Systems of polynomial equations defining hyperelliptic d-osculating covers (original) (raw)

On double coverings of hyperelliptic curves

Journal of Pure and Applied Algebra, 2006

We prove various properties of varieties of special linear systems on double coverings of hyperelliptic curves. We show and determine the irreducibility, generically reducedness and singular loci of the variety W r d for bi-elliptic curves and double coverings of genus two curves. Similar results for double coverings of hyperelliptic curves of genus h ≥ 3 are also presented.

Curves on Rational Surfaces with Hyperelliptic Hyperplane Sections

In this article we study, given a pair of integers (d, g), the problem of existence of a smooth, irreducible, non-degenerate curve in P n of degree d and genus g (the Halphen-Castelnuovo problem). We define two domains from the (d, g)-plane, D n 1 and D n 2 , and we prove that there is no gap in D n 1 . This follows constructing curves on some rational surfaces with hyperelliptic hyperplane sections and from some previous theorems of Ciliberto, Sernesi and of the author. Moreover, in the last section, based on some results of Horrowitz, Ciliberto, Harris, Eisenbud, we conjecture that D n 2 is the right lacunary domain.

On unramified normal coverings of hyperelliptic curves

Journal of Pure and Applied Algebra, 2007

It is well known that the number of unramified normal coverings of an irreducible complex algebraic curve C with a group of covering transformations isomorphic to Z 2 ⊕ Z 2 is (2 4g − 3 • 2 2g + 2)/6. Assume that C is hyperelliptic, say C : y 2 = 2g+2 d=1 (x − µ d). Horiouchi has given the explicit algebraic equations of the subset of those covers which turn out to be hyperelliptic themselves. There are 2g+2 3 of this particular type. In this article, we provide algebraic equations for the remaining ones.

Hyperelliptic curves among cyclic coverings of the projective line, I

Archiv der Mathematik, 2013

In this note, we prove a necessary and sufficient condition for whether a d-cyclic covering of the complex projective line has gonality 2 (i.e., is elliptic or hyperelliptic), where d is a positive integer. The case of 3 branch points has been solved in our previous paper (see [3]).

From Separable Polynomials to Nonexistence of Rational Points on Certain Hyperelliptic Curves

Journal of the Australian Mathematical Society, 2014

We give a separability criterion for the polynomials of the form \begin{equation*} ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e). \end{equation*}Usingthisseparabilitycriterion,weproveasufficientconditionusingtheBrauer–ManinobstructionunderwhichcurvesoftheformUsing this separability criterion, we prove a sufficient condition using the Brauer–Manin obstruction under which curves of the formUsingthisseparabilitycriterion,weproveasufficientconditionusingtheBrauer–Maninobstructionunderwhichcurvesoftheform\begin{equation*} z^2 = ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e) \end{equation*}$$ have no rational points. As an illustration, using the sufficient condition, we study the arithmetic of hyperelliptic curves of the above form and show that there are infinitely many curves of the above form that are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction.

Rational Points on Certain Hyperelliptic Curves over Finite Fields

Bulletin of the Polish Academy of Sciences Mathematics, 2007

Let K be a field, a, b ∈ K and ab = 0. Let us consider the polynomials g1(x) = x n + ax + b, g2(x) = x n + ax 2 + bx, where n is a fixed positive integer. In this paper we show that for each k ≥ 2 the hypersurface given by the equation

Hyperelliptic Curves With Given A-number

Arxiv preprint math.NT/0401008, 2004

In this paper, we show that there exist families of curves (defined over an algebraically closed field k of characteristic p) whose Jacobians have interesting ptorsion. For example, we produce families of curves of large dimension so that the p-torsion of the Jacobian of each fibre contains multiple copies of the group scheme α p. The method is to study curves which admit a (Z/2) n-cover of the projective line. As a result, some of these families intersect the hyperelliptic locus H g. For small values of the a-number these families of curves intersect the hyperelliptic locus H g , resulting in the following corollaries. Corollary 1.2. Suppose g ≥ 2 and p ≥ 5. There exists a (g − 2)-dimensional family of smooth hyperelliptic curves of genus g all of whose fibres have a-number at least 2. Corollary 1.3. Suppose g ≥ 3 is odd and p ≥ 7. There exists a (g − 5)/2-dimensional family of smooth hyperelliptic curves of genus g whose fibres have a-number at least 3. Consider the group scheme k[F,V ]/(F 3 ,V 3 , F 2 +V 2). This group scheme corresponds to the p-torsion of an abelian variety of dimension 2 which has a-number 1 and prank 0. Corollary 1.4. For all g ≥ 4, there exists a (g − 2)-dimensional family of smooth curves X in H g,2 so that Jac(X)[p] ≃ k[F,V ]/(F 3 ,V 3 , F 2 +V 2) ⊕ (µ p ⊕ Z/p) g−2. In fact, we show that when k ≥ 2 and 2 k ≤ g < 2 k + 2 k−1 then there exists a hyperelliptic curve X with Jac(X)[p] ≃ k[F,V ]/(F 3 ,V 3 , F 2 +V 2) ⊕ (µ p ⊕ Z/p) g−2. Our method is to analyze the curves in the locus H g,n in terms of fibre products of hyperelliptic curves. We extend results of Kani and Rosen [6] to compare the p-torsion of the Jacobian of a curve X in H g,n to the p-torsion of the Jacobians of its Z/2Z-quotients up to isomorphism. In particular, we use Yui's description of the p-torsion of the Jacobian of a hyperelliptic curve in terms of the branch locus [15]. In some cases, this reduces the study of the p-torsion of the Jacobian of X to the study of the intersection of some subvarieties in the configuration space of branch points. The results above on families of curves having prescribed group schemes in the ptorsion of their Jacobians are all found in Section 4. In Section 2, we describe properties of the locus H g,n which are used in Section 4, including the proof that a curve X in this locus is the fibre product of n hyperelliptic curves and the comparison of the Jacobian of X with the Jacobians of its quotients. As an unrelated geometric result, we show that H g,2 is connected if and only if g ≤ 3 in Theorem 2.13. In Section 3, we study the the branch loci corresponding to non-ordinary hyperelliptic curves and prove some intersection results for this subvariety of the configuration space which are again used in Section 4. The limitation of the approach is to understand the p-torsion of Jacobians of hyperelliptic curves, especially in terms of their branch loci. Any progress in this direction can be used to extend these results. For example, we use recent work of [4] to prove Corollary 4.13 on the dimension of the intersection of H g,2 with the locus of curves of genus g having prank exactly f. We would like to thank E. Goren for suggesting the topic of this paper and E. Kani for help with Proposition 2.5. 2 Fibre products of hyperelliptic curves Let k be an algebraically closed field of characteristic p = 2. Let G be an elementary abelian 2-group of order 2 n. In this section, we describe G-Galois covers f : X → P 1 k

Double covers of smooth hyperquadrics as ample and very ample divisors

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1994

In this paper we deal with the following problem. Let Qn c pn+l be an ndimensional smooth hyperquadric of the complex projective (n + 1)-space and let n:A --* Q" be a double cover branched along a smooth hypersurface of Q". Classify all smooth projective (n + l)-folds X c IP u admitting A among their hyperplane sections. This is the same as (*) classifying pairs (X, L) where X is a smooth complex projective (n + 1)fold and L E Pic(X) is a very ample line bundle, whose complete linear system ILl contains a smooth element A as above.

Pencils of quadrics and the arithmetic of hyperelliptic curves

2013

A hyperelliptic curve over mathbbQ\mathbb QmathbbQ is called "locally soluble" if it has a point over every completion of mathbbQ\mathbb QmathbbQ. In this paper, we prove that a positive proportion of hyperelliptic curves over mathbbQ\mathbb QmathbbQ of genus ggeq1g\geq 1ggeq1 are locally soluble but have no points over any odd degree extension of mathbbQ\mathbb QmathbbQ. We also obtain a number of related results. For example, we prove that for any fixed odd integer k>0k > 0k>0, the proportion of locally soluble hyperelliptic curves over mathbbQ\mathbb QmathbbQ of genus ggg having no points over any odd degree extension of mathbbQ\mathbb QmathbbQ of degree at most kkk tends to 1 as ggg tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the Brauer-Manin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to 2, together with suitable arguments from the geometry of numbers.