The object of this paper is to prove some general results about rational idempotents for a finite group $G$ and deduce from them geometric information about the components that appear in the decomposition of the Jacobian variety of a curve with $G-$action.
We give an algorithm to find explicit primitive rational idempotents for any $G$, as well as for rational projectors invariant under any given subgroup. These explicit constructions allow geometric descriptions of the factors appearing in the decomposition of a Jacobian with group action: from them we deduce the decomposition of any Prym or Jacobian variety of an intermediate cover, in the case of a Jacobian with $G-$action. In particular, we give a necessary and sufficient condition for a Prym variety of an intermediate cover to be such a factor.
We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of dimension 2.
Let $X$ be a complex smooth projective variety of dimension $d$. Under some assumption on the cohomology of $X$, we construct mutually orthogonal idempotents in $CH_d(X \times X) \otimes \Q$ whose action on algebraically trivial cycles coincides with the Abel-Jacobi map. Such a construction generalizes Murre's construction of the Albanese and Picard idempotents and makes it possible to give new examples of varieties admitting a self-dual Chow-Künneth decomposition satisfying the motivic Lefschetz conjecture as well as new examples of varieties having a Kimura finite dimensional Chow motive. Fo…
Jacobian conjectures (that nonsingular implies a global inverse) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The birational case is proved and the Galois case clarified. Two known special cases of the Strong Real Jacobian Conjecture (SRJC) are generalized to the rational map context. For an invertible map, the associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. That disqualifies the Pinchuk counter examples to the SRJC a…
We consider two cycles on the moduli space of compact type curves and prove that they coincide. The first is defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized rational curve, while the second is obtained as the intersection of the Abel section of the universal Jacobian with the zero section. Our comparison extends results of Cavalieri-Marcus-Wise where the same identity was proved over on the locus of rational tails curves.
We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free g^r_d of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of P^1 whose ramification points are all of order d, and then for hyperelliptic curves.
Let K be an algebraically closed field of characteristic zero and let f(x,y) be a nonzero polynomial of K[x,y]. We prove that if the generic element of the family $(f-λ)\_λ$ is a rational polynomial, and if the Jacobian J(f,g) is a nonzero constant for some polynomial g in K[x,y], then K[f,g] =K[x,y].
It is shown that a polynomial map $(P,Q)\in \mathbb{Q}[x,y]^2$ with $P_xQ_y-P_yQ_x \equiv 1$ has an inverse map in $\mathbb{Q}[x,y]^2$ if the fiber $P=0$ contains an infinite subset of $ d^{-1}\mathbb{Z}^2$ for an integer $d$.
We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.
Jacobian conjectures (that nonsingular implies invertible) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. The extensions for the Pinchuk counterexamples to the strong real Jacobian conjecture have no nontrivial automorphisms, but are of degree six. The birational case is proved, the Galois case is clarified but the general case of odd degree remains open. However, cert…
We produce new explicit examples of genus-2 curves over the rational numbers whose Jacobian varieties have rational torsion points of large order. In particular, we produce a family of genus-2 curves over Q whose Jacobians have a rational point of order 48, parametrized by a rank-2 elliptic curve over Q, and we exhibit a single genus-2 curve over Q whose Jacobian has a rational point of order 70, the largest order known. We also give new examples of genus-2 Jacobians with rational points of order 27, 28, and 39. Most of our examples are produced by `gluing' two elliptic curves together along t…
Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call Jacobian discrepancy, is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper is a formula measuring the gap between the dualizing sheaf and the Grauert-Rieme…
Any counterexample to the two-dimensional Jacobian Conjecture gives a rational map from one projective plane to another. We use some ideas of the Minimal Model Program to study the combinatorial structure of a rational surface, that is obtained by resolving this map. Several structural results are proven, revealing a rather orderly structure of the graph of the curves at infinity. We also exhibit and discuss a graph that may lead to a counterexample to the Jacobian Conjecture.
We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve $X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it. When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational torsion points $\widetilde{J}_0(N)(\mathbb{Q})_{\mathrm{Tor}}$ tends to be much smaller than the classical Jacobian.
We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\geq 3$ with a rational Weierstrass point have fewer than 20 rational points.
We show that the Euler number of the compactified Jacobian of a rational curve $C$ with locally planar singularities is equal to the multiplicity of the $δ$-constant stratum in the base of a semi-universal deformation of $C$. In particular, the multiplicity assigned by Yau, Zaslow and Beauville to a rational curve on a K3 surface $S$ coincides with the multiplicity of the normalisation map in the moduli space of stable maps to $S$.
A rational map $ϕ: \mathbb{P}_k^m \dashrightarrow \mathbb{P}_k^n$ is defined by homogeneous polynomials of a common degree $d$. We establish a linear bound in terms of $d$ for the number of $(m-1)$-dimensional fibers of $ϕ$, by using ideals of minors of the Jacobian matrix. In particular, we answer affirmatively Question~11 in arXiv:1511.02933v2.
We give effective bounds for the class number of any algebraic function field of genus $g$ defined over a finite field. These bounds depend on the possibly partial information on the number of places on each degree $\leq g$. Such bounds are especially useful for estimating the class number of function fields in towers of function fields over finite fields. We give examples in the case of asymptotically good towers. In particular we estimate the class number of function fields which are steps of towers having one or several positive Tsfasman-Vladut invariants. Note that the study is not done as…
We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained result shows that such polynomial maps $F$ is invertible if $(0,0)$ is a regular value of $F$ or if the Jacobian condition holds.
A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ in terms of $K$ and its genus $g$. A recent paper by Stoll proved uniform bounds for the number of $K$-rational points on a hyperelliptic curve $X$ provided that the rank of the Jacobian of $X$ is at most $g - 3$. Katz, Rabinoff and Zureick-Brown generalized his result to arbitrary curves satisfying the same rank condition. In this paper, we prove conditional uniform bounds on the number of rational points on the symmetric square of $X$ outside its algebraic special set, pr…
We prove by explicit example that the compactified jacobian can be nonreduced. The example is a rational space curve of arithmetic genus 4. This answers a question posed by Cyril D'Souza in 1979.
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao's conjecture on the freeness of a line arrangement being determined by its combinatorics.
In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.
This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptiona…
In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in the proof are essentially those from the explicit asymptotic theory of global fields. We thus provide a concrete application of effective results from the asymptotic theory of global fields and their zeta-functions.
Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves. In particular, we show that the Weil- and the Tate-pairing are non-degenerate over the same field extension of the ground field. From this generalization we get a complete description of the l-torsion subgroups of Jacobians of su…
Let $J$ be the Jacobian of a smooth projective curve. We define a natural action of the Lie algebra of polynomial Hamiltonian vector fields on the plane, vanishing at the origin, on the Chow group of $J$ with rational coefficients. Using this action we obtain some relations between tautological cycles on $J$.
We present relations between cycles with rational coefficients modulo algebraic equivalence on the Jacobian of a curve. These relations depend on the linear systems the curve admits. They are obtained in the tautological ring, the smallest subspace containing (an embedding of) the curve and closed under the basic operations of intersection, Pontryagin product and the pullback and pushdown induced by homotheties.
It is well-known that abelian varieties are projective, and so that there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational functions describing these varieties or their group laws in dimensions greater than two. One exception can be found in Mumford's classic "Lectures on Theta", where he describes how to obtain an explicit model for hyperelliptic Jacobians as the union of several affine pieces described as the vanishing locus of explicit polynomial equations. In this article, …
We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invarian…
Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let Λbe its value group. Given a smooth, proper, connected K-curve X and a skeleton Γof the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(Γ) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classic…
We introduce an algorithm to compute the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points ov…
Let $G$ denote a finite group and $π: Z \to Y$ a Galois covering of smooth projective curves with Galois group $G$. For every subgroup $H$ of $G$ there is a canonical action of the corresponding Hecke algebra $\mathbb{Q}[H \backslash G/H]$ on the Jacobian of the curve $X = Z/H$. To each rational irreducible representation $\mathcal{W}$ of $G$ we associate an idempotent in the Hecke algebra, which induces a correspondence of the curve $X$ and thus an abelian subvariety $P$ of the Jacobian $JX$. We give sufficient conditions on $\mathcal{W}$, $H$, and the action of $G$ on $Z$, which imply $P$ to…
In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational Weierstrass point. They showed that the average size of the 2-Selmer group of the Jacobians of these curves, when ordered by height, is 3. In this paper, we consider the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational non-Weierstrass point. We show that when these curves are ordered by height, the average size of the 2-Selmer group of their Jacobians is 6. This yields an upper bound of 5/2 on the average rank of the Mo…
We discuss the structure of the endomorphism algebras of the jacobians of superelliptic curves y^q=f(x) where q is a prime power and f(x) is an irreducible cubic or quartic polynomial over the field of rational functions k(t) in characteristic zero, whose Galois group is the full symmetric group.
Consider a finite group $G$ acting on a Riemann surface $S$, and the associated branched Galois cover $π_G:S \to Y=S/G$. We introduce the concept of geometric signature for the action of $G$, and we show that it captures the information of the geometric structure of the lattice of intermediate covers, the information about the isotypical decomposition of the rational representation of the group $G$ acting on the Jacobian variety $JS$ of $S$, and the dimension of the subvarieties of the isogeny decomposition of $JS$. We also give a version of Riemann's existence theorem, adjusted to the present…
The geometric motivic Poincaré series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series has a rational form. We describe it in the case of an affine toric variety of arbitrary dimension. The result, which provides an explicit set of candidate poles, is expressed in terms of the sequence of Newton polyhedra of certain monomial ideals, which we call logarithmic jacobian ideals, associated to the modules of differential forms with logarithmic poles…
The main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the Jacobian of a Shimura curve over the field of rational numbers at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem of Cerednik and Drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant. As a corollary, we offer a proof of a conjecture formulated by M. Greenberg in his paper on Stark-Heegner points and quaternionic Shimura curves, thus making Greenberg's construction of local points on …
We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities r…
Let $\ell$ be a rational prime and $k$ a number field. Given a superelliptic curve $C/k$ of $\ell$-power degree, we describe the field generated by the $\ell$-power torsion of the Jacobian variety in terms of the branch set and reduction type of $C$ (and hence, in terms of data determined by a suitable affine model of $C$). If the Jacobian is good away from $\ell$ and the branch set is defined over a pro-$\ell$ extension of $k(μ_{\ell^\infty})$ unramified away from $\ell$, then the $\ell$-power torsion of the Jacobian is rational over the maximal such extension. By decomposing the covering int…
Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell--Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the second- and third-named authors. In addition we obtain uniform bounds for the number of torsion in the Jacobian that lie each fiber o…
We continue the study of the rational-slope generalized $q,t$-Catalan numbers $c_{m,n}(q,t)$. We describe generalizations of the bijective constructions of J. Haglund and N. Loehr and use them to prove a weak symmetry property $c_{m,n}(q,1)=c_{m,n}(1,q)$ for $m=kn\pm 1$. We give a bijective proof of the full symmetry $c_{m,n}(q,t)=c_{m,n}(t,q)$ for $\min(m,n)\le 3$. As a corollary of these combinatorial constructions, we give a simple formula for the Poincaré polynomials of compactified Jacobians of plane curve singularities $x^{kn\pm 1}=y^n$. We also give a geometric interpretation of a relat…
A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension. We show that $G$-equivariant counterexamples for infinite group $G$ exist if and only if $G=\mathbb{C}^*$ and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant étale endomorphisms. We prove also that for every integers $r\geq …
The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent theory of (Jacobians of) noncommutative Chow motives, we give an alternative proof of this conjecture in the case of a complete intersection of either two quadrics or three odd-dimensional quadrics. Moreover, without the use of the powerful Lefschetz theorem, we prove that in these cases the unique non-trivial algebraic Jacobian is the middle one. As an applic…
