singular pencil of quadrics in with Segre symbol [2...21...1], where the number of 2's is equal to the number of nodes. We show that the compactified Jacobian ofY is isomorphic to the spaceR of (g−1) dimensional linear subspaces of which are contained in the intersectionQ of quadrics of the pencil. We also prove that (under this isomorphism) the generalized Jacobian ofY is isomorphic to the open subset ofR consisting of the (g−1) dimensional subspaces not passing through any singular point ofQ.     

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 of the moduli space M₃ 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 M₃.