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Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A normal variety which contains an algebraic torus as an open dense subset, such that the natural action of the torus on itself extends to the whole variety, is called a toric variety. Toric varieties form an important class of algebraic varieties, whose geometry can be totally determined by combinatorial data. For instance, let \(\mathbb{T}=\mathbb{G}_m^n\) be a split torus over a field \(K\). Let \(N=\text{Hom}(\mathbb{G}_m,\mathbb{T})\simeq \mathbb{Z}^n\) be the lattice of one-parameter subgroups of \(\mathbb{T}\) and \(M=N^\vee\) the dual lattice of characters of \(\mathbb{T}\). Set \(N_\mathbb{R}=N\otimes_{\mathbb{Z}}\mathbb{R}\) and \(M_\mathbb{R}=M\otimes_{\mathbb{Z}}\mathbb{R}\). Then the fans \(\Sigma\) on \(N_\mathbb{R}\) correspond to the toric varieties \(X_\Sigma\) of dimension \(n\) over \(K\). \(X_\Sigma\) is proper if \(\Sigma\) is complete. A virtual support function on \(\Sigma\) is a continuous function \(\Psi: N_\mathbb{R}\to \mathbb{R}\) whose restriction to each cone of \(\Sigma\) is an element of \(M\). Such a function determines a \(\mathbb{T}\)-Cartier divisor \(D_\Psi\) and a toric line bundle \(L_\Psi=\mathcal{O}(D_\Psi)\) with a canonical toric section \(s_\Psi\) such that \(\text{div}(s_\Psi)=D_\Psi\). \(L_\Psi\) is generated by global sections if and only if \(\Psi\) is a concave function. In this case, the lattice polytope
\[
\Delta_\Psi:=\{x\in M_\mathbb{R}: \langle x,u\rangle \geq \Psi(u)\text{ for all }a\in N_\mathbb{R}\}\subset M_\mathbb{R}
\]
conveys all the information about the pair \((X_\Sigma,L_\Psi)\), and one has the degree formula
\[
\text{deg}_{L_\Psi}(X_\Sigma)=n!\text{vol}_M(\Delta_\Psi)
\]
where \(\text{vol}_M\) is the Haar measure on \(M_\mathbb{R}\) normalized so that \(M\) has covolume \(1\).
The content of the monograph under review is to develop an arithmetic analogue of this degree formula for heights of toric varieties. The authors have shown that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. And more generally, they established a close relation between the arithmetic geometry of toric varieties and the convex analysis. This generalizes the relation between the algebraic geometry of toric varieties and the convex geometry that we described at the beginning of this review.
To do this, the authors study the Arakelov geometry of toric varieties, including models of toric varieties over a discrete valuation ring, metrized line bundles, and their associated measures and (local/global) heights. Roughly speaking, for the global case (say \(K=\mathbb{Q}\)), given a family of concave functions \((\psi_v)_{v\in \mathcal{M}_\mathbb{Q}}\) such that \(\mid \psi_v-\Psi\mid\) is bounded for all places \(v\) and such that \(\psi_v=\Psi\) for all but a finite number of \(v\), one may endow the toric line bundle \(L:=L_\Psi\) a family of semipositive toric metrics \(\parallel\cdot\parallel_{\psi_v}\) on analytifications \((L^v)^{an}\) for all places \(v\). By definition, \(\overline{L}=(L,(\parallel\cdot\parallel_{\psi_v})_v)\) becomes a semipositive quasi-algebraic metrized toric line bundle. Moreover, every semipositive quasi-algebraic toric metric on \(L\) arises in this way. To all places \(v\), the associated roof functions \(\vartheta_{\overline{L}^v,s}: \Delta_\Psi\to \mathbb{R}\) are zero except for a finite number of places. Then, the global height of \(X\) with respect to \(\overline{L}\) can be computed as
\[
h_{\overline{L}}(X)=\sum_{v\in \mathcal{M}_\mathbb{Q}}h_{\overline{L}^v}^{\text{tor}}(X_v)=(n+1)!\sum_{v\in \mathcal{M}_\mathbb{Q}}\int_{\Delta_\Psi}\vartheta_{\overline{L}^v,s}d\text{vol}_M.
\]
Apart from this formula, the authors also presented a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This is helpful in computing the height of toric varieties with respect to some interesting metrics arising from polytopes. As applications, they computed the height of toric projective curves with respect to the Fubini-Study metric, and of some toric bundles. Toric variety; Berkovich space; integral model; metrized line bundle; height of a variety; concave function; Legendre-Fenchel dual; real Monge-Ampere measure José Ignacio Burgos Gil, Patrice Philippon & Martín Sombra, Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque 360, Société Mathématique de France, 2014 Toric varieties, Newton polyhedra, Okounkov bodies, Arithmetic varieties and schemes; Arakelov theory; heights, Convex functions and convex programs in convex geometry Arithmetic geometry of toric varieties. Metrics, measures and heights | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(L\) be an ample line bundle of polarization type \((1, d_ 2, \dots, d_ n)\) on the abelian \(n\)-fold \(X\). It is known that if \(n = 2\) then \(L\) is very ample iff \(d = d_ 2 \geq 5\), and \(L\) is globally generated (free) iff \(d \geq 3\). If \(n \geq 3\) not much is known.
In this paper the authors treat the case \(n = 3\). The general Ein-Lazarsfeld criterion for freeness of an ample line bundle \(L\) on a 3-fold \(X\) assumes the non-existence of some special curves on \(X\). The question is to replace this criterion by an effective one in the case of abelian 3-folds. Example 4.2 shows that no numerical criterion guarantees the freeness of \(L\): see proposition 4.3. However, the pair \((X,L)\) defines at least one principally polarized abelian 3-fold \(Y\) isogenous to \(X\). In this paper, the authors find such sufficient conditions for freeness of \(L\) only in terms of the fixed isogeny \(X \to Y\), and depending of the decomposition type of \(Y\) (being a product of 1, 2, or 3 irreducible abelian varieties): see theorem 1. -- Next, the authors study sufficient conditions ensuring very ampleness of \(L\), by using the method of Comessatti for Jacobians with two principal polarizations, and Sakai's version of Reider's theorem yielding very ampleness for a large class of line bundles \(L\): see theorem 2. In particular, if the polarization is of type \((1,1,d)\), \(d \geq 13\), \(d \neq14\), then \(L\) is very ample. abelian variety; freeness of an ample line bundle on a 3-fold; ample line bundle; isogeny; very ampleness Birkenhake, Ch., Lange, H., Ramanan, S.: Primitive line bundles on abelian threefolds. Manusc. Math.81, 299--310 (1993) \(3\)-folds, Abelian varieties and schemes, Divisors, linear systems, invertible sheaves, \(n\)-folds (\(n>4\)), Vector bundles on surfaces and higher-dimensional varieties, and their moduli Primitive line bundles on abelian threefolds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The aim of this paper is to give a variant of the Beauville-Narasimhan-Ramanan correspondence for irregular parabolic Higgs bundles on smooth projective curves with fixed semi-simple irregular part and to show that it defines a Poisson isomorphism between certain irregular Dolbeault moduli spaces and relative Picard bundles of families of ruled surfaces over the curve. The paper is organized as follows: the first section is an introduction to the subject. In Section 2 the author sets some notation concerning irregular Higgs bundles and parabolic sheaves. In Section 3 he gives the correspondence in the case of Higgs bundles with regular singularities. Then he turns to the irregular case in Section 4 and extends the correspondence to this case. Finally, in Section 5 the author spells out the natural Poisson structures on the relative Dolbeault moduli space and the relative Picard bundle and shows that the correspondence is a Poisson isomorphism between dense open subsets smooth projective curve; irregular Higgs bundle; spectral sheaf; ruled surface; relative Picard bundle; holomorphic Poisson structure Szabó, Sz., The birational geometry of unramified irregular Higgs bundles on curves, Internat. J. Math., 28, 6, (2017) Relationships between algebraic curves and integrable systems, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Rational and birational maps The birational geometry of unramified irregular Higgs bundles on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Fix a point \(x_0\) of a compact connected Riemann surface \(X\). Let \(n\) and \(d\) be two mutually coprime integers with \(n\geq 1\). Let \(E\) be a holomorphic vector bundle of rank \(n\) and degree \(d\) over \(X\).
The authors prove that the following three conditions on \(E\) are equivalent: (1) There is a logarithmic connection \(\mathcal D\) on \(E\) singular exactly over \(x_0\) such that the residue of \(\mathcal D\) at \(x_0\), which is an endomorphism of the fiber \(E_{x_0}\), is of the form \(\lambda-\text{Id}_{E_{x_0}}\), where \(\lambda\in\mathbb C\). (2) The vector bundle \(E\) is indecomposable, i.e., if \(E\) is holomorphically isomorphic to \(E_1\oplus E_2\), then either \(E_1 = 0\) or \(E_2 = 0\). (3) The projective bundle \(\mathbb P(E)\) over \(X\) admits a holomorphic connection.
Furthermore, any logarithmic connection on \(E\), singular exactly over \(x_0\), with residue of the form \(\lambda\cdot\) Id\(_{E_{x_0}}\) is actually an irreducible logarithmic connection. The given condition that the degree and rank of \(E\) are \(d\) and \(n\), respectively, implies that the above constant \(\lambda\) is \(-d/n\).
Assume that genus\((X) \geq 2\). Let \(\mathcal M(n,d)\) denote the moduli space of all stable vector bundles over \(X\) of rank \(n\) and degree \(d\). Let \(\mathcal M'_D(n)\) denote the moduli space of all pairs of the form \((E,\mathcal D)\), where \(E\in\mathcal M(n,d)\) and \(\mathcal D\) is a logarithmic connection on the vector bundle \(E\), singular exactly over \(x_0\), such that the residue of \(\mathcal D\) at \(x_0\) is \(-\frac{d}{n}\operatorname{Id}_{E_{x_0}}\). Consider the projection of \(\mathcal M'_D(n)\) to \(\mathcal M(n,d)\) defined by sending any pair \((E,\mathcal D)\) to \(E\). It is proved that the pullback of line bundles defined over \(\mathcal M(n,d)\) to \(\mathcal M'_D(n)\) by this map gives an isomorphism of Picard groups. Furthermore, the inclusion of \(\mathcal M'_D (n)\) in the moduli space of all logarithmic connections of rank \(n\) over \(X\), singular exactly over \(x_0\) with residue \(-\frac{d}{n}\operatorname{Id}\) at \(x_0\), gives an isomorphism of Picard groups.
Fix a holomorphic line bundle \(L\) over \(X\) of degree \(d\), and fix a logarithmic connection \(D_L\) over \(L\) which is singular exactly over \(x_0\). Let \(\mathcal N(n, L)\subset\mathcal M(n,d)\) denote the moduli space of all stable vector bundles \(E\) with \(\bigwedge^n E \cong L\). Similarly, define \(\mathcal N'_p(L)\subset\mathcal M'_D(n)\) to be the moduli space of all pairs \((E,\mathcal D)\) with \(E\in\mathcal N(n, L)\) and \(\mathcal D\) a logarithmic connection on \(E\) singular over \(x_0\) such that the logarithmic connection on \(\bigwedge^n E \cong L\) induced by \(\mathcal D\) coincides with the given logarithmic connection \(D_L\) on~ \(L\).
Let \(p_0:\mathcal N'_d(L)\rightarrow\mathcal N_d(L)\) be the projection defined by \((E,\mathcal D)\mapsto E\). The corresponding homomorphism \(\xi\mapsto p_0^\ast\xi\), where \(\xi\in\text{Pic}(\mathcal N(n, L)) =\mathbb Z\), identifies the Picard group of \(\mathcal M'_D(L)\) with \(\mathbb Z\).
Let \(\Theta\) be the ample generator of Pic\((\mathcal N(n, L))\). It is proved that \(H^0(\mathcal N'_D(L),p_0^\ast\Theta^{\otimes m})=0\) for all \(m < 0\), and \(H^0(\mathcal N'_D(L),\mathcal O_{\mathcal N'_D(L)})=~\mathbb C\). Let \(\mathcal N_D(L)\) denote the moduli space of all pairs of the form \((E,\mathcal D)\), where \(E\) is a holomorphic vector bundle over \(X\) of rank \(n\) with \(\bigwedge^n E \cong L\) and \(\mathcal D\) a loga\-ri\-th\-mic connection on \(E\) singular exactly over \(x_0\) with residue \(-\frac{d}{n}\) Id such that the logarithmic connection on \(\bigwedge^n E\) induced by \(\mathcal D\) coincides with the given logarithmic connection \(D_L\) on \(L\). So \(\mathcal N'_D(L)\) is a Zariski open dense subset of \(\mathcal N_D(L)\). Since \(\mathcal N'_D(L)\) does not admit any nonconstant algebraic functions, it follows immediately that \(\mathcal N_D(L)\) does not admit any nonconstant algebraic functions. The moduli space \(\mathcal N_D(L)\) is biholomorphic to an affine variety, and it is quasi-complete.
For any point \(z\in\mathcal N(n,L)\) the fiber \(p_0^{-1}(z)\subset\mathcal N'_D(L)\) is canonically an affine space for the holomorphic cotangent space \(T_z^\ast\mathcal N(n,L)\). In other words, \(\mathcal N'_D(L)\) is a \(T^\ast\mathcal N(n,L)\)-torsor over \(\mathcal N(n, L)\). Using this torsor structure, the variety \(\mathcal N'_D(L)\) has a natural smooth compactification; the compactifying divisor is identified with the total space of \(\mathbb P(T\mathcal N(n,L))\), the space of hyperplanes in the fibers of \(T\mathcal N(n,L)\). This compactification is denoted by \(\overline{\mathcal N'_D(L)}\), and the divisor at infinity, namely the complement \(\overline{\mathcal N'_D(L)}\setminus\mathcal N'_D(L)\), is denoted by \(\Delta\). Since \(\mathcal N'_D(L)\) does not admit any nonconstant function, it follows immediately that \(\Delta\) is not ample. In fact \(\Delta\) is of Kodaira-Iitaka dimension zero. However, one can still ask if \(\Delta\) is numerically effective (which corresponds to the closure of the ample cone). The authors show that the tangent bundle \(T\mathcal N(n,L)\) is numerically effective iff the effective divisor \(\Delta\subset\overline{\mathcal N'_D(L)}\) is numerically effective. Using a classification of Fano three-folds with numerically effective tangent bundle it follows that \(T\mathcal N(n,L)\) is not numerically effective if genus\((X) = 2 = n\). If a conjecture given in [\textit{F.~Campana} and \textit{T.~Peternell} [Math. Ann. 289, No. 1, 169--187 (1991; Zbl 0729.14032)] is true, then \(T\mathcal N(n, L)\) is never numerically effective. vector bundles over a compact Riemann surface; residue; vector bundles admitting a logarithmic connection; pullback of line bundles; Picard group of moduli space of logarithmic connection; functions on the moduli space; compactification; torsor; Kodaira-Iitaka dimension; Fano three-folds; numerically effective tangent bundle I. Biswas, N. Raghavendra, Line bundles over a moduli space of logarithmic connections on a Riemann surface, Geom. Funct. Anal., in press Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (analytic), Compact Riemann surfaces and uniformization, Complex-analytic moduli problems Line bundles over a moduli space of logarithmic connections on a Riemann surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X \subset \mathbb P^N\), \(m > 2N/3\), be a smooth \(m\)-dimensional projective submanifold. In 1974 Hartshorne conjectured that \(X\) is a complete intersection. A far weaker conjecture is that the dual variety of \(X\) is a hypersurface; this is known only if \(m= N-2\). Here the author raises this weaker conjecture (ascribing it to A. Landman) and gives a new proof of the case \(m=N-2\) using the spannedness of \(N_X(-1)\), where \(N_X\) is the normal bundle of \(X\). complete intersection; low codimensional submanifold; dual variety; normal bundle; Chern classes; positive vector bundle; defect of a projective variety Holme, A., A combinatorial proof of the duality defect conjecture in codimension 2, Discrete math., 241, 1-3, 363-378, (2001) Low codimension problems in algebraic geometry, Complete intersections, Projective techniques in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry A combinatorial proof of the duality defect conjecture in codimension 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians An orthogonal representation of a graph G is an assignment of the vertices in G to vectors in \({\mathbb{R}}^ d\) such that non adjacent vertices are represented by orthogonal vectors. An orthogonal representation of G with the further property that any d representing vectors are linearly independent is called a general-position orthogonal representation of G (or GPOR of G). The paper shows that an n vertex graph G has a GPOR in \({\mathbb{R}}^ d\) if and only if G is (n-d) connected. An algorithm is given for finding the GPOR. The closure of the set of all \(GPOR^ s\) of a given graph is shown to be an irreducible algebraic variety. k-connected graphs; orthogonal representation of a graph; irreducible algebraic variety Lovász, L.; Saks, M.; Schrijver, A., Orthogonal representations and connectivity of graphs, Linear Algebra Appl., 114/115, 439-454, (1989) Connectivity, Varieties and morphisms, Vector spaces, linear dependence, rank, lineability Orthogonal representations and connectivity of graphs | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We give an example of a vector bundle \({\mathcal E}\) on a relative curve \(C\to\text{Spec}\,\mathbb Z\) such that the restriction to the generic fiber in characteristic zero is semistable but such that the restriction to positive characteristic \(p\) is not strongly semistable for infinitely many prime numbers \(p\). Moreover, under the hypothesis that there exist infinitely many Sophie Germain primes, there are also examples such that the density of primes with nonstrongly semistable reduction is arbitrarily close to one. vector bundle on a relative curve; Sophie Germain primes; density H. Brenner. On a problem of Miyaoka. In Number Fields and Function Fields--Two Parallel Worlds, Progress in Math. 239, Birkhäuser, 51-59 (2005). Vector bundles on curves and their moduli On a problem of Miyaoka | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this article, the author gives an extensive survey of some of the spectacular recent advances in the algebraic theory of quadratic forms and the algebro-geometric methods that have been developed to achieve these advances. The paper focuses mainly on the contributions by Rost, Voevodsky, Vishik, Karpenko, Merkurjev, Brosnan and some others. It is divided into two parts. The first part develops the algebraic theory of quadratic forms, starting with Witt's theory. Pfister forms and the filtration of the Witt ring are introduced, with mention of the Arason-Pfister Hauptsatz. The basic theory of function fields and Knebusch's notion of generic splitting are explained, including the definitions of height, Pfister neighbors and excellent quadratic forms. The author then introduces the notion of stable birational equivalence of quadrics and mentions the reviewer's theorem on anisotropy of one quadratic form over the function field of another when their dimensions are separated by a \(2\)-power. In the last two sections of part one, the author states four of the most striking results on quadratic forms that have recently been proved and gives a few applications. The first of these four results concerns gaps in the dimensions of anisotropic forms in \(I^n\) and is due to Vishik (unpublished) and Karpenko. The second result concerns the so-called essential dimension of a quadratic form (in the sense of Izhboldin) and a necessary condition relating the essential dimensions of two anisotropic forms if one becomes isotropic over the function field of the other. This result, conjectured by Izhboldin, was proved by Karpenko-Merkurjev. The third and fourth result concern the possible values of the first Witt index of a quadratic form in terms of its dimension (that relation has been conjectured by the reviewer) and relations between the higher Witt indices of a quadratic form, respectively. Both these results have been proved by Karpenko, and they together imply the first result. Among the known applications of these results, it is shown how one can deduce the dimensions of so-called forms of height \(2\), how to get a bound on the Witt index of a quadratic form over the function field of another, and how to determine the minimal transcendence degree of a generic isotropy field for a quadratic form.
Part two develops the algebro-geometric background needed in the proofs of the above results. First, the category of Chow motifs and Tate motifs are introduced. Next, the decomposition of the Chow motif of a projective homogeneous variety is studied. Rost's nilpotence theorem is mentioned as a special case of a far reaching result due to Chernousov-Gille-Merkurjev. The author then introduces the Steenrod operations with particular emphasis on Steenrod operations on modulo \(2\) Chow groups as constructed by Brosnan. He then studies in more detail the structure of the Chow motif of a quadric, in particular its decomposition into a direct sum of indecomposable motifs. As examples, the structure of a split quadric and Rost's result on the motif of a Pfister quadric in terms of so-called Rost motifs are given, and Vishik's major contribution to the vast generalization of that theory of motivic decomposition of quadrics is sketched. As an application, Vishik's result on motivic equivalence of quadrics is mentioned. Furthermore, binary motifs are defined and their connection with excellent quadratic forms is explained. The final section of part two is dedicated to the proofs of the results on essential dimension and on the possible values of the first Witt index (only the latter requires the use of Steenrod operations).
Most of the results are stated without proof. In a few cases, the author provides sketches of proofs or gives his own sometimes a little simpler variations of the original proofs. Even when no proof is given, it is often explained which ingredients go into it, and many enlightening comments are added that put the results into context. With its 70 references, this well-written article will be an important reference for those who want to learn more about the exciting developments the algebraic theory of quadratic forms has experienced in recent years and for specialists in these topics alike. quadratic form; Witt ring; fundamental ideal; Pfister form; Witt index; higher Witt indices; generic splitting; height of a quadratic form; stable birational equivalence; Chow motif; Tate motif; Rost motif; homogeneous variety; Rost's nilpotence theorem; Steenrod operation; motivic equivalence Kahn, B., \textit{formes quadratiques et cycles algébriques [d'après rost, Voevodsky, vishik, karpenko ...], exposé bourbaki no. 941}, Astérisque, 307, 113-163, (2006) Quadratic forms over general fields, Algebraic theory of quadratic forms; Witt groups and rings, (Equivariant) Chow groups and rings; motives, Algebraic cycles, Rational and birational maps, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Operations and obstructions in algebraic topology Quadratic forms and algebraic cycles (after Rost, Voevodsky, Vishik, Karpenko, \dots) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The (rational) algebraic cohomology ring \(H^*_A(X)\) of a smooth projective variety \(X\) is the subring of the ordinary ring \(H^*(X,{\mathbb Q})\) spanned by the algebraic cycles. Accordingly the algebraic Poincaré polynomial is defined by the formula \(P_A(X;t)=\sum_i \dim H_A^i(X)t^i\). Let \({\mathcal N}_C\) be the moduli stack of rank \(2\) stable holomorphic vector bundles on a smooth curve \(C\) of genus \(g\geq 2\) with fixed determinant of odd degree. Let \(J_C\) be the Jacobian of \(C\). The authors describe generators for \(H^*_A({\mathcal N}_C)\) and the algebraic Poincaré polynomial of \({\mathcal N}_C\). More precisely they show that
\[
P_A({\mathcal N}_C;t)={{P_A(J_C;t^3)-t^g P_A(J_C;t)}\over{(1-t)(1-t^2)}}
\]
which is analogous to the classical formula for the ordinary Poincaré polynomial. The proof of this formula follows an idea of \textit{M. Thaddeus} [Invent. Math. 117, No. 2, 317-353 (1994; Zbl 0882.14003)], who relates a projective bundle over \({\mathcal N}_C\) to a projective space through a chain of smooth flips whose centers are symmetric powers \(S^kC\).
Hence in order to prove this formula the authors firstly determine how the algebraic Poincaré polynomials (and later even the Chow groups) transform under a smooth flip in the sense of Thaddeus. They secondly determine the algebraic Poincaré polynomial of \(S^kC\) by using Collino's description of the Chow ring of \(S^kC\). Both these tools are developed very clearly and are nice to read.
By a well known theorem of Newstead, the ordinary cohomology ring of \({\mathcal N}_C\) is generated by two classes \(\alpha\in H^2({\mathcal N}_C)\) , \(\beta\in H^4({\mathcal N}_C)\) and moreover by \(H^3({\mathcal N}_C)\) which corresponds in a natural way to \(H^1(J_C)\) where \(J_C\) is the Jacobian of \(C\). Hence Newstead's theorem can be expressed by the following statement:
There is a surjective ring homomorphism \(\nu\colon {\mathcal Q}[\alpha, \beta ]\otimes H^*(J_C)\to H^*({\mathcal N}_C)\). The authors prove that \(\nu\) takes algebraic classes on \(J_C\) to algebraic classes on \({\mathcal N}_C\). It follows that \(H_A^*({\mathcal N}_C)= \nu( {\mathcal Q}[\alpha, \beta ]\otimes H_A^*(J_C))\).
The authors remark that their arguments can be extended to prove that if the Hodge conjecture holds for \(J_C\) then it holds also for \({\mathcal N}_C\). A paper of \textit{A. D. King} and \textit{P. E. Newstead} about \(H^*({\mathcal N}_C)\) has appeared in Topology 37, No. 2, 407-418 (1998; Zbl 0913.14008)]. algebraic cycles; algebraic Poincaré polynomial; moduli stack; algebraic cohomology; vector bundles on a smooth curve; Jacobian; smooth flip; Hodge conjecture Balaji, V., King, A.D., Newstead, P.E.: Algebraic cohomology of the moduli space of rank 2 vector bundles on a curve. Topology 36, 567--577 (1997) Families, moduli of curves (algebraic), (Co)homology theory in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Jacobians, Prym varieties Algebraic cohomology of the moduli space of rank 2 vector bundles on a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this article the authors propose an effective measure of rational chain connection on projective algebraic varieties. A few years back, \textit{F. Campana} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 25, No. 5, 539-545 (1992; Zbl 0783.14022)] and \textit{J. Kollár, Y. Miyaoka} and \textit{S. Mori} [J. Algebr. Geom. 1, No. 3, 429-448 (1992; Zbl 0780.14026), J. Differ. Geom. 36, No. 3, 765-779 (1992; Zbl 0759.14032) and in: Classification of irrgular varieties, minimal models and abelian varieties, Proc. Conf., Trento 1990, Lect. Notes Math. 1515, 100-105 (1992; Zbl 0776.14012)], have introduced the concept of rational chain connection: A projective algebraic variety \(X\) is said to be rationally connected, if for two generally chosen points \(x\), \(y\) of \(X\) there exists a rational curve \(C = C_{x, y}\) inside \(X\) which passes through \(x\) and \(y\). Among others they proved that a non-singular Fano variety satisfies this property.
There are at least two main reasons by which this result had called significant attention: First, the non-singular Fano varieties in each dimension were proved to form a bounded moduli, which essentially relied on this rational connectedness theorem. Second, it was discovered that some of Fano varieties are irrational [e.g. \textit{V. A. Iskovskikh} and \textit{Yu. I. Manin}, Math. USSR, Sb. 15, 141-166 (1972); translation from Mat. Sb., Nov. Ser. 86(128), 140-166 (1971; Zbl 0222.14009)] and \textit{C. H. Clemens} and \textit{P. A. Griffiths} [Ann. Math., II. Ser. 95, 281-356 (1972; Zbl 0214.48302)], and accordingly the concept of rational connection was considered as a reasonable compensation for such failure to rationality. Thus, it was considered by all means natural to attempt to study in detail the class of rationally connected varieties [the book of \textit{J. Kollár}: ``Rational curves on algebraic varieties'' (1995; Zbl 0877.14012)].
In the present article, the authors propose to introduce a hierarchy into the category of rationally connected varieties, by means of measuring the length of rational curves connecting points with respect to a polarization of \(X\). Namely, let \(X\) be a normal projective variety, and assume that for an \(r\)-tuple of generally chosen points \(x_1, \cdots, x_r\) in \(X\) there exists a rational curve \(C = C_{x_1, \cdots, x_r}\) in \(X\) which passes through \(x_1, \cdots, x_r\). When \(r = 2\), this assumption is equivalent to the rational connection.
The authors consider the following problem: Assuming that \(X\) has an ample line bundle \(L\), such that \((L . C) = \int_C c_1(L) \leq d\), extract any geometric properties one can find out from \(X\), for each (reasonable) pair of positive integers \((d, r)\). Classify them when \(d\) and \(r\) are small. In Math. Ann. 297, No. 2, 191-198 (1993; Zbl 0789.14011), \textit{M. Andreatta, E. Ballico} and \textit{J. A. Wiśniewski} formerly proved a result, which can be interpreted in the above framework as follows: When \((d,r)=(1,2)\), then such \(X\) is the projective space \(\mathbb P^n\), in pursuit of investigating the structure of extremal contractions in higher dimensions, based on the observation of \textit{T. Fujita} [e.g. Nagoya Math. J. 115, 105-123 (1989; Zbl 0699.14002)]. An exposition by \textit{M. Andreatta} and \textit{J. A. Wiśniewski} [in: Algebraic Geometry, Proc. Summer Res. Inst., Santa Cruz 1995, Proc. Symp. Pure Math. 62, No. 1, 153-183 (1997)] covers much of their perspective. In the present article the authors generalize this to the case \((d, r) = (2, 3)\); they prove that such \(X\) is isomorphic either to \(\mathbb P^n\), or a quadric \(Q^n\) in \(\mathbb P^{n+1}\). This, along with Andreatta-Ballico-Wiśniewski's theorem indeed generalizes the Kobayashi-Ochiai theorem on the characterization of \(\mathbb P^n\) and \(Q^n\) as Fano varieties having the two highest Fano indices [cf. \textit{S. Kobayashi} and \textit{T. Ochiai}, J. Math. Kyoto Univ. 13, 31-47 (1973; Zbl 0261.32013)].
Methodologywise, the present article extensively considers the pro-algebraic relation arisen from the Hilbert scheme of the variety, after Campana and Kollár. They prove the existence of the algebraic quotient for such pro-algebraic relation under the circumstance that the members of the prescribed family form a complete intersection of divisors (codimension 1 cycles) of certain type, by making use of the intersection theory. As a consequence, the authors derive the existence of a flat family \(\{S_z\}_{z \in X}\) of divisors which are ruled by rational curves of length \(1\) passing through the common point \(z\), where (morally) the set of closed points of \(X\) itself serves as the space of its parameters. With this, along with the base-change theorem, and the inversion of Mori theory [\textit{S. Mori}, Ann. Math., II. Ser. 110, 593-606 (1979; Zbl 0423.14006)] or the so-called algebraic Sard theorem, the authors prove that the Picard number (or the second Betti number) of \(X\) is equal to \(1\). This way they obtain \(X \simeq \mathbb P^n\) or \(Q^n\). (This part is subtle, indeed a complete rework will become necessary when the characteristic of the base field is no longer assumed zero, as is shown recently by Kachi and Kollár.)
The authors remark also that their result remains to hold true even when \(X\) has considerably general class of singularities, by virtue of the argument of Fujita and Mella. Overall, the present article is a step toward the understanding of the birational and biregular behavior of Fano and uniruled varieties from a slightly unconventional angle, and its scope may have potential to lead to a search of further clarifying the geography of those varieties in a more unified framework. small degrees; rational chain connection; length of rational curves; extremal contractions; divisors; algebraic Sard theorem; Picard number ] Yasuyuki Kachi, Characterization of Veronese varieties and the Hodge index theorem, (unpublished), 1999. Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves, Rational and birational maps, \(4\)-folds Polarized varieties whose points are joined by rational curves of small degress | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let G be a semisimple complex group, B be a Borel subgroup, X be the closure in G/B of a general orbit of a maximal torus. The author describes a basis of the homologies \(H_*(X,{\mathbb{Z}})\) and the image of the restriction homomorphism \(i^*: H^*(G/B,{\mathbb{Q}})\to H^*(X,{\mathbb{Q}})\). This image is the algebra of invariants \(H(X,{\mathbb{Q}})^ W\) with respect of the action induced by an action of the Weyl group W on X. There is an explicit presentation of generators and defining relations for this algebra of invariants. The \(i^*\)-images of the generalized Schubert cocycles are expressed in the terms of these generators. The note contains some applications to the case \(G=GL_ n\) and to usual Schubert cells. orbits of a maximal torus on a flag space; basis of the homologies; algebra of invariants; Weyl group; generalized Schubert cocycles; Schubert cells Klyachko, A, Orbits of a maximal torus on a flag space, Funktsional. Anal. i Prilozhen., 19, 77-78, (1985) Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Geometric invariant theory, Group actions on varieties or schemes (quotients) Orbits of a maximal torus on a flag space | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A subspace in the ambient space of a Veronese variety is said to be invariant if it is fixed under the group of automorphic collineations of the variety. Among the invariant subspaces are the nuclei of Veronesean; they arise as intersection of all osculating subspaces of a fixed type.
In the excellently written paper the author gives a survey on recent results about nuclei of Veronese varieties and invariant subspaces of normal rational curves. If the characteristic of the ground field is zero, then there are only the trivial invariant subspaces. If the characteristic is a prime \(p\), then there are in general many such subspaces, and there is a close relationship to the array of multinomial coefficients modulo \(p\).
The paper is completed by precisely cited references. Veronese variety; Normal rational curve; Nucleus; Pascal's triangle; Multinomial coefficients; survey; nuclei of Veronese varieties; invariant subspaces of normal rational curves Havlicek H (2003) Veronese varieties over fields with non-zero characteristic: a survey. Discrete Math 267:159--173 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Grassmannians, Schubert varieties, flag manifolds Veronese varieties over fields with non-zero characteristic: A survey | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study the nef cone and the pseudoeffective cone of a product $X$ of two projective bundles ${\mathbb P}(E_1)$ and ${\mathbb P}(E_2)$ over a smooth irreducible projective curve $C$. Several cases are considered: \begin{itemize} \item[1.] If both $E_1$ and $E_2$ are semistable vector bundles then one shows that the nef cone and pseudoeffective cone coincide. \item[2.] If both $E_1$ and $E_2$ have rank $2$ the nef cone and pseudoeffective cone are computed, in the assumption that none of $E_1$ and $E_2$ is semistable or if one of them is semistable. \item[3.] If both vector bundles have rank $2$ one shows that if the nef cone and pseudoeffective cone coincide then both vector bundles are semistable.\end{itemize} nef cone; pseudoeffective cone; semistability; cone of curves; fibre product of projective bundle over a curve Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(3\)-folds, Minimal model program (Mori theory, extremal rays) Nef and pseudoeffective cones of product of projective bundles over a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be a number field. The author's aim is to study the rank of \(A(K')/\tau(B'(k))\), where \(K=k(\mathcal{C})\) and \(K'= k(\mathcal{C}')\) are the function fields on the curves \(\mathcal{C}\) and \(\mathcal{C}'\) such that \(\mathcal{C}' \to \mathcal{C}\) is a finite geometrically abelian cover of a smooth projective irreducible curve \(\mathcal{C}\) of genus \(g\) defined over \(k\), \(A/K\) an abelian variety of dimension \(d\), and \((\tau',B')\) the \(K'/k\) trace of \(A\).
Let \(\phi \colon \mathcal{A} \to \mathcal{C}\) be a proper flat morphism defined over \(k\) from a smooth projective irreducible variety \(\mathcal{A}\) defined over \(k\) with generic fibre \(A/K\), \((\tau,B)\) the \(K/k\)-trace of \(A\). By the Lang-Néron theorem \(A(K)/\tau B(k)\) and \(A(\overline{k}(\mathcal{C}))/\tau B(\overline{k})\) are finitely generated abelian groups. Ogg proved that \(2d(2g-2)+f_A+4 \dim(B)\) is an upper bound for the rank of the group \(A(\overline{k}(\mathcal{\mathcal{C}}))/\tau B(\overline{k})\), where \(f_A\) is the degree of the conductor divisor of \(A\) on \(\mathcal{C}\).
The author considers a finite cover \(\pi \colon \mathcal{C}' \to \mathcal{C}\) defined over \(k\) which is geometrically abelian with geometric automorphism group \(\mathcal{G}= \mathrm{Aut}\, (\mathcal{C}'/\mathcal{C})\). Set \(\mathcal{A}'=\mathcal{A} \times_\mathcal{C} \mathcal{C}, \; A'=A\times_K K'\). Under the assumptions that Tate's conjecture is true for \(\mathcal{A}'/k\) and certain monodromy representations are irreducible, the author gives an upper bound for the rank of \(A(K')/\tau'B'(k)\) improving Ogg's bound in the case where \(A\) is a Jacobian variety and \(\pi\) is unramified. More precisely, the main result of the paper states that
\[
\mathrm{rank} \left( \frac{A(K')}{\tau' B'(k)} \right) \leq \frac{\sharp \mathfrak{O}_{G_k}(\mathcal{G})}{\sharp \mathcal{G}}(d(2d+1)(2g'-2))+ \sharp \mathfrak{O}_{G_k}(\mathcal{G})2df_A,
\]
where \(\mathfrak{O}_{G_k}(\mathcal{G})\) is the set of \(G_k\)-orbits of \(\mathcal{G}\) with respect to the natural action of absolute Galois group \(G_k\) on \(\mathcal{G}\), and \(g'\) is the genus of \(\mathcal{C}'\). Also the author examines how varies the rank of \(A(K')/\tau' B'(k)\) in towers of function fields over \(K\). rank of abelian variety; function fields; elliptic curve Pacheco, A.: The rank of abelian varieties over function fields. Manuscripta Math. 118, 361--381 (2005) Abelian varieties of dimension \(> 1\), Rational points, Arithmetic ground fields for curves, Arithmetic ground fields for abelian varieties On the rank of abelian varieties over function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author extends the work of his earlier paper [Math. Ann., 287, No. 1, 135--150 (1990; Zbl 0668.14019)] to the wild case. The models in question are regular models of the smooth proper geometrically connected curves \(X/K\) of genus \(g > 0\) over quasi-local field (complete discrete valuation field \(K\) with ring of integer \(O_K\) and algebraically closed residue field \(k\) of characteristic \(p > 0\)) obtained by desingularizing of wild quotient singularities of the smooth models of \(X_{L}/L\), \([L:K] = p\).
``Assume that \(X/K\) does not have good reduction and that it obtains good reduction over a Galois extension \(L/K\).''
Let \({\mathcal Y}/{\mathcal O}_L\) be the smooth model of \(X_{L}/L\). Let \(H := Gal(L/K)\), and let \({\mathcal Z}/{\mathcal O}_K\) denote the quotient \({\mathcal Y}/H\) with singular points \(Q_{1}, \dots, Q_{d}\) and \(d \geq 1\). Let \({\mathcal X}/{\mathcal O}_K\) be a regular model of \(X/K\). The regular model can be obtained by resolving the singularities of the scheme \({\mathcal Z}\). Let \({\mathcal X}/k := \sum_{i=1}^{v} r_{i}C_{i}\) denote the special fibre of \({\mathcal X}\).
Let \(\sigma\) denote a generator of \(H\).
The main theorem (6.8) says that if all ramification points of \({\mathcal Y}_{k} \to {\mathcal Y}_{k}/{\langle \sigma \rangle} \) are weakly ramified, then, for all \(i = 1, \dots, d,\) we have \( r_{i} = p\), and the graph \(G_{Q_{i}}\) of the desingularization of \( Q_{i}\) is a graph with a single node \(C_{i}\) of degree 3. The intersection matrix \(N(p,\alpha_{i},r_{1}(i))\) of the resolution of \( Q_{i}\) is uniquely determined by the two integers \(\alpha_{i}\) and \( r_{1}(i)\) with \(1 \leq r_{1}(i) < p\). The integer \( r_{1}(i)\) is number of vertices of self-intersection \(-2\) (including the node \(C_{i}\)) on the chain of \(G_{Q_{i}}\) connecting the node \(C_{0}\) to the single node \(C_{i}\) of \(G_{Q_{i}}\), and this integer \(\alpha_{i}\) is divisible by \(p\).
The proof is based on author's results on arithmetical graphs (Theorem 6.4 and Proposition 4.3 of the paper under review) and on computations and properties of Smith group of the intersection matrix by the author [Math. Z. 275, No. 1--2, 211--232 (2013; Zbl 1309.14003)].
Let \(A/K\) be the Jacobian of \(X/K\) of genus \(g\). Let \({\mathcal A}/{\mathcal O}_K\) be its Néron model. The following are corollaries to the theorem in the case \(g > 1\): (I) If all ramification points of \({\mathcal Y}_{k} \to {\mathcal Y}_{k}/{\langle \sigma \rangle} \) are weakly ramified, then: (a) \(X(K) \neq \emptyset\). (b) The unipotent part \(U/k\) of the connected component of the identity in \({\mathcal A}_{k}/k\) is a product of additive groups \({\mathbb G}_{a,k}\). (c) The group of components \(\Phi_{A,K}\) of the Néron model is isomorphic to \(({\mathbb Z}/p{\mathbb Z})^{2d - 2}\). (II) If all ramification points of \({\mathcal Y}_{k} \to {\mathcal Y}_{k}/{\langle \sigma \rangle} \) are weakly ramified, then \(\Phi_{A,K}\) is a \({\mathbb Z}/p{\mathbb Z}\)-vector space of dimension \(2d - 2\), and \(\Phi_{A,K}^{0}\) is a subspace of dimension \(d - 1\). Moreover, \(\Phi_{A,K}^{0} = (\Phi_{A,K}^{0})^{\perp}\).
Properties of groups of components of Néron models of abelian varieties and of algebraic tori over discrete valuation fields with non-separable closed residue fields have been investigated by \textit{S. Bosch} and \textit{Q. Liu} [Manuscr. Math. 98, No. 3, 275--293 (1999; Zbl 0934.14029)].
The paper under review contains further interesting results. Section 2 includes Proposition 2.5 which exhibits a key difference between the tame and wild cases. For smooth projective connected hyperelliptic curves corollaries and examples are given. Results about models of superelliptic curves has obtained by \textit{C. Greither} [J. Number Theory 154, 292--323 (2015; Zbl 1326.11051)].
The third section of the paper under review deals with the arithmetical graphs \(G\) and in the fourth section the author introduces and investigates a measure of how ``omplicated'' certain graphs are.
Section five discusses the quotient construction. Let \(\alpha: {\mathcal Y} \to {\mathcal Z}\) denote the quotient map and let \({\mathcal X} \to {\mathcal Z}\) obtained from \( {\mathcal Z}\) by minimal desingularization. ``After finitely many blow-ups \({\mathcal X'} \to {\mathcal X}\) we can assume that the model \({\mathcal X'}\) is such that \({\mathcal X'}_{k}\) has smooth components and normal crossings and is minimal with this properties. Let \(f\) denote the composition \({\mathcal X'} \to {\mathcal Z}\).'' The main result of the section is Theorem 5.3. It says that if \(G_{Q_{i}}\) denote the graph associated with the curve \(f^{-1}(Q)\) and if \(G\) denote the graph associated with the special fiber \({\mathcal X'}_{k}\), then for all \(i = 1, \dots, d,\) the graph \(G_{Q_{i}}\) contains a node of \(G\) and \(p\) divides \(r_{1}\).
Finally the author gives two remarks concerning intersection matrix and graph associated with the resolution of a \({\mathbb Z}/p{\mathbb Z}\)-singularity and the graph associated with resolution of a singular point \(Q_{i}\) in the quotient \({\mathcal Z}\). model of a curve; ordinary curve; cyclic quotient singularity; wild ramification; arithmetical tree; resolution graph; component group Néron model D. Lorenzini, Wild models of curves. Algebra Number Theory 8(2), 331-367 (2014) Local ground fields in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Arithmetic ground fields for abelian varieties, Singularities of surfaces or higher-dimensional varieties Wild models of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result of the paper is a wall-crossing formula for the degree of a central projection defined on a smooth submanifold (not necessarily algebraic) of a real projective space. More precisely, let \(V,W\) be real vector spaces, \(X\subset{\mathbb P}(V)\) a smooth variety of dimension \(\dim W-1\), \(f\in\text{Hom}(V,W)\) be such that \({\mathbb P}(\text{Ker}(f))\cap X=\emptyset\). This data induces a finite map \(X\to{\mathbb P}(W)\), which relatively orientable if \(w_1(X)=\dim W\cdot w_1(\lambda_{V,X})\), where \(\lambda_{V,X}\) is the restriction of the tautological line bundle \(\lambda_V\) on \(X\). Under the latter condition, one obtains a well-defined \({\mathbb Z}\)-valued degree as \(f\) varies in \(\text{Hom}(V,W)\setminus{\mathcal W}_X\), where \({\mathcal W}_X= \{f\;|\;{\mathbb P}(\text{Ker}(f))\cap X\neq\emptyset\}\) is the wall. The authors show that any two chambers in \(\text{Hom}(V,W)\setminus{\mathcal W}_X\) can be joined by a smooth path crossing the wall only in finitely many regular points, and they suggest a wall-crossing formula for the jump of the degree at a regular intersection point with \({\mathcal W}_X\). Several interesting examples (the pole placement map, the Wronski map, a version of the real subspace problem) illustrate the general result. real projective variety; degree; central projection; relative orientation; wall-crossing; conservation of numbers Topology of real algebraic varieties, Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus A wall-crossing formula for degrees of real central projections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth curve of genus \(g\), \(J(C)\) its Jacobian and \(\Theta\) the Theta divisor on \(J(C)\). It is a well known result that the dimension of the singularity locus \(\text{sing} (\Theta)\) is \(g-4\) for nonhyperelliptic \(C\) and \(g-3\) for hyperelliptic curves \(C\). This statement is part of a criterion characterizing Jacobians of hyperelliptic curves among all principally polarized abelian varieties. The main theorem of the paper under consideration is:
A principally polarized abelian variety \((X,\Theta)\) of dimension \(g\) is the Jacobian of a hyperelliptic curve if and only if \(\dim \text{sing} (\Theta) \leq g-3\) and there exists a global section of the sheaf \({\mathcal O}_ \Theta (\Theta)\) whose zero locus admits nonreduced components. Theta divisor on Jacobian; dimension of the singularity locus; characterizing Jacobians of hyperelliptic curves among all principally polarized abelian varieties; principally polarized abelian variety Theta functions and curves; Schottky problem, Jacobians, Prym varieties, Theta functions and abelian varieties A characterization of hyperelliptic Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We consider the question what can be said about the rank of the Picard group \(\text{Pic } X_ \Sigma\) of a compact toric variety \(X_ \Sigma\) if we know only the combinatorial type of the associated fan \(\Sigma\). We establish upper and lower bounds for the rank of \(\text{Pic } X_ \Sigma\) and give conditions for \(\text{Pic } X_ \Sigma\) to be determined by the combinatorial type of \(\Sigma\). Furthermore, we show that for simple fans \(\text{Pic } X_ \Sigma\) is necessarily isomorphic to \(\{0\}\) or \(\mathbb{Z}\) and give an example for a compact toric variety having a trivial Picard group. Moreover in the projective case we study the relation between addition of \(T\)-invariant Cartier divisors on \(X_ \Sigma\), taking the tensor product of elements of \(\text{Pic } X_ \Sigma\) and piecewise linear functions on \(\Sigma\) with Minkowski- addition of polytopes, where the latter operation is extended to a group operation. Finally, we explain the relation to strong cohomology in the projective case. associated fan of toric variety; rank of the Picard group M. Eikelberg, Picard groups of compact toric varieties and combinatorial classes of fans, Results Math. 23 (1993), 251--293. Toric varieties, Newton polyhedra, Okounkov bodies, Picard groups, Grassmannians, Schubert varieties, flag manifolds Picard groups of compact toric varieties and combinatorial classes of fans | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author obtains the normal form of the Hasse-Witt matrix of a Fermat variety. He gets conditions for this matrix to be zero for Fermat curves and surfaces. Fermat surfaces; Hasse-Witt matrix of a Fermat variety; Fermat curves K. TOKI, On Hasse-Witt matrices of Fermat varieties, Hiroshima Math. J. 18 (1988), 95-111 Finite ground fields in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Global ground fields in algebraic geometry, Witt vectors and related rings On Hasse-Witt matrices of Fermat varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X=SU_C (2,2d)\) denote the moduli space of semistable rank-2 vector bundles with fixed determinant of even degree over a smooth projective curve \(C\) of genus \(g \geq 2\). It is well-known that \(\text{Pic} (X)\simeq \mathbb{Z}\). The generalized theta divisor is by definition the ample generator \({\mathcal L}\) of \(\text{Pic} (X)\). Let \(\varphi_{\mathcal L}: X\to\mathbb{P} H^0 ({\mathcal L})^*\) be the map associated to \({\mathcal L}\). \textit{A. Beauville} showed [Bull. Soc. Math. Fr. 116, No. 4, 431-448 (1988; Zbl 0691.14016)] that \(\varphi_{\mathcal L}\) is of degree \(\leq 2\) onto its image and \(\varphi_{\mathcal L}\) has degree 2 if and only if \(C\) is hyperelliptic. \textit{Y. Laszlo} showed [Math. Ann. 299, No. 4, 597-608 (1994; Zbl 0846.14011)] that \(\varphi_{\mathcal L}\) is an embedding for a general nonhyperelliptic curve. -- In the present paper the following result is given:
Suppose \(C\) is nonhyperelliptic. (1) \(\varphi_{\mathcal L}\) is injective, (2) \(d\varphi_{{\mathcal L}, x}\) is injective if \(X\) represents a stable vector bundle.
It is proven by relating the map \(\varphi_{\mathcal L}\) to the geometry of quadrics containing the projective embeddings of \(C\) as a curve of degree \(2g+2\). The method is interesting in itself. Picard group; nonhyperelliptic curve; generalized theta divisor Brivio, Sonia; Verra, Alessandro, The theta divisor of \({\mathrm SU}_C(2,2d)^s\) is very ample if \(C\) is not hyperelliptic, Duke Math. J., 82, 3, 503-552, (1996) Vector bundles on curves and their moduli, Theta functions and abelian varieties The theta divisor of \(SU_ C(2,2d)^ s\) is very ample if \(C\) is not hyperelliptic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians If \(\pi :Y\rightarrow X\) is an unramified double cover of a smooth curve of genus \(g\), then the Prym variety \(P_\pi\) is a principally polarized abelian variety of dimension \(g- 1\). When \(X\) is defined over an algebraically closed field \(k\) of characteristic \(p\), it is not known in general which \(p\)-ranks can occur for \(P_\pi\) under restrictions on the \(p\)-rank of \(X\). In this paper, when \(X\) is a non-hyperelliptic curve of genus \(g = 3\), we analyze the relationship between the Hasse-Witt matrices of \(X\) and \(P_\pi\). As an application, when \(p\equiv 5 \mod 6\), we prove that there exists a curve \(X\) of genus 3 and \(p\)-rank \(f = 3\) having an unramified double cover \(\pi :Y\rightarrow X\) for which \(P_\pi\) has \(p\)-rank 0 (and is thus supersingular); for \(3\leq p\leq 19\), we verify the same for each \(0\leq f \leq 3\). Using theoretical results about \(p\)-rank stratifications of moduli spaces, we prove, for small \(p\) and arbitrary \(g\geq 3\), that there exists an unramified double cover \(\pi :Y\rightarrow X\) such that both \(X\) and \(P_\pi\) have small \(p\)-rank. curve; Jacobian; Prym variety; abelian variety; \(p\)-rank; supersingular; moduli space; Kummer surface Jacobians, Prym varieties, Curves over finite and local fields, Abelian varieties of dimension \(> 1\), Positive characteristic ground fields in algebraic geometry, Arithmetic ground fields for abelian varieties Non-ordinary curves with a Prym variety of low \(p\)-rank | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0702.14009. moduli space of semistable vector bundles of rank two on a non-singular curve; Brill-Noether theory Teixidor i Bigas, M.: Brill-Noether theory for vector bundles of rank 2. Tôhoku Math. J.43, 123--126 (1991) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles Brill-Noether theory for vector bundles of rank 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors of this note investigate the character \(\lambda =\text{Tr}(G| H^ 1(X,{\mathbb{Q}}_{\ell}))\) of the action of a cyclic group \(G\) of finite order \(n\) on the first \(\ell\)-adic cohomology group of an algebraic curve \(X\) over an algebraically closed field of arbitrary characteristic. They use the Lefschetz fixed point formula to show that \(a_ e\geq a_ d- 2\delta_{d,1}\) for divisors \(d,e\) of \(n\) with \(d| e\), if \(\lambda\) is written in the standard form as \(\lambda =\sum_{d| n}a_ d\chi_ d.\) As a corollary they obtain estimates \(n\leq 4g+2\) and \(n\leq 4g\), if \(n\neq 4g+2\) where g is the genus of \(X\), and the explicit structure of \(\lambda\), in the extreme cases. Based on a previous result of the second author [J. Math. Soc. Japan 39, 269-286 (1987; Zbl 0623.14009)] they also obtain a characterization of those characters of G which can be realized as \(\text{Tr}(G| H^ 0(X,\Omega_ X))\). action of a cyclic group; \(\ell \)-adic cohomology group of an algebraic curve; Lefschetz fixed point formula Group actions on varieties or schemes (quotients), Curves in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology On the action of automorphisms of a curve on the first \(\ell\)-adic cohomology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians ''The aim of this paper is to show that there is an effective way to determine the defining equations of a large class of projective monomial curves. That is based on a degree bound for the elements of all minimal bases of the defining ideal of such curves.''
Let V be a projective monomial curve given parametrically by a set \(M_ V\) of monomials of some degree, say d, in two indeterminates t and s such that \(t^ d,t^{d-1}s,ts^{d-1},s^ d\in M_ V.\) Let S denote the multiplicative monoid generated by the monomials of \(M_ V\). Then it is easily seen that there exist positive integers n such that \(t^{nd- a}s^ a\in S\) for all \(a=0,1,...,nd\). Let \(n_ V\) denote the minimum of such integers n.
Main result: There is a basis for the defining prime ideal I(V) of V consisting of binomials of degree \(\leq n_ V+1\). - By a binomial we understand a difference of two monomials of the same degree.
Clearly the theorem yields an effective way to determine the defining equation of V because one can easily establish an algorithm to compute \(n_ V\) and all binomials of degree \(\leq n_ V+1\) vanishing on V. - Moreover the author shows, with examples, that a basis found by this way may be, or not, a minimal basis of I(V) and that \(n_ V\) is not always the maximal degree appearing in every minimal basis of I(V). minimal basis of the defining prime ideal of a curve; defining equations; projective monomial curves; binomials Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Special algebraic curves and curves of low genus, Relevant commutative algebra Degree bound for the defining equations of projective monomial curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0657.00005.]
The author shows that, in the second half of the 17th century, when mathematicians wanted to find a solution of a geometrical problem, they were not satisfied by reducing it to finding the roots of an algebraic (or even transcendental) equation, but wanted to obtain it by a geometrical construction. However, for problems not solvable ``by ruler and compass'' (which was generally the case) how was one to find such constructions, and even a ``best'' construction? Descartes emphasized constructions by intersection of curves; for example, the trisection of an angle is solved by the intersection of a circle and a parabola. The curves used in this method were to be as ``simple'' as possible. But one had to ``construct'' these curves, and with exception of conics, one could only construct separate points of the curve. The author gives an example of such a construction given by Leibniz in a letter to Huygens; the construction of dense set of points of the curve can still be done by ruler and compass. But other examples such as the elastica and the paracentric isochrone, due to the Bernoullis, require the evaluation of areas limited by auxiliary curves, or lengths of arcs of such curves...
As there was no way to describe a general method which could apply to all these ``geometric'' constructions, they were rapidly abandoned after 1700 in favour of numerical approximations. But the author notes that they had at least contributed to the study of new curves and of relations between them, such as those which led Fagnano to the remarkable properties of elliptic integrals. construction of a curve History of algebraic geometry, History of mathematics in the 17th century, Curves in algebraic geometry The concept of construction and the representation of curves in seventeenth-century mathematics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians All the known violations of the Hasse-principle for a variety \(X/\mathbb{Q}\) are accounted for by an obstruction defined by Manin, coming from the Brauer-Grothendieck group (referred as the Brauer-Manin obstruction). In this note are given explicit hypersurfaces \(X_k\) in \(\mathbb{P}^4\) which have real and \(p\)-adic points for all \(p\) and for which the Brauer-Manin obstruction vanishes. Moreover either some \(X_k\) has no rational points or a related variety \(X\) in \(\mathbb{P}^5\) is a counterexample to Lang's conjecture that any hyperbolic variety defined over \(\mathbb{Q}\) has only finitely many rational points. number of rational points; Hasse-principle for a variety; Brauer-Manin obstruction P. Sarnak and L. Wang, ''Some hypersurfaces in \({\mathbf P}^4\) and the Hasse-principle,'' C. R. Acad. Sci. Paris Sér. I Math., vol. 321, iss. 3, pp. 319-322, 1995. Rational points, Hypersurfaces and algebraic geometry, Projective techniques in algebraic geometry, \(3\)-folds Some hypersurfaces in \(\mathbb{P}^ 4\) and the Hasse-principle | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians As in part I [Int. Math. Res. Not. 1994, No. 8, 343-361 (1994; Zbl 0813.52017)], the author studies the Mordell-Weil lattice \(L\) of a constant elliptic curve over the function field of a hyperelliptic curve \(C\), both attaining Weil's upper bound on the number of rational points over the finite field with \(q^2\) elements. He now takes \(q=64\) and \(C\) of genus \(g=6\). Then, by the general theory, \(L\) is an even lattice of rank \(n=4g=24\), and the product of \(\det L\) with the order of the Tate-Shafarevich group is \(2^n\). The author shows that here the group is trivial, any nonzero vector of \(L\) has at least norm 8, and all norms are multiples of 4. This proves that \(L\) is similar to Leech's lattice. Finally, the 196560 minimal vectors are found, and this enumeration is used to complete the proof in an alternative way. The author also announces a subsequent paper studying Mordell-Weil lattices of 2-power ranks up to \(n=32\). Leech lattice; Mordell-Weil lattice; elliptic curves; function field of a hyperelliptic curve; minimal vectors N. D. Elkies, ''Mordell-Weil lattices in characteristic 2, II: The Leech lattice as a Mordell-Weil lattice,'' Invent. Math., vol. 128, iss. 1, pp. 1-8, 1997. Lattice packing and covering (number-theoretic aspects), Curves of arbitrary genus or genus \(\ne 1\) over global fields, Elliptic curves, Packing and covering in \(n\) dimensions (aspects of discrete geometry) Mordell-Weil lattices in characteristic 2. II: The Leech lattice as a Mordell-Weil lattice | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let S be a closed Riemann surface of genus \(g>0\) and denote by J(S) the Jacobian variety of S, defined as the set of equivalence classes of divisors of degree zero. For \(\alpha\in J(S)\) and N a natural number, denote by \(\alpha_ N\) the set of divisors of type N in \(\alpha\). Furthermore, set \(J_ N(S)=\{\alpha_ N:\alpha\in J(S)\}\). The author shows:
Theorem 1: Let \(F_ N(\alpha,\beta)\) be a non-empty family of quasiconformal automorphisms of S, carrying a divisor from \(\alpha_ N\) to \(\beta_ N\). Then \(F_ N(\alpha,\beta)\) contains an extremal quasiconformal map \(f_ 0\) that minimizes K[f]. \(f_ 0\) is the Teichmüller map of S, and its Beltrami coefficient is \(k\phi\) (z)/\(| \phi (z)|\) where \(0\leq k<1\) and \(\phi\) (z) is a quadratic differential that is regular on S except in the points of the divisor associated with \(f_ 0\) at which \(\phi\) (z) has simple poles.
Let \(\alpha\),\(\beta\in J(S)\). If \(F_ N(\alpha,\beta)\neq \emptyset\), set \(\rho_ N(\alpha,\beta)=\ell n K[f_ 0]\) where \(f_ 0\) is an extremal map in \(F_ N(\alpha,\beta)\) that is homotopic to identity; otherwise set \(\rho_ N(\alpha,\beta)=\infty\). \(\rho_ N\) is a metric.
Theorem \(2: \rho_ N\) is finite in some neighborhood of \(\alpha\in J(S)\) if and only if \(\alpha_ N\) contains a divisor \(\sum \{P_ j-Q_ j:\) \(j=1,...,N\}\) such that there is no non-zero abelian differential whose divisor is \(\geq \sum \{P_ j+Q_ j:\) \(j=1,...,N\}.\)
Theorem 3: For \(N\geq g\), the canonical map \(J_ N(S)\to J(S)\) is a continuous surjection. Jacobian variety; divisors; extremal quasiconformal map; Teichmüller map; Beltrami coefficient Extremal problems for conformal and quasiconformal mappings, other methods, Quasiconformal mappings in the complex plane, Compact Riemann surfaces and uniformization, Jacobians, Prym varieties Extremal quasiconformal mappings and classes of divisors on Riemann surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We consider complex algebraic varieties endowed with an action of a torus \({\mathbb{T}}\) and we define the \({\mathbb{T}}\)-equivariant Euler class for any, smooth or not, isolated fixed point \(x\in{\mathbb{X}}^{\mathbb{T}}\). This class is a rational fraction on a finite number of variables and when \(x\) is a rationally smooth point of \({\mathbb{X}}\), it is a polynomial function canonically identified to the usual equivariant Euler class. We give sufficient conditions for an isolated fixed point of \({\mathbb{X}}\) having a polynomial equivariant Euler class to be smooth or rationally smooth. Finally, we apply these ideas to prove a rational smoothness criterion for the points of a Schubert variety. pseudomanifold; equivariant cohomology; equivariant Thom-Gysin morphism; rational smoothness; singularity; Schubert variety; action of a torus; equivariant Euler class Arabia, A., Classes d'euler équivariantes et points rationnellement lisses, Ann. Inst. Fourier (Grenoble), 48, 3, 861-912, (1998) Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Singularities in algebraic geometry, Homogeneous spaces and generalizations, Topological properties in algebraic geometry Classes d'Euler équivariantes et points rationnellement lisses. (Equivariant Euler classes and rationally smooth points.) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this interesting paper the author uses Morse theoretical arguments to study algebraic curves in \(\mathbb{C}^2\). The main strategy is as follows: Consider an algebraic curve \(C\subset \mathbb{C}^2\) and intersect it with spheres with fixed origin and growing radii, then describe how the embedded type of the intersection changes when passing a singular point of \(C\).
The paper is organized into six sections dealing with the following aspects : handles related to singular points, number of non-transversality points, signature of a link and its properties, change of signature upon an addition of a handle, application of Tristram-Levine signatures. The paper ends with a bibliography containing \(39\) suggestive references. Other papers by the author directly connected to this topic are [\textit{M. Borodzik} and \textit{H. Żoładek}, Pac. J. Math. 229, No. 2, 307--338 (2007; Zbl 1153.14026)], [J. Math. Kyoto Univ. 48, No. 3, 529--570 (2008; Zbl 1174.14028)], [J. Differ. Equations 245, No. 9, 2522--2533 (2008; Zbl 1160.34026)], [Isr. J. Math. 175, 301--347 (2010; Zbl 1202.14031)] (all jointly with H. Żołcadek). algebraic curve in \(\mathbb{C}^2\); singular point; critical value; multiplicity of a singular point; transversality; Milnor number; signature of a link; Tristram-Levine signatures Maciej Borodzik, Morse theory for plane algebraic curves, J. Topol. 5 (2012), no. 2, 341 -- 365. Knots and links in the 3-sphere, Critical points and critical submanifolds in differential topology, Plane and space curves, Local complex singularities, Singularities of curves, local rings Morse theory for plane algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let C be a smooth complete curve of genus g over an algebraically closed field k. Consider the d-th Picard variety \(Pic_ d\) of C[1] and a Poincaré sheaf \({\mathcal P}_ d\) on \(C\times Pic_ d\). If \(d>2g-2\) and \(\pi\) is the projection from \(C\times Pic_ d\) to \(Pic_ d\), then \(W_ d=\pi ({\mathcal P}_ d)\) is locally free. In a not too difficult fashion, the author proves that \(W_{2g-1}\) is a stable bundle with respect to polarization by the theta divisor. moduli space of curves of genus g; Picard variety; Poincaré sheaf; stable bundle; polarization by the theta divisor Kempf, G., Rank \(g\) Picard bundles are stable, Am. J. Math., 112, 397-401, (1990) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Picard groups, Families, moduli of curves (algebraic) Rank g Picard bundles are stable | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal M}\) be the moduli space of stable principal \(G\)-bundle over a compact Riemann surface \(C\), with \(G\) a reductive algebraic complex group and denote by \(K\) the canonical bundle over \(C\). In Duke Math. J. 54, 90- 114 (1987; Zbl 0627.14024), \textit{N. Hitchin} defined a map \({\mathcal H}\) from the cotangent bundle \(T^* {\mathcal M}\) to the ``characteristic space'' \({\mathcal K}\) by associating to each \(G\)-bundle \(P\) and section \(s \in H^0 (C,\text{ad}P \otimes K)\) the spectral invariants of \(s\). Next he described in the classical cases \((G = Gl(n), SO(n), Sp(n))\) the generic fibre of \({\mathcal H}\) in terms of a suitable abelian variety inside the Jacobian of a spectral curve covering \(C\). Let \(T \subset G\) be some fixed maximal torus with Weyl group \(W\) and denote by \(X(T)\) the group of characters on \(T\). One may associate to each generic \(\varphi \in {\mathcal K}\) a \(W\)-Galois covering \(\widetilde C\) of \(C\) and consider the ``generalized Prym variety'' \({\mathcal P} = \Hom_W (X(T), J(\widetilde C))\).
In this paper the author explicitly defines a map \({\mathcal F} : {\mathcal H}^{-1} (\varphi) \to {\mathcal P}_0\), \({\mathcal P}_0\) being the connected component in \({\mathcal P}\) containing the identity element. Such \({\mathcal F}\) depends in fact on the choice of one basis of roots \(\Delta \subset R(G,T)\). The author proved [``An elementary approach to the abelianization of the Hitchin system for arbitrary reductive groups'' (preprint 1994)] that the restriction of \({\mathcal F}\) to each connected component of \({\mathcal H}^{-1} (\varphi)\) has finite fibres. In the present paper, for each dominant weight \(\lambda \in \Lambda (T)^+\) (equivalently for each irreducible representation of \(G)\) the author defines by suitable divisorial correspondences a Prym-Tjurin variety \(P_\lambda \subset J (\widetilde C)\) and shows that there is an isogeny: \({\mathcal P} \to P_\lambda\). Her construction of \(P_\lambda\) is a natural generalization of the one given by \textit{V. Kanev} [in: Algebraic geometry, Proc. Conf., Sitges 1983, Lect. Notes Math. 1124, 166-215 (1985; Zbl 0575.14037)].
Next the author considers an irreducible spectral covering \(C_\lambda\) of \(C\) and, in analogy with the above, defines a Prym-Tjurin variety \({\mathcal A}_\lambda \subset J (C_\lambda)\). In the classical cases, when \(\lambda\) corresponds to the natural representation \({\mathcal A}_\lambda\) is exactly the abelian variety found by Hitchin. Following Hitchin's procedure, the author defines a map \(h_\lambda : {\mathcal H}^{-1} (\varphi) \to {\mathcal A}_\lambda\) and using her previous results and the fact that \(P_\lambda\) is isogenous to \({\mathcal A}_\lambda\) proves that the restriction of \(h_\lambda\) to each connected component of \({\mathcal H}^{-1} (\varphi)\) has finite fibres.
Similar results concerning isogenies between a generalized Prym variety not depending on the representation of \(G\) and spectral Prym-Tjurins are also given by \textit{R. Donagi} [in: Journées de géométrie algébrique, Orsay 1992, Astérisque 218, 145-175 (1993; Zbl 0820.14031)]. isogenies between a generalized Prym variety and spectral Prym-Tjurins; moduli space; Riemann surface; Prym-Tjurin variety R. Scognamillo, Prym - Tjurin varieties and the Hitchin map, Math. Ann. 303 (no. 1) (1995), 47--62. Picard schemes, higher Jacobians, Jacobians, Prym varieties, Holomorphic bundles and generalizations, Simple, semisimple, reductive (super)algebras, Group actions on varieties or schemes (quotients), Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Prym-Tjurin varieties and the Hitchin map | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f\) be a modular form of weight \(k\geq 4\) and level \(N\), and assume that there is a prime \(p\) exactly dividing \(N\). Let \(K\) be a quadratic imaginary field. Let \(L_p(f,K,s)\) be the anti-cyclotomic \(p\)-adic \(L\)-function constructed by \textit{M. Bertolini}, \textit{H. Darmon}, \textit{A. Iovita} and \textit{M. Spiess} [Am. J. Math. 124, No. 2, 411--449 (2002; Zbl 1079.11036)]. This \(p\)-adic \(L\)-function interpolates the central critical values (or derivatives) of the \(L\)-series \(L(f/K, s)\) twisted by Hecke characters of infinity type \((j-k/2,k/2-j)\) with \(1\leq j\leq k-1\).
The author proves a \(p\)-adic formula of Gross-Zagier type relating the \(p\)-adic Abel-Jacobi image of a generalized Heegner cycle on a Kuga-Sato-like variety fibered over a Shimura curve, to the special value of a \(p\)-adic \(L\)-function \(L_p(f,K,s)\) via the \(p\)-adic analytic uniformisation theory of Cherednik-Drinfeld. This result generalizes the results of \textit{M. Bertolini} and \textit{H. Darmon} [Duke Math. J. 98, 305--334 (1999; Zbl 1037.11045)], and \textit{A. lovita} and \textit{M. Spiess} [Invent. Math. 154, No. 2, 333--384 (2003; Zbl 1099.11032)], which considered only the central value \(s= k/2\). The main idea in the proof is to consider generalized Heegner cycles instead of the more standard Heegner cycles that appear in all the previous papers. CM cycles; Shimura curve; \(p\)-adic integration; anti-cyclotomic \(p\)-adic \(L\)-function; \(p\)-adic formula of Gross-Zagier type; generalized Heegner cycles; \(p\)-adic Abel-Jacobi map M. Masdeu, CM cycles on Shimura curves, and \textit{p}-adic \textit{L}-functions, Compos. Math. 148 (2012), no. 4, 1003-1032. \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Holomorphic modular forms of integral weight, Arithmetic aspects of modular and Shimura varieties, Algebraic cycles CM cycles on Shimura curves, and \(p\)-adic \(L\)-functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The computation of the \(\zeta\)-function of a variety \(V\) can always be carried through, if one motivically can split the cohomology into one-dimensional pieces. The method is essentially to write down a finite number of possibilities for the \(\zeta\)-function and test them by reduction \(\text{mod } p\) until all but one is excluded. In the article under review, the author considers motives of rank two over \(\mathbb{Q}\) with a non-degenerate symmetric pairing, and shows how in this case one may reduce the computation to the first simple case.
He applies this to the computation of the \(\zeta\)-function of the diagonal quartic \(\sum X_i= \sum X_i^{- 1}= 0\) K-3 surface, earlier computed by \textit{C. Peters}, \textit{J. Top} and \textit{M. van der Vlugt} [J. Reine Angew. Math. 432, 151-176 (1992; Zbl 0749.14037)], getting a simpler algorithm only needing to reduce for \(p= 2\). The proofs make essential use of \(p\)-adic Hodge theory as explained by \textit{G. Faltings} [J. Am. Math. Soc. 1, No. 1, 255-299 (1988; Zbl 0764.14012)]. motivic orthogonal two-dimensional representations; motives of rank two with a non-degenerate symmetric pairing; diagonal quartic K-3 surface; \(\zeta\)-function of a variety; algorithm; \(p\)-adic Hodge theory Livn\.e, R., Motivic orthogonal two-dimension representations of Gal\(\left(\overline{{\mathbf{Q}}}/{\mathbf{Q}}\right)\), Israel J. Math, 92, 149-156, (1995) Arithmetic aspects of modular and Shimura varieties, Representation-theoretic methods; automorphic representations over local and global fields, Local ground fields in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology Motivic orthogonal two-dimensional representations of \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let k be a perfect field of positive characteristic, X a smooth projective variety over k and \({\mathcal C}\) the category of augmented artinian local k-algebras. For \(A\in {\mathcal C}\) let \({\mathcal K}^ s_{n,X\times A/X}\) be the sub-sheaf of the sheaf of Quillen's K-groups on \(X\times_{spec k}spec A\) which is locally generated by Steinberg symbols with at least one factor \(\equiv 1 mod {\mathcal O}_ X\otimes_ k \max id A.\) In this paper the cohomology groups of these sheaves are analysed; more precisely, one studies the functor \(H^*(X,{\mathcal K}^ s_{*,X\times /X}):{\mathcal C}\to\) \{bigraded abelian groups\}.
It is shown that this functor admits a sort of Künneth decomposition with one factor solely depending on X and the other factor a functor on \({\mathcal C}\) independent of X. This decomposition gives an analogue of Cartier's formula relating a formal group to its Cartier-Dieudonné module. This result is remarkable because \(H^*(X,K^ s_{*,X\times /X})\) is usually far from being pro-representable. The analogue of a Cartier-Dieudonné module for this functor turns out to be the \(E_ 1\)- term \(H^*(X,W\Omega^._ X)\) of the slope spectral sequence for the crystalline cohomology of X [cf. \textit{L. Illusie} and \textit{M. Raynaud,} Publ. Math., Inst. Hautes Etud. Sci. 57, 73-212 (1983; Zbl 0538.14012)]. This result generalises the well-known connection between the formal Picard group, or more generally the Artin-Mazur groups [cf. \textit{M. Artin} and \textit{B. Mazur}, Ann. Sci. Éc. Norm. Super, IV. Ser. 10, 87- 132 (1977; Zbl 0351.14023)] and Witt vector cohomology. Because of Bloch's formula \(CH^ n(X)=H^ n(X,{\mathcal K}_{n,X})\) [cf. \textit{D. Quillen} in Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)] we like to think of \(H^*(X,{\mathcal K}^ s_{*,X\times /X})\) as the formal completion of the Chow group of codimension n cycles modulo rational equivalence on X, but it is not yet clear what our theorem about the structure of \(H^*(X,{\mathcal K}^ s_{*,X\times /X})\) really means for the Chow groups. As a first step in understanding the connection we present a formal analogue of the Abel-Jacobi map \(CH_ 0(X)\to Alb(X).\) As another step in understanding our result better we give, in case X is a K3- surface, a presentation for the groups \(H^ 2(X,{\mathcal K}_{2,X\times A/X})\) in terms of generators and relations, which is an intriguing mixture of a formal group law and the well-known presentation for \(K_ 2\). In the course of our analysis of \(H^*(X,{\mathcal K}^ s_{*,X\times /X})\) we give a new K-theoretic construction of the De Rham-Witt complex. Cartier-Dieudonné theory; sheaf of K-groups; augmented artinian local k-algebras; crystalline cohomology; formal Picard group; Artin-Mazur groups; Witt vector cohomology; Chow group Stienstra, J.: Cartier-Dieudonné theory for Chow groups. J. Reine Angew. Math. \textbf{355}, 1-66 (1985) (correction. J. Reine Angew. Math. \textbf{362}, 218-220 (1985)) Applications of methods of algebraic \(K\)-theory in algebraic geometry, (Equivariant) Chow groups and rings; motives, Formal groups, \(p\)-divisible groups, \(p\)-adic cohomology, crystalline cohomology, Cycles and subschemes, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) Cartier-Dieudonné theory for Chow groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author considers a smooth curve \(C\) lying on a (possibly singular) surface \(F\) in projective 3-space. If there exists a surface \(G\) intersecting \(F\) \((t+1)\)-fold along the curve \(C\), then \(C\) is called a curve of contact on \(F\). The main result of the paper gives necessary and sufficient conditions for \(C\) to be a curve of contact. The conditions involve ideals \(J_ s\) defining locally Cohen-Macaulay curves \(C_ s\) of \(C\) on \(F\) (if \(F\) is smooth along \(C\), then \(C_ s\) is just \(sC)\). curve on a surface; curve of contact; locally Cohen-Macaulay curves M. Boratyński, On the curves of contact on surfaces in a projective space, Algebraic \?-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989) Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 1 -- 8. Plane and space curves, Surfaces and higher-dimensional varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the curves of contact on surfaces in a projective space | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a non-singular projective curve of genus \(g\geq 2\) defined over the complex numbers, and let \(M_\xi\) denote the moduli space of stable bundles of rank \(n\) and determinant \(\xi\) on \(C\), where \(\xi\) is a line bundle of degree \(d\) on \(C\) and \(n\) and \(d\) are coprime. It is shown that a universal bundle \({\mathcal U}_\xi\) on \(C\times M_\xi\) is stable with respect to any polarisation on \(C\times M_\xi\). Similar results are obtained for the case where the determinant is not fixed and for the bundles associated to the universal bundles by irreducible representations of \(\text{GL} (n,\mathbb{C})\). It is shown further that the connected component of the moduli space of bundles with the same Hilbert polynomials as \({\mathcal U}_\xi\) on \(C\times M_\xi\) containing \({\mathcal U}_\xi\) is isomorphic to the Jacobian of \(C\). stability of the Poincaré bundle; determinantal variety; moduli space; Hilert polynomials; Jacobian Balaji, V.; Brambila-Paz, L.; Newstead, P. E., Stability of the Poincaré bundle, Math. Nachr., 188, 5-15, (1997) Vector bundles on curves and their moduli, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Stability of the Poincaré bundle | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We consider both the arithmetic and geometry of the curve in the title. Let two points of a curve be equivalent if the image of their difference in the Jacobian of the curve has finite order. An equivalence class is called a torsion packet. The Weierstrass points form a torsion packet and they are exactly the \(\mathbb{Q} (\zeta_{12})\)-rational points on this curve. The latter result is obtained from the fact that the Mordell-Weil group of the Jacobian over the field \(\mathbb{Q} (\zeta_{12})\) is finite. Since the Mordell-Weil group over the rationals is also finite, we can describe all solutions of the equation in fields of degree 3 or less over the rationals. In addition, we find bases for the 2- and 3-torsion of the Jacobian and describe an isogeny from the Jacobian to the product of three CM elliptic curves. The finiteness of the Mordell-Weil group was shown using a 3-descent on the Jacobian that did not make use of this isogeny. Jacobian variety; quartic curve; Weierstrass points; Mordell-Weil group; isogeny; CM elliptic curve Matthew J. Klassen and Edward F. Schaefer, Arithmetic and geometry of the curve \?³+1=\?\(^{4}\), Acta Arith. 74 (1996), no. 3, 241 -- 257. Rational points, Jacobians, Prym varieties, Cubic and quartic Diophantine equations, Riemann surfaces; Weierstrass points; gap sequences Arithmetic and geometry of the curve \(y^ 3+1=x^ 4\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The intersection \(W\) of the Grassmannian \(G\) of lines in \(\mathbb{P}^ 3\), considered as a quadric in \(\mathbb{P}^ 5\), with a quadric hypersurface \(H\) in \(\mathbb{P}^ 5\) is classically known as the quadric line complex. Associated to \(W\) is a Kummer surface \({\mathcal K}\) defined by the foci of pencils of lines in \(W\) [see e.g. \textit{P. Griffiths} and \textit{J. Harris}, ``Principles of algebraic geometry (New York 1978; Zbl 0408.14001)]. In the paper under consideration this construction is extended to a general ground field \(k\) of characteristic \(\neq 2\). The main theorem is:
Let \(W\) denote the complete intersection of 2 quadrics \(Q_ 1\) and \(Q_ 2\) in \(\mathbb{P}^ 5\). The variety \({\mathcal A} = \{ \text{lines in } W \}\) has a natural structure of a principal homogeneous space over the Jacobian of the genus 2 curve \({\mathcal E} = {\mathcal E} (Q_ 1, Q_ 2)\) defined by the equation: \(\text{det} (\lambda_ 1 Q_ 1 + \lambda_ 2 Q_ 2) = -\mu^ 2\).
Further it is shown that, if \({\mathcal A}\) contains a rational element, then \({\mathcal A}\) is in fact an abelian variety, and that given a genus 2 curve \({\mathcal C}\) defined by a degree 6 polynomial \(f\) with leading coefficient 1 there is a quadric \(H\) (explicitly given in terms of the coefficients of \(f)\) such that \({\mathcal C} = {\mathcal E} (G,H)\). quadric line complex; Kummer surface; Jacobian of the genus 2 curve \(K3\) surfaces and Enriques surfaces, Jacobians, Prym varieties, Complete intersections Jacobian in genus 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0539.00007.]
The authors discuss some of the interactions, resulting from the notion of the monodromy group, between group theory on the one hand and the study of Riemann surfaces and differential equations on the other hand. The paper has been written with group theory in mind and hence it contains some background material about Riemann surfaces and differential equations. The authors state that the paper is neither a survey article nor a historical account of monodromy groups, in particular, it contains new results: the introduction and application of combinatorial algorithms.
The paper provides a good introduction to the monodromy group of a branched covering and of the Hurwitz system used to obtain a cellular decomposition of it. It is described how to obtain the intersection matrix for the 1-cycles on the branched covering from the Hurwitz system. Furthermore, an algorithm is provided for calculating the intersection matrix, and it is shown how to determine a homology basis for it. There is a brief survey of results about monodromy groups of conformal self- mappings of Riemann surfaces, and a new combinatorial proof of a theorem of Hurwitz on biholomorphic self-mappings of finite order of a Riemann surface. The results are applied to determine the periods and quadratic periods of Abelian integrals on the Klein-Hurwitz curve and on the Fermat curve. Finally, there is a section on the monodromy group of a homogeneous linear differential equation. monodromy group of a branched covering; Hurwitz system; cellular decomposition; intersection matrix for the 1-cycles; homology basis; conformal self-mappings of Riemann surfaces; periods; quadratic periods; Abelian integrals; Klein-Hurwitz curve; Fermat curve; homogeneous linear differential equation Tretkoff, [Tretkoff and Tretkoff 84] C. L.; Tretkoff, M. D., Combinatorial group theory, Riemann surfaces and differential equations. contributions to group theory, 467--519, contemp. math., 33., \textit{Providence, RI: Amer. Math. Soc.}, (1984) Differentials on Riemann surfaces, Compact Riemann surfaces and uniformization, Coverings of curves, fundamental group, Low-dimensional topology of special (e.g., branched) coverings, Generators, relations, and presentations of groups, Transformation and reduction of ordinary differential equations and systems, normal forms, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Combinatorial group theory, Riemann surfaces and differential equations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be a field, complete with respect to a discrete valuation, and \({\mathcal O}_K\) its ring of integers. Let \(A/K\) be an abelian variety and \({\mathcal A}/{\mathcal O}_K\) its Néron model. The special fibre of \({\mathcal A}\) is an extension of a connected group scheme by a finite abelian group \(\Phi_K\), called the group of components of \({\mathcal A}\). The authors study the kernel \(\Psi_{K,M}\) of the natural map \(\Phi_K\to \Phi_M\) if \(M/K\) is a finite separable field extension. Their main result (which generalizes an unpublished result of \textit{McCallum}) is that the order of every element of \(\Psi_{K,M}\) divides the order of \(\text{Gal} (M/K)\). The proof makes use of the Weil restriction of the Néron model of \(A_M\) to \({\mathcal O}_K\) and relies on earlier results of \textit{B. Edixhoven} [Compos. Math. 81, No. 3, 291-306 (1992; Zbl 0759.14033)].
In the (larger) remainder of the paper, the authors present an example where \(\Psi_{K,M}\) is not annihilated by the exponent of \(\text{Gal} (M/K)\). Since this property was known previously to hold for the prime-to-\(p\)-part of \(\Phi_K\), this is in particular a result on the \(p\)-part of \(\Phi_K\), as the title promises. The example is the Jacobian of a special hyperelliptic curve; the main work consists in an explicit calculation of the stable reduction of this curve. stable reduction; discrete valuation field; Jacobian of hyperelliptic curve; Néron model; group of components; separable field extension; Weil restriction 7. B. Edixhoven, Q. Liu and D. Lorenzini, The p-part of the group of components of a Néron model, J. Alg. Geom.5(4) (1996) 801-813. Local ground fields in algebraic geometry, Arithmetic ground fields for abelian varieties The \(p\)-part of the group of components of a Néron model | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present book is a thorough presentation of the theory of function fields of curves over a real closed field, their Witt rings and Witt equivalence between such fields, where fields are called Witt equivalent if their Witt rings are isomorphic, and injectivity and surjectivity questions when passing from a Witt ring of a domain to that of certain ring extensions. There are many related aspects that are also presented in detail such as Witt equivalence of real holomorphy rings. Some of the sections consist of work published earlier by the author, some contain new material.
Section 1 gives an introduction to the basic notions and relevant known results such as Witt rings, orderings and valuations, and the Knebusch-Milnor exact sequence of Witt rings
\[
0\to WP\to WK\to \bigoplus WK(\mathfrak{p})
\]
where \(P\) is a Dedekind domain, \(K\) its quotient field, \(K(\mathfrak{p})\) the residue field at a maximal ideal \(\mathfrak{p}\), and where the sum ranges over all maximal ideals of \(P\). A short introduction to real curves is also included. In the situation where \(K\) is the function field of a real curve \(\gamma\) over a real closed field and \(R\) is the ring of regular functions on \(\gamma\), it is shown how the Knebusch-Milnor exact sequence extends to Knebusch's exact sequence
\[
0\to WR\to WK\to \bigoplus WK(\mathfrak{p})\to\mathbb{Z}^N\to 0
\]
where \(N\) is the number of semi-algebraically connected components of \(\gamma\). In Section 4, this exact sequence is revisited and it is shown that it splits, more precisely, that there is an exact sequence going from right to left in with the maps are suitably compatible with the ones in the original Knebusch sequence, i.e. in homological language, that the Knebusch sequence slices into and is patched by two split short exact sequences. It is furthermore shown that if \(P\) is the ring of polynomial functions on \(\gamma\) and if \(\gamma\) is affine semi-algebraically compact and semi-algebraically connected, then \(WP\) is a direct summand of \(WR\). These results from section 4 have appeared in earlier papers by the author [J. Algebra 301, No. 2, 616--626 (2006; Zbl 1158.11019); Int. J. Pure Appl. Math. 45, No. 1, 5--11 (2008; Zbl 1142.11329)].
Section 2 contains new results. It deals with versions of the Scharlau transfer and Scharlau's norm theorem in the case of bilinear forms over a local domain \(P\) with \(2\) invertible and where the extension is given by \(P[\sqrt{d}\,]\) for \(d\in P\) a non-square. The main problem in defining a transfer is the fact that \(P[\sqrt{d}\,]\) need not be free as a \(P\)-module. The transfer thus only yields a map \(s_*:WP[\sqrt{d}\,]\to WP/\mathfrak{c}\) where \(\mathfrak{c}=\{ a\in P\,|\,aP[\sqrt{d}\,]\subseteq P\}\) is the conductor. This can be used to show that the map \(WP\to WP[\sqrt{d}\,]\) need not be an epimorphism. One also obtains a version of Scharlau's norm theorem stating that if \(\xi\) is a bilinear form over \(P\), then \(a+b\sqrt{d}\in P[\sqrt{d}\,]\) is a similarity factor of \(\xi\otimes P[\sqrt{d}\,]\) if and only if the norm \(N(a+b\sqrt{d})\) (defined as an element in \(P/\mathfrak{c}\)) is a similarity factor of the bilinear form \(\overline{\xi}\) over \(P/\mathfrak{c}\).
Section 3 also contains new results. The question here is as follows. Let \(P\) be a domain and \(R\) be its integral closure. What can one say about the injectivity of \(WP\to WR\)? The main result of this section reads as follows. Let \(P\) be a Noetherian domain of dimension \(1\), \(R\) its integral closure, and suppose further that the units of \(R\) are in \(P\) and that the kernel of the natural Picard group map \(\text{Pic}(P)\to\text{Pic}(R)\) contains elements of order \(2\), then \(WP\to WR\) is not a monomorphism. It is shown how this result applies to explicit situations where \(P\) is the coordinate ring of a certain curve.
In the final Section 5, the author gives a proof of a result by the reviewer and Grenier-Boley [Forum Math., to appear] stating that two real fields \(K\) and \(L\) with \(u\)-invariant \(\leq 2\) (e.g. function fields of real curves over a real closed field) have isomorphic Witt rings if and only if \(K\) and \(L\) are \(\mathcal{X}\)-equivalent, i.e. there is a homeomorphism \(T:\mathcal{X}_K\to \mathcal{X}_L\) of the respective spaces of orderings and an isomorphism of square class groups \(t:\dot{K}/\dot{K}^2\to \dot{L}/\dot{L}^2\) such that for all \(\alpha\in\mathcal{X}_K\) one has \(a>0\) at \(\alpha\) iff \(t(a)>0\) at \(T(\alpha)\). This result is then extended to the real holomorphy rings \(\mathcal{H}_K\) and \(\mathcal{H}_L\) of \(K\) and \(L\), respectively, and it is shown that in the case where \(\mathcal{H}_K\) and \(\mathcal{H}_L\) are Dedekind domains, then they are Witt equivalent if \(K\) and \(L\) are tamely \(\mathcal{X}\)-equivalent, where ``tame'' means that the \(\mathcal{X}\)-equivalence is compatible with real valuations in a suitable sense. The section concludes with an appendix in which the author summarizes his earlier results on Witt equivalence between function fields of real curves over real closed fields and Witt equivalence between their rings of regular functions. Witt ring; Witt functor; Witt equivalence; real closed field; function field of a curve; Scharlau transfer; Scharlau's norm principle; Knebusch-Milnor exact sequence; real holomorphy ring Algebraic theory of quadratic forms; Witt groups and rings, Forms over real fields, Research exposition (monographs, survey articles) pertaining to number theory, Algebraic functions and function fields in algebraic geometry Witt morphisms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper the author gives an example of a simple 2-dimensional abelian variety over \(\mathbb{Q}\) with rank at least 20. This is done using the methods of an earlier paper by the author [Two simple 2-dimensional abelian varieties defined over \(\mathbb{Q}\) with Mordell-Weil group at least 19, C. R. Acad. Sci., Paris, Sér. I 321, No. 10, 1341-1345 (1995; Zbl 0859.11033)]. The idea is to produce 20 \(\mathbb{Q}\)-rational points of a curve \(C\) of genus two over \(\mathbb{Q}\), fix one of these points, say \(P_0\) and consider the classes of \(P'-P_0\) in the Jacobian variety \(J\) of \(C\), where \(P'\) runs through the other 19 points. Then one constructs a homomorphism \(\Phi:J (\mathbb{Q}) \to \mathbb{F}^N_2\) such that \(\Phi (P'-P_0)\) are \(\mathbb{F}_2\)-linearly independent and shows also that \(J(\mathbb{Q})\) has no torsion. This gives 19 as a lower bound for the rank of \(J(\mathbb{Q})\). The kernel of \(\Phi\) has an extra independent generator (since \(C\) satisfies a certain Galois property). This increases the lower bound for the rank of \(J(\mathbb{Q})\) by 1. The simplicity of \(J\) is proved in the same way as in the cited paper by the author. curve of genus two; simple 2-dimensional abelian variety; rank of Mordell-Weil group Abelian varieties of dimension \(> 1\), Arithmetic ground fields for abelian varieties An example of a simple 2-dimensional abelian variety defined over \(\mathbb{Q}\) with Mordell-Weil group of rank at least 20 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A well known theorem of Noether-Enriques-Petri says that a canonical curve C (i.e.: \(C\subseteq {\mathbb{P}}^{g-1}; g(C)=g; \omega_ C\simeq {\mathcal O}_ C(1))\) is projectively normal and is intersection of quadrics unless C is trigonal. Furthermore if C is trigonal the intersection of the quadrics through C is a rational ruled surface or the Veronese surface [\textit{P. Griffiths} and \textit{J. Harris,} ''Principles of algebraic geometry'' (1978; Zbl 0408.14001), p. 535]. In the paper under review the author studies a similar problem for curves of genus three embedded by complete linear series. Let L be a very ample line bundle of degree d on the curve X of genus three. Denote by \(\phi_ L(X)\) the image of X in the embedding: \(\phi_ L:X\hookrightarrow {\mathbb{P}}(H^ 0(L)^{\nu})=:{\mathbb{P}}^ n.\) Note that if \(d\geq 7\) then \(\phi_ L(X)\) is projectively normal [\textit{D. Mumford}, CIME \(3^ o\) Ciclo Varenna 1969, Quest. algebr. Varieties, 29-100 (1970; Zbl 0198.258)]. The cases \(d\leq 6\) have been previously studied by the author [Tsukuba J. Math. 4, 269-279 (1980; Zbl 0473.14015)]. If \(d\geq 8\) it follows from a theorem of \textit{B. Saint-Donat} [C. R. Acad. Sci., Paris, Sér. A 274, 324-327 (1972; Zbl 0234.14012)] that \(\phi_ L(X)\) is intersection of quadrics. So the only case left is \(d=7\). In the first part of the paper, the author describes the scheme \(Q(\phi_ L(X))\) defined by the ideal generated by the quadrics through \(\phi_ L(X)\). The last section contains a comment on the general case. From the result of Saint-Donat quoted above it follows that if g(X)\(\geq 2\) and \(\deg(L)=2g+1,\) then the homogeneous ideal of \(\phi_ L(X)\) can be generated by its elements of degree 2 and 3. The author makes the following conjecture: ''Assume X non hyperelliptic then: (1) if \(h^ 0(L\otimes \omega_ X^{-1})=2, Q(\phi_ L(X))\) is a rational ruled surface; (2) if \(h^ 0(L\otimes \omega_ X^{-1})=1, Q(\phi_ L(X))\) is the union of \(\phi_ L(X)\) and a line: (3) if \(h^ 0(L\otimes \omega_ X^{-1})=0, Q(\phi_ L(X))=\phi_ L(X).\)'' - Section 1 proves the conjecture for \(g=3\) and the paper ends with the proof of (1) for every g. very ample invertible sheaf on a curve of genus three; Noether-Enriques- Petri theorem; intersection of quadrics; curves of genus three embedded by complete linear series Special algebraic curves and curves of low genus, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic) Theorem of Enriques-Petri type for a very ample invertible sheaf on a curve of genus three | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This short introductory survey covers basic topics of toric geometry paying special attention to blowing-up and to resolution of singularities. Starting with the standard definition of a toric variety by gluing together affine toric varieties corresponding to the cones of a fan, the author discusses the relationship between orbital structure of toric varieties and combinatorics of their fans. The criteria of completeness and smoothness of a toric variety in terms of its fan are given. Further, the theory of Weil and Cartier divisors on a toric variety is considered. Support functions of invariant Cartier divisors are defined and it is shown that a divisor on a projective toric variety is globally generated (ample) iff its support function is (strictly) convex. On a smooth toric variety, every ample divisor is very ample (Demazure), and on any \(n\)-dimensional toric variety (\(n>1\)), the \((n-1)\)-th multiple of any ample divisor is very ample [see \textit{G. Ewald} and \textit{W. Wessels}, Result. Math. 19, No. 3/4, 275-278 (1991; 739.14031)]. Global sections of Cartier divisors are described, and it is shown that a projective toric variety \(X\) is recovered from the polytope of global sections of an ample divisor on \(X\).
Alternative constructions of toric varieties are given: as \(\text{Proj}\) of a graded ring (whose \(\text{Spec}\) is the affine cone over a projective toric variety), as a quotient of an open subset of an affine space by a quasitorus [see \textit{D. A. Cox}, J. Algebr. Geom. 4, No. 1, 17-50 (1995; Zbl 0846.14032)], and by toric ideals (in the affine case). Non-normal toric varieties are discussed. In terms of subdivisions of fans, the blow-up of a smooth toric variety along an invariant subvariety is described, and the resolution of singularities of any toric variety \(X\), bijective over \(X^{\text{reg}}\), is constructed. Finally, rationality of singularities of any toric variety is proven.
The survey contains no proofs, except for the construction of a toric desingularization and the rationality of singularities, where detailed proofs are given. Instead, simple instructive examples illustrate definitions and theorems of toric geometry are given. toric variety; fan; toric ideal; blow-up; desingularization; rational singularity; resolution of singularities; divisors David A. Cox, Toric varieties and toric resolutions, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 259 -- 284. Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves Toric varieties and toric resolutions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0741.00049.]
In the introduction of this paper the authors confess: ``This paper is intended as a tribute to the outstanding algebraic geometers F. Enriques and B. Segre \((\dots)\); it consists in a survey report on some recent results on linear systems of plane curves inspired by or related with a minor but interesting part of their work, and shows how one can find plenty of ideas also in little things.'' There are two theorems of B. Segre and one of F. Enriques which are involved in this interesting and very actual report. The first theorem of B. Segre refers to complete intersections and the postulation problem: ``Let \(Z\) be a closed 0- dimensional subscheme of a nonsingular curve \(C\) of degree \(d\geq 3\), such that the degree of \(Z\) is \(\delta=e\cdot d\) and \(H^ 1({\mathcal I}_ Z(e+d-3))\neq 0\); then there exists a curve \(C'\) of degree \(e\) such that \(C\cap C'=Z''\). This theorem was first completely proved by \textit{E. Davis} [in Curves Semin. Queen's, Vol. 3, Kingston/Can. 1983, Queen's Pap. Pure Appl. Math. 67, Exposé D (1984; Zbl 0596.14037)]. In a paper announced by \textit{S. Greco} it will be proved the following result: if \(C\) is a complete integral curve in a projective space, of degree \(d\) and genus \(g\), \(Z\subset C\) a zero-dimensional subscheme, if \(\lambda=\min\{h^ 0(\text{Coker}f);f:{\mathcal I}_ 2\to\omega_ c\) is injective\} and \(n>(\delta-\lambda+2g-2)/d\), then \(h^ 1({\mathcal I}_ Z(n))=0\).
The second theorem of B. Segre discusses the bound of regularity of the linear system of all plane curves of degree \(t\) containing a fat point \(Z\) as a subscheme. Many types of fat points are considered.
In the last section it is discussed the following theorem of \textit{F. Enriques}: every plane integral curve can be obtained as a projection of a nonsingular curve in \(\mathbb{P}^ 4\) from a straight line. It is announced a complete proof due to \textit{L. Caire} [cf. Manuscr. Math. 67, No. 4, 433-450 (1990; Zbl 0728.14030)]. linear systems of plane curves; fat point; projection of a nonsingular curve Maria Virginia Catalisano and Silvio Greco, Linear systems: developments of some results by F. Enriques and B. Segre, Geometry and complex variables (Bologna, 1988/1990) Lecture Notes in Pure and Appl. Math., vol. 132, Dekker, New York, 1991, pp. 41 -- 57. Divisors, linear systems, invertible sheaves, History of algebraic geometry, Plane and space curves, Projective techniques in algebraic geometry, History of mathematics in the 19th century, History of mathematics in the 20th century Linear systems: Developments of some results by F. Enriques and B. Segre | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be a number field and \(C\) be an algebraic curve of genus \(g\geq 2\) defined over \(K\). Put
\[
N(g,K)=\lim \sup_C\# C(K) \text{ and }N (g)= \max_KN(g,K).
\]
J. F. Mestre and Brumer independently obtained that \(N(g) \geq 16(g+1)\). In this paper, the author improves this result for \(g=2,3,5,6\) and 7. He proves, by means of some infinite families of curves passing through more than \(16(g+1)\) rational points or using a number field of degree smaller than \(\varphi (g+1)\), that \(N(2,\mathbb{Q})\geq 66\), \(N(3,\mathbb{Q})\geq 72\), \(N(5,\mathbb{Q}) \geq 96\), \(N(6,\mathbb{Q})\geq 96\), \(N(7,\mathbb{Q})\geq 128\), \(N(7,\mathbb{Q}(\sqrt 2))\geq 128\), \(N(11,\mathbb{Q}(\sqrt 3))\geq 192\). genus \(\neq 1\); hyperelliptic involution; group of automorphisms of a curve; elliptic curves; isogenies; many rational points; algebraic curve; families of curves Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rational points Algebraic curves of genus \(\geq 2\) having numerous rational points | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\varphi :\Sigma_1\longrightarrow{\mathbb{P}}^2\) be a blow up at a point on \({\mathbb{P}}^2\). Let \(C\) be the proper transform of a smooth plane curve of degree \(d\ge 4\) by \(\varphi\), and let \(P\) be a point on \(C\). Let \(\pi :{\tilde{C}}\longrightarrow C\) be a double covering branched along the reduced divisor on \(C\) obtained as the intersection of \(C\) and a reduced divisor in \(|-2K_{\Sigma_1}|\) containing \(P\). In this paper, we investigate the Weierstrass semigroup \(H({\tilde{P}})\) at the ramification point \({\tilde{P}}\) of \(\pi\) over \(P\), in the case where the intersection multiplicity at \(\varphi (P)\) of \(\varphi (C)\) and the tangent line at \(\varphi (P)\) of \(\varphi (C)\) is \(d-1\). Weierstrass semigroup; double covering of a curve; Hirzebruch surface; normalization of a curve Rational and ruled surfaces, Special divisors on curves (gonality, Brill-Noether theory), Riemann surfaces; Weierstrass points; gap sequences New examples of Weierstrass semigroups associated with a double covering of a curve on a Hirzebruch surface of degree one | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an absolutely irreducible normal variety defined over a finite field \(k=\mathbb{F}_q\) with \(q=p^a\) elements. It is proved in this paper that the characteristic roots of the \(L\)-function \(L(t,R,X)\) associated to the Artin-Schreier cover \(Y\colon Z^p - Z=R(x)\) of \(X\) differ from the characteristic roots of the zeta function \(Z(t,Y,q)\) of \(YY\) relative to the field of definition \(k\) by roots of unity. This gives an affirmative answer to a conjecture of \textit{E. Bombieri} [Am. J. Math. 88, 71--105 (1966; Zbl 0171.41504)]. normal variety defined over a finite field; characteristic roots of the L-function; characteristic roots of the zeta function Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Exponential sums Exponential sums and forms for varieties over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the review in Zbl 0455.32012. generic embedding of an elliptic curve in a complex surface Iljashenko, Y. and Pyartli, A. , Neighborhoods of zero type in embedded complex tori . In: Topics in Modern Mathematics. Transl. from Petrovskii Sem. No. 5 (ed. O.A. Oleinik), New York 1985. Deformations of submanifolds and subspaces, Elliptic curves, Embeddings in algebraic geometry Neighborhoods of zero type in embedded complex tori | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A Kuga fiber variety is a family of abelian varieties parametrized by an arithmetic variety and constructed from a symplectic representation of an algebraic group. A lower bound for the field of definition of a complex algebraic variety \(X\) is given by Bot\(X\), the strong bottom field; this is a field \(k\) such that for any automorphism \(\sigma\) of \(\mathbb{C}\), \(X^ \sigma\cong X\) if and only if \(\sigma\) is the identity on \(k\). The principal result of this paper is that if \(A\to V\) is a Kuga fiber variety defined by a \(\mathbb{Q}\)-irreducible representation satisfying a certain rigidity condition, and if the generic fibers are principal abelian varieties, then Bot\(A\) is an abelian extension of Bot\(V\). In fact, the Galois group of Bot\(A\) over Bot\(V\) is embedded into the class group of a maximal order in a simple algebra. Kuga fiber variety; arithmetic variety; field of definition; strong bottom field; class group of a maximal order Abdulali, S.: Conjugation of Kuga fiber varieties, Math. Ann.294, 225-234 (1992). Algebraic theory of abelian varieties, Relevant commutative algebra, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Conjugation of Kuga fiber varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this interesting paper, the authors extend considerably our knowledge about local-global principles (or their failure) in Galois cohomology over certain types of base fields. The general set-up is as follows. Let \(F\) be a field and \((F_v)_{v\in \Omega}\) a collection of overfields \(F_v\supseteq F\). In our context, \(\Omega\) will always be a set of valuations on \(F\) and \(F_v\) will denote the completion of \(F\) with respect to \(v\in\Omega\). Consider furthermore a field extension \(E/F\) which may be finite, in which case one puts \(E_v=E\otimes_F F_v\), or \(E=F(Y)\) is the function field of a geometrically integral algebraic variety over \(F\), in which case one puts \(E_v=F_v(Y_{F_v})\). If \(\mu\) is now a \(\hbox{Gal}(\overline{E}/E)\)-module (where \(\overline{E}\) denotes an algebraic closure of \(E\)), one considers the group \[{Ш}_{\Omega}^n(E,\mu)= \ker\left[\,H^n(E,\mu)\to\prod_{v\in\Omega}H^n(E_v,\mu)\,\right],\] (where \(H^n(\cdot,\cdot)\) denotes Galois cohomology). If this group is trivial, then one says that a local-global principle holds for the given data \(F,E,F_v,\Omega,\mu\). For example, the Albert-Brauer-Hasse-Noether theorem implies that if \(F\) is a number field, \(\Omega\) is the set of places of \(F\), \(F_v\) are the respective completions, \(E\) is a finite extension of \(F\) and \(\ell\) is a prime and \(\mu_\ell\) denotes the \(\ell\)-th roots of unity in \(\overline{E}\), then \({Ш}_{\Omega}^2(E,\mu_\ell)=0\). In this case, \(H^2(E,\mu_\ell)\) is nothing else but the \(\ell\)-torsion of the Brauer group \(\hbox{Br}(E)\). While this generally will no longer hold when \(E=F(Y)\) for a curve \(Y\) over a global field \(F\), \textit{K. Kato} [J. Reine Angew. Math. 366, 142--183 (1986; Zbl 0576.12012)] showed that in this situation one still gets \({Ш}_{\Omega}^3(F(Y),\mu_\ell^{\otimes 2})=0\).
The authors are mainly interested in getting new results in the case where \(F\) is a so-called semi-global field, i.e., a function field of transcendence degree \(1\) over a complete discretely valued field \(K\), and \(E=F(Y)\) for some curve \(Y\) over \(F\). In this setting, let \(m>1\) be an integer prime to the residue characteristic of \(K\), let \(\Omega_F\) denote the set of all so-called divisorial discrete valuations on \(F\) and let \(F_v\) be the completion at \(v\in\Omega_F\). Then the authors show (Theorem 5.2) that in the case \(E=F(t)\), one obtains that \({Ш}_{\Omega_F}^n(E,\mu_m^{\otimes n-1})\) is trivial for \(n\geq 3\), nontrivial for \(n=2\), and they provide a necessary and sufficient criterion for triviality for \(n=1\). The triviality for \(n\geq 3\) still holds when \(E\) is the function field of a regular projective curve over \(F\) under the assumption that \(K\) is a non-Archimedean local field (Theorem 5.5).
The authors then consider only the set \(\Omega_{F,0}\) of those discrete valuations on \(F\) that are trivial on \(K\). Assuming that \(F\) is the function field of a smooth projective curve over a non-Archimedean local field \(K\) and \(E=F(t)\), it is shown that \({Ш}_{\Omega_{F,0}}^n(E,\mu_m^{\otimes n-1})\) is nontrivial for \(n=2,4\), and a necessary and sufficient criterion for triviality for \(n=3\) is also provided (Theorem 5.12). Galois cohomology; local-global principle; semi-global field; function field of a curve; discrete valuation Galois cohomology of linear algebraic groups, Arithmetic ground fields for curves, Linear algebraic groups over arbitrary fields, Curves over finite and local fields, Galois cohomology, Other nonalgebraically closed ground fields in algebraic geometry Local-global principles for curves over semi-global fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0607.00004.]
If X denotes a projective threefold and \(Z\to F\) a family of algebraic 1- cycles on X, F a smooth projective variety, then one obtains the corresponding Abel-Jacobi map Alb(F)\(\to J(X)\), J(X) the intermediate Jacobian. - Grothendieck's generalized Hodge conjecture says that given any polarized Hodge substructure \(K\subset H^ 3(X,{\mathbb{Q}})\) perpendicular to \(H^{3,0}(X)\) there exists a corresponding abelian subvariety of J(X) that can be parametrized by algebraic cycles.
This preprint deals with the case \(H^{3,0}(X)\neq 0\). In a later paper the author will prove that for the universal family of smooth complete intersections of four quadrics in \({\mathbb{P}}^ 7\) with a fixed point free involution \(\sigma\) Grothendieck's conjecture is true for a generic element of that family. (In this case \(K=H^ 3(X,{\mathbb{Q}})^-:=\{x\in H^ 3(X,{\mathbb{Q}})| \quad \sigma x=-x\}\) corresponds to an abelian subvariety \(J^-(X)\) of J(X) that can be parametrized by 1-cycles. threefold; family of algebraic 1-cycles; Abel-Jacobi map; intermediate Jacobian; Grothendieck's generalized Hodge conjecture Parametrization (Chow and Hilbert schemes), \(3\)-folds, Transcendental methods, Hodge theory (algebro-geometric aspects) On Grothendieck's generalized Hodge conjecture for a family of threefolds with geometric genus one. (A study of the Abel-Jacobi map) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus \(g\geq 2\) whose Jacobian has Mordell-Weil rank \(r<g\). This paper shows that the hypothesis on the Mordell-Weil rank is essential. This is done by exhibiting for each prime \(p\geq 5\) an explicit family of curves of genus \((p-1)/2\) and rank at least \((p-1)/2\) for which this bound is violated. The examples in question are given by
\[
y^2=(x-a_0) (x-a_1) \dots (x-a_{p-1}) +p^2d^2,
\]
where \(p\geq 5\) is prime, \(d\) is any non-zero integer and \(a_0, \dots, a_{p-1}\) are any integers pairwise distinct modulo \(p\). curve of genus \(g\geq 2\); Coleman-Chabauty bound; upper bound; number of rational points; Jacobian; Mordell-Weil rank Curves over finite and local fields, Jacobians, Prym varieties, Arithmetic ground fields for curves Sur la borne de Coleman-Chabauty | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A. Nadel proved that for a Fano fourfold \(X\) with \(b_ 2(X)=1\) one has \(c_ 1(X)^ 4 \leq 20^ 4\). The proof was based on an analytic theorem to the effect that the non-integrability loci of certain almost plurisubharmonic functions on families of Riemann spheres are saturated. In the note under review the author generalizes this theorem to families of Riemann surfaces of arbitrary genus. This yields the following main theorem: If \(X\) is a smooth projective variety, \(\dim X=n\), \(L\) is an ample line bundle on \(X\) and \(C\) is a ``sufficiently movable'' curve ``immersed'' in \(X\) such that \((L^ n)^{1/n}>n(LC)\), then \(X\) contains an effective divisor \(E\) such that \((EC)=0\) (in particular, the Picard number of \(X\) is greater than 1).
Since, according to Mori, a Fano variety \(X\) contains a movable rational curve of degree \(d(X)\) not exceeding \(n+1\), from this it follows that if \(X\) is a Fano variety with \(b_ 2(X)=1\), then \(c_ 1(X)^ n \leq \bigl( nd(X) \bigr)^ n \leq \bigl( n(n+1) \bigr)^ n\). This generalizes Nadel's result for fourfolds and yields finiteness of the number of deformation families of Fano varieties of fixed dimension with \(b_ 2(X)=1\) (this last result was also proved by Nadel and Tsuji by different methods).
(For the detailed version see the paper in Bull. Soc. Math. Fr. 119, No. 4, 479-493 (1991) mentioned above). product theorem of Nadel; bound for Chern class; Fano variety; movable rational curve Campana, F.: Une version géométrìque généralisée du théorème de produit de Nadel. C.R. Acad. Sci. Paris312, 853-856 (1991) Fano varieties, Characteristic classes and numbers in differential topology, Formal methods and deformations in algebraic geometry Une version géométrique généralisée du théorème du produit de Nadel. (A generalized geometric version of the product theorem of Nadel) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(N\) be a square-free positive integer. This paper proves an estimate comparing the Arakelov metric for the smooth projective curve \(X_0(N)\) with the Poincaré metric \((dx\wedge dy)/y^2\) on its universal covering space, the upper half plane \(\mathcal H\). More specifically, for \(N\) as above let \(g\) be the genus of \(X_0(N)\), let \(f_1,\dots,f_g\) be an orthonormal basis for \(\mathcal S(2,\Gamma_0(N))\) under the Petersson inner product, and let \(F_N=\frac{y^2}g\sum_{i=1}^g| f_i| ^2\). This is the ratio between the metrics of Arakelov and Poincaré. The paper shows that for all \(\varepsilon>0\) there exists \(A_\varepsilon>0\) such that for all \(N\) as above and all \(z\in\mathcal H\), the inequality \(F_N(z)\leq A_\varepsilon N^{1+\varepsilon}\) holds.
An intermediate result in the above proof is the following. For all \(\varepsilon>0\) there is a constant \(C_\varepsilon>0\) such that for all \(N\) as above, all newforms \(f\) for \(\Gamma_0(N)\) (required to be normalized eigenforms for the Hecke operators), and all \(z\in\mathcal H\), the inequality \(| yf(z)| \leq C_\varepsilon N^{1/2+\varepsilon}\) holds.
Finally, as a corollary of the main theorem, the following two assertions are equivalent:
(i) there is an upper bound, polynomial in the conductor, for the degree of a strong modular parametrization of a semi-stable elliptic curve \(E\) over \(\mathbb{Q}\) (strong means that the induced map \(H_1(X_0(N),\mathbb{Z})\to H_1(E,\mathbb{Z})\) is surjective); and
(ii) the modular height of such an elliptic curve satisfies an upper bound that is linear in \(\log N\). modular height; Arakelov metric; Poincaré metric; modular parametrization of a semi-stable elliptic curve; modular curve Abbes, A.; Ullmo, E., Comparaison des métriques d'arakelov et de Poincaré sur \(X_0(N)\), Duke Math. J., 80, 2, 295-307, (1995) Arithmetic varieties and schemes; Arakelov theory; heights, Arithmetic aspects of modular and Shimura varieties, Hecke-Petersson operators, differential operators (one variable), Modular and Shimura varieties, Arithmetic ground fields for curves Comparison of Arakelov and Poincaré metrics on \(X_0(N)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Les auteurs étudient la structure galoisienne des anneaux d'entiers des extensions \(N\) d'un corps de nombres \(K\) qui sont engendrées par les points de division rationnels sur \(K\) d'une courbe elliptique définie sur le corps \(K\).
Dans une première partie algébrique, ils rappellent la définition de l'invariant de Picard d'un espace homogène principal pour un ordre de Hopf, ce qui leur permet de traduire en termes géométriques les problèmes de structure galoisienne, et s'intéressent ensuite dans une seconde partie de nature arithmétique au cas des ordres de Hopf provenant des schémas en groupes associés aux courbes elliptiques. Pour les points d'ordre fini, les résultats de \textit{A. Srivastav} et \textit{M. J. Taylor} [Invent. Math. 99, 165-184 (1990; Zbl 0705.14031)] et de \textit{K. Bouklou} [Arithmétique d'espaces homogènes principaux associés à une courbe elliptique, Thèse Bordeaux (1996)] montrent que l'invariant de Picard est généralement trivial. Pour le points d'ordre infini à invariant de Picard trivial, la théorie d'Iwasawa permet de montrer que les classes des espaces homogènes principaux libres sur l'ordre maximal proviennent du sous-groupe de Greenberg du groupe de Mordell-Weil complété. Les auteurs présentent alors leur résultat obtenu dans un travail sur la détermination de générateurs explicites de ces modules libres [Ann. Inst. Fourier 44, 631-661 (1994; Zbl 0810.11039)] et mettent plus particulièrement l'accent sur le rôle déterminant de la fonction \(L\) \(p\)-adique de la courbe et des unités elliptiques dans cette construction. Galois structure; elliptic curves; Hopf orders; principal homogeneous space; explicit determination of generators; group schemes; \(p\)-adic \(L\)-function of an elliptic curve; elliptic units; Picard invariant Cassou-Noguès, Structures galoisiennes et courbes elliptiques, J. Théor. Nombres Bordeaux 7 pp 307-- (1995) Integral representations related to algebraic numbers; Galois module structure of rings of integers, Elliptic curves over global fields, Group schemes Galois structure and elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The problem is considered of classifying all univariate polynomials defined over a domain \(k\) whose derivatives (up to a certain degree) have their roots also in \(k\). Usually, \(k\) is the field \({\mathbb Q}\) of rationals or its ring \({\mathbb Z}\) of integers. The domain \(k ={\mathbb Q}(\sqrt{m})\) of a quadratic field or its ring \({\mathcal O}\) of integers is also treated. The problem is of interest in algebraic geometry also since it is intimately related to that of finding rational points on elliptic or hyperelliptic curves. For instance, the Jacobian \(J({\mathbb Q})\) of the hyperelliptic curve
\[
C: z^2 = 9w^6+195w^4+975w^2+1125
\]
is isogenous to a degenerate algebraic surface, namely the product of two elliptic curves. It finally turns out to be isomorphic to the product of the Kleinian four group and two copies of \({\mathbb Z}\):
\[
J({\mathbb Q}) \cong {\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}^2.
\]
In particular, it has rank 2. (Additonally, the elliptic curves
\[
E_m: mz^2 = w(w-6)(w+18)
\]
for various \(m \in {\mathbb Z}\), \(m > 0\), have rank 1 or 2 over the quadratic fields \({\mathbb Q}(\sqrt{m})\).) However, the classification problem is not completely solved. For let \(p_{(m_1,m_2,\ldots,m_r)} (x) \in {\mathbb Z}[x]\) be a polynomial of degree \(n\) with \(r\) distinct roots, where the \(i\)-th root has multiplicity \(m_i\) and \(n = m_1+m_2+ \cdots + m_r\). Then the problem is unsolved for the quartic polynomial \(p_{(1,1,1,1)}(x)\) and the quintic polynomials \(p_{(1,1,1,1,1)}(x)\), \(p_{(2,1,1,1)}(x)\) and \(p_{(3,1,1)}(x)\). But it is solved modulo the truth of two conjectures. These conjectures take care of the higher cases too. (We mention that for genus one curves Tate's Haverford College Notes [see \textit{J. H. Silverman} and \textit{J. Tate}, Rational points on elliptic curves. New York: Springer (1992; Zbl 0752.14034)] and for genus two curves \textit{J. W. S. Cassels}'s and \textit{E. V. Flynn}'s book [Prolegomena to a middlebrow arithmetic of curves of genus 2. Cambridge Univ. Press (1996; Zbl 0857.14018)] is cited.) polynomial; derivative; quartic Diophantine equation; elliptic curve; hyperelliptic curve; Jacobian variety; algebraic surface R. H. Buchholz, J. A. MacDougall, When Newton met Diophantus: A study of rational-derived polynomials and their extension to quadratic fields, J. Number Theory 81 no. 2 (2000) 210-233. Cubic and quartic Diophantine equations, Higher degree equations; Fermat's equation, Polynomials in number theory, Elliptic curves, Polynomials (irreducibility, etc.), Special surfaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Jacobians, Prym varieties, Elliptic curves over global fields When Newton met Diophantus: a study of rational-derived polynomials and their extension to quadratic fields. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author states an estimate about the number of integral points on a Thue curve and a property of the rational points. The proofs of these results are given in [\textit{M. Fujimori}, Tôhoku Math. J., II. Ser. 46, 523-539 (1994; Zbl 0828.11015)]. estimate; number of integral points on a Thue curve; rational points Higher degree equations; Fermat's equation, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rational points, Arithmetic ground fields for curves On the solutions of Thue equations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article is a nice review of recent and not quite so recent results on Shimura curves: their complex uniformization and modular interpretation, the theorem of Cherednik-Drinfeld, which says that Shimura curves are, for suitable primes \(p\), Mumford curves and describe explicitly their \(p\)-adic uniformization, and finally the theorem of Ribet, who found a surprising relation between the Jacobians of a \(p\)-adic Shimura curve and certain \(q\)-adic modular curves, for different primes \(p\) and \(q\). complex uniformization; Mumford curves; Jacobians of a \(p\)-adic Shimura curve; \(q\)-adic modular curves Modular and Shimura varieties, Arithmetic ground fields for curves, Arithmetic ground fields for abelian varieties Shimura curves. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Der Verfasser hat eine grosse Arbeit, die alle in dieser Note befindlichen Resultate enthält, im Jahre 1872 veröffentlicht. Siehe F. d. M. IV p. 329, JFM 04.0329.02. Theorems of Pascal and Brianchon; extension; plane curves; surfaces of the third order; Cartesian higher geometriy; conjugated polygons; inscribed into a curve; conic section Desarguesian and Pappian geometries, Euclidean geometries (general) and generalizations, Plane and space curves, Families, moduli, classification: algebraic theory, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Curves in Euclidean and related spaces, Polyhedra and polytopes; regular figures, division of spaces, Euclidean analytic geometry Note on extending the theorems of Pascal and Brianchon to the plane curves and to the surfaces of \(3^rd\) class. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0651.00010.]
Let L be an algebraic link, i.e. the intersection of a small sphere about the origin in \({\mathbb{C}}^ 2\) with a complex algebraic surface \(f(z,w)=0\) in \({\mathbb{C}}^ 2\) which has an isolated singular point at the origin. \textit{D. Eisenbud} and \textit{W. Neumann} [Three-dimensional link theory and invariance of plane curve singularities, Ann. Math. Stud. 110 (Princeton, New Jersey) (1985; Zbl 0628.57002)], devised a procedure, based on the polynomial f, for representing L as an ``iterated torus link'' and in particular as a positive closed braid (``positive'' means that all crossings are of the same sign). In the present article the author proves that this braid is actually ``very positive'', i.e. contains the full twist as a factor, and is therefore a minimal closed braid representative of L, by a theorem of \textit{J. Franks} and the author [Trans. Am. Math. Soc. 303, 97-108 (1987; Zbl 0647.57002)] and \textit{H. R. Morton} [Closed braid representatives for a link and its Jones-Conway polynomial, Preprint (1985)]. This means that one can calculate the braid index of L directly from f. isolated singularity of a complex curve; algebraic link; plane curve singularities; iterated torus link; positive closed braid; minimal closed braid representative Knots and links in the 3-sphere, Singularities in algebraic geometry, Local complex singularities The braid index of an algebraic link | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Fix a prime \(p\) and an integer \(t\ge 2\). Set \(q_0:= p^t\) and \(q:= p^{2t-1}\). Let \(A_p\) the smooth projective model of the plane \(y^q-y=x^{q_0}(x^q-x)\). \(A_2\) is the famous Suzuki curve which is maximal over \(\mathbb {F}_{q_4}\). This paper introduces and study the curve \(A_p\), \(p\) odd, calling it Generalized Suzuki functions. The authors compute its zeta function (over \(\mathbb {F}_{q^n}\) it is a maximal curve for its genus \(q_0(q-1)/2\) if and only if \(p\equiv 3\mod{4}\) and \(n\equiv 2\pmod{4}\)). They compute their L-function and their automorphism group, which has order \(q^2(q-1)\). They compute the automorphism group of more general curves, the smooth model of the plane curves \(y^q-y=x^{q_0}(x^q-x)\) with \(q=p^m\) and \(q_0=p^t\) with \(2t>m\le t\), solving a conjecture by \textit{M. Giulietti} and \textit{G. Korchmáros} [Radon Ser. Comput. Appl. Math. 16, 93--120 (2014; Zbl 1332.14037)]. curve over a finite field; Suzuki group; automorphism group of a curve; maximal curve Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Curves over finite and local fields, Rational points, Automorphisms of curves On the zeta function and the automorphism group of the generalized Suzuki curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0671.00022.]
The authors improve the algorithms to determine the topology of a real plane curve (given by a polynomial F(X,Y)\(\in {\mathbb{Z}}[X,Y])\) proposed by several authors [cf. \textit{P. Gianni} and \textit{G. Traverso}, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 29, 87-109 (1983; Zbl 0557.14011) and \textit{D. S. Arnon} and \textit{S. McCallum}, J. Symb. Comput. 5, No.1/2, 213-236 (1988; Zbl 0664.14017)]. The authors compute the analytic components of the given curve: the output consists of the planar graph (which is homeomorphic to \(C=\{(x,y): F(x,y)=0\}\in {\mathbb{R}}^ 2)\) with the edges numbered in such a way that each number labels a distinguished global real analytic component of C. To do so, the authors calculate the discriminant locus, the points of the curve lying over this set and for each of those points the half-branches match together. Then the edges belonging to the same component are collected together by just a pursuit of the component of the graph. topology of a real plane curve; analytic components; real analytic component Cucker, F., Pardo, L. M., Raimondo, M., Recio, T., Roy, M. F.: On local and global analytic branches of a real algebraic curve. Lecture Notes in Computer Science, Vol. 356, 161--182, Berlin, Heidelberg, New York: Springer 1989 Software, source code, etc. for problems pertaining to algebraic geometry, Topological properties in algebraic geometry, Curves in algebraic geometry, Real algebraic and real-analytic geometry On the computation of the local and global analytic branches of a real algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be the Grassmann variety of lines in \(\mathbb{P}^4\), and embed it in \(\mathbb{P}^9= \mathbb{P}(\bigwedge^2V)\) by the Plücker embedding. Let \(X\) be a quadratic complex of lines in \(\mathbb{P}^4\), namely the intersection of \(G\) with a quadric \(Q\subset\mathbb{P}^9\). Then \(X\) is a Fano fivefold of index 3 whose cohomology group \(H^5(X)\) carries a Hodge structure of level one with \(h^{2,3} (X)=10\). The main theorem of this paper states that the generalized Hodge conjecture GHC\((X,5,2)\) holds if \(X\) is general. For the proof, the author employs the Fano variety \(F_X\) of two-planes contained in \(X\), which is shown to be a smooth connected curve of genus 161 when \(X\) is general, and shows that the cylinder homomorphism \(H_1(F_X,\mathbb{Z})\to H_5(X,\mathbb{Z})\) associated to the family \(F_X\) is surjective. Grassmann variety; Plücker embedding; complex of lines; generalized Hodge conjecture; Fano variety Nagel, J., \textit{the generalized Hodge conjecture for the quadratic complex of lines in projective four-space}, Math. Ann., 312, 387-401, (1998) Transcendental methods, Hodge theory (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Pencils, nets, webs in algebraic geometry The generalized Hodge conjecture for the quadratic complex of lines in projective four-space | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0518.00004.]
The Néron-Severi group of a (nonsingular projective) variety is, by definition, the group of divisors modulo algebraic equivalence, which is known to be a finitely generated abelian group. Its rank is called the Picard number of the variety. Thus the Néron-Severi group is defined in purely algebro-geometric terms, but it is a rather delicate invariant of arithmetic nature. Perhaps, because of this reason, it usually requires some nontrivial work before one can determine the Picard number of a given variety, let alone the full structure of its Néron-Severi group. This is the case even for algebraic surfaces over the field of complex numbers, where it can be regarded as the subgroup of the cohomology group \(H^ 2(X,{\mathbb{Z}})\) characterized by the Lefschetz criterion. Now the purpose of the present paper is to find certain explicitly defined curves on the complex Fermat surface \((1.1)\quad X^ 2_ m:\quad x^ m+y^ m+z^ m+w^ m=0\) whose cohomology classes (equivalently, algebraic equivalence classes) form (a part of) generators of the Néron-Severi group \(NS(X^ 2_ m)\otimes {\mathbb{Q}}\). divisors modulo algebraic equivalence; Picard number; complex Fermat surface; generators of the Néron-Severi group N. Aoki and T. Shioda, ``Generators of the Néron-Severi group of a Fermat surface'' in Arithmetic and Geometry, Vol. I , Progr. Math. 35 , Birkhäuser Boston, Boston, 1983, 1-12. Surfaces and higher-dimensional varieties, Special surfaces, (Equivariant) Chow groups and rings; motives, Picard groups Generators of the Néron-Severi group of a Fermat surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(S= \mathbb{K}[x_0,\dots, x_n]\) where \(\mathbb{K}\) is an algebraically closed field, and let \(M\) be a graded module over \(S\). We say that \(M\) is \(r\)-regular if the \(i\)th syzygy module of \(M\) is generated in degrees less than or equal to \(r+ i\) for all \(i\). The regularity of \(M\), denoted \(\text{reg}(M)\), is defined to be the infimum of all \(r\)'s such that \(M\) is \(r\)-regular. If \(X\) is a subscheme of \(\mathbb{P}^n\), then we define the regularity of \(X\), denoted \(\text{reg}(X)\), to be the regularity of the saturated homogeneous ideal of \(X\), \(I_X\).
\textit{D. Eisenbud} and \textit{S. Goto} [J. Algebra 88, No. 1, 89--133 (1984; Zbl 0531.13015)] conjectured that if \(X\) is a reduced nondegenerate connected in codimension 1 subscheme of \(\mathbb{P}^n\), then the regularity of \(X\) is at most the degree of \(X\) minus the codimension of \(X\) plus 1. Various weakenings of this conjecture are known to be true in low dimension, but the general conjecture is still open. In this paper we prove that this conjecture is true when \(X\) is a curve. We denote by \(\text{Span}(C)\) the linear span of the curve \(C\).
Main Theorem. If \(C\) is a connected reduced curve in \(\mathbb{P}^n\), then
\[
\text{reg}(C)\leq \text{deg}(C)-\dim(\text{Span}(C))+ 2.\tag{1}
\]
\textit{L. Gruson, R. Lazarsfeld} and \textit{C. Peskine} [Invent. Math. 72, No. 3, 491--506 (1983; Zbl 0565.14014)] proved this result in the case when \(C\) is assumed to be irreducible. We prove our theorem by using their theorem as a base case and inducting on the number of irreducible components of \(C\). The main lemma that allows this to work is a result of G. Caviglia. We also investigate the structure of connected curves for which in (1) the equality holds. In particular, we construct connected curves of arbitrarily high degree in \(\mathbb{P}^4\) having maximal regularity, but no extremal secants. We also show that any connected curve in \(\mathbb{P}^3\) of degree at least 5 with maximal regularity and no linear components has an extremal secant. Eisenbud-Goto conjecture; syzygy module; regularity of a curve D. Giaimo, On the Castelnuovo--Mumford regularity of connected curves, Trans. Am. Math. Soc. 358(1), 267--284 (2006) (electronic). Curves in algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the Castelnuovo-Mumford regularity of connected curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be a field of characteristic \(p > 0\), let \(X\) be an integer scheme over \(k\) which admits a nontrivial \(\alpha_p\) or \(\mu_p\) action and let \(\pi: X \to Y\) be the quotient under this action. In the paper, which is divided into seven sections, the author study the structure of the map \(\pi\) and of the quotient \((Y,\pi)\). The main result of the paper under review is an adjunction formula (Theorem 6.1).
The paper is concerned with the following problems and results: {\parindent=8mm \begin{itemize}\item[(i)] the existence of the quotient \(\pi: X \to Y\) and inseparability of \(\pi\) by \textit{D. Mumford} [Abelian varieties. London: Oxford University Press (1970; Zbl 0223.14022)] and by \textit{T. Ekedahl} [Publ. Math., Inst. Hautes Étud. Sci. 67, 97--144 (1988; Zbl 0674.14028)]; \item[(ii)] vector fields and inseparable morphisms of algebraic surfaces by \textit{A. N. Rudakov} and \textit{I. R. Shafarevich} [Izv. Akad. Nauk SSSR, Ser. Mat. 40, 1269--1307 (1976; Zbl 0365.14008)]; \item[(iii)] canonically polarized varieties and schemes by \textit{T. Matsusaka} [Am. J. Math. 92, 283--292 (1970; Zbl 0195.22802)] and by others; \item[(iv)] non-smooth Picard scheme by \textit{J.-i. Igusa} [Proc. Natl. Acad. Sci. USA 41, 964--967 (1955; Zbl 0067.39102)], by \textit{M. Raynaud} [Astérisque 64, 87--148 (1979; Zbl 0434.14024)], by \textit{C. Liedtke} [Math. Z. 259, No. 4, 775--797 (2008; Zbl 1157.14023)] and by others.
\end{itemize}} Sections 1 and 2 of the paper under review are on the introduction and basic definitions. Section 3 is on existence and basic properties of the quotient, Sections 4 and 5 on quotients by \(\mu_p\) and \(\alpha_p\) actions.
Section 6 deals with adjunction formulas. The author's method of proof of Theorem 6.1 is as follows: {\parindent=6mm \begin{itemize}\item[(a)] using properness of \(X\) and properties of the quotient map \(\pi: X \to Y\) he obtains the existance of a dualizing sheaf \(\omega_Y\) on \(Y\); \item[(b)] next he demonstrates that \(Y\) has Gorenstein singularities in codimension 1; \item[(c)] from that and from Serre's condition \(S_2\) for \(X\) he obtains that \(\omega_X = {\pi}^{*}\cdot \omega_Y \otimes {\pi}^{!} {\mathcal O}_Y\); \item[(d)] next he proves that \(({\pi}^{!} {\mathcal O}_Y)^* = I_{\mathrm{fix}}^{[p-1]}\) and concludes the proof of the theorem.
\end{itemize}} The last section contains interesting observations on non-smooth Picard scheme of smooth canonically polarized surfaces over algebraically closed fields of characteristics \(p = 2\) and \(p>2\). integral scheme; field of positive characteristic; canonically polarized surface; quotient of an integral scheme by a nontrivial action; adjunction formula; Picard scheme Group actions on varieties or schemes (quotients), Group schemes, Moduli, classification: analytic theory; relations with modular forms, Automorphisms of surfaces and higher-dimensional varieties Quotients of schemes by \(\alpha _p\) or \(\mu _{p}\) actions in characteristic \(p>0\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This note is a résumé of a joint paper of the author with \textit{M. van der Put} which is to appear. The main result gives conditions for an analytic quotient G/\(\Lambda\) to be an abelian variety, where G is an extension of an abelian variety by a torus and L is a lattice in G. This partly generalizes and simplifies related results of Mumford, Gerritzen, Faltings and Chai. rigid analytic torus; quotient of an analytic group by a lattice; extension of an abelian variety Arithmetic ground fields for abelian varieties, Analytic theory of abelian varieties; abelian integrals and differentials, Local ground fields in algebraic geometry Construction analytique (rigide) de variétés abéliennes. ((Rigid) analytic construction of abelian varieties) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians One (complicated) way to recover a smooth nonhyperelliptic projective curve from its Jacobian \((J,\Theta)\) is to take the intersection in the tangent space at the origin of \(J\) of the (transferred) tangent cones at points of multiplicity 2 of \(\Theta\). To each double étale cover \(\pi: \widetilde{C}\to C\) of connected smooth projective curves, one can associate a principally polarized abelian variety \((P,\Xi)\), its Prym variety. To follow the procedure above to recover the double cover from its Prym, one needs to know first that the tangent cones at points of multiplicity 2 of \(\Xi\) generate the space of quadrics containing the Prym-canonical curve \(C_0\) (this is the image of the morphism associated with the linear system \(|\omega_C \otimes\eta|\), where \(\eta\) is the half-period on \(C\) that determines \(\pi\)), and second that \(C_0\) can be recovered from the quadrics that contain it. Both steps have been carried out by the reviewer for a generic curve \(C\) of genus \(\geq 7\). The authors show that the second step always works if the Clifford index of \(C\) is at least 3 (all known counter-example to the Torelli theorem for Pryms have Clifford index 2). When \(\text{Cliff} (C)\geq 5\), it follows from results of Green and Lazarsfeld that \(C_0\) is intersection of quadrics. For \(\text{Cliff} (C)=3\) or 4, a more detailed analysis, based on the same principles, is needed to show that the intersection of quadrics containing \(C_0\) is the union of \(C_0\) and proper linear subspaces of \(|\omega_C \otimes\eta|^*\). Torelli problem; Jacobian; principally polarized abelian variety; Prym variety; Clifford index; intersection of quadrics LANGE (H.) , SERNESI (E.) . - Quadrics containing a Prym-canonical curve , J. Alg. Geom., t. 5, 1996 , n^\circ 2, p. 387-399. MR 96k:14021 | Zbl 0859.14009 Jacobians, Prym varieties, Picard schemes, higher Jacobians, Theta functions and curves; Schottky problem Quadrics containing a Prym-canonical curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is an extended version of the author's report to the Warsaw Congress [Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 603-619 (1984; Zbl 0571.14012)]. It is devoted to a circle of problems popularized by Hilbert in his 16-th problem from his famous talk at the Paris (1900) congress. Traditionally, the main question here is the determination of possibly mutual positions of connected components of a nonsingular plane real algebraic curve of given degree and similar problems in higher dimensions. Now this purely real picture of a real algebraic curve is enriched by some data coming from the complex domain. This important modification of the viewpoint is due to Rokhlin. Naturally, this point of view is the one adopted in the paper. The paper gives a rather complete and up to data survey (without proofs) of known facts about nonsingular plane real algebraic curves. In particular, in the last section the author's method of construction of curves with prescribed topological properties is sketched. This method leads a complete classification of curves of degree 7- the most famous contribution of the author to the field. Now this method became the main research tool in the field. The complex point of view hopefully can be extended from curves to higher dimensional varieties. At the moment, only the case of surfaces is fairly well understood. This theme is also presented in the paper. Also, Kharlamov's results on the classification of surfaces of degree 4 are covered.
The flowering of this field in the last fifteen years is due mainly to the late V. A. Rokhlin. He not only contributed many important ideas and results, but also created an active team of researchers. The author is one of the leaders of this team now. His authoritative survey of this field is of primarily importance for all interested in the topology of real algebraic varieties. real algebraic surfaces; Hilbert's 16-th problem; complexification; mutual positions of connected components of a nonsingular plane real algebraic curve of given degree O. Ya Viro, Progress in the topology of real algebraic varieties over the last six years, \textit{Uspekhi Mat. Nauk.}\textbf{41} (1986) 45-67 (in Russian); \textit{Russian Math. Surveys}\textbf{41} (1986) 55-82. Topological properties in algebraic geometry, History of algebraic geometry, Projective techniques in algebraic geometry, Real algebraic and real-analytic geometry, Special surfaces, Curves in algebraic geometry Progress in the topology of real algebraic varieties over the last six years | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We prove a general ``diagram method'' theorem valid for a quite large class of 3-folds with \(\mathbb{Q}\)-factorial singularities (see the basic theorems 1.3.2 and 3.2 and also theorem 2.2.6). This generalizes our results on Fano 3-folds with \(\mathbb{Q}\)-factorial terminal singularities [\textit{V. V. Nikulin}, J. Math. Kyoto Univ. 34, No. 3, 495-529 (1994; Zbl 0839.14030)]. -- As an application, we get the following result about Calabi-Yau 3-folds \(X\):
Assume that the Picard number \(\rho(X)>40\). Then one of two cases (i) or (ii) holds:
(i) There exists a small extremal ray on \(X\).
(ii) There exists a nef element \(h\) such that \(h^3=0\) (thus, the nef cone \(\text{NEF}(X)\) and the cubic intersection hypersurface \({\mathcal W}_X\) have a common point; here, we do not claim that \(h\) is rational!).
As a corollary, we get: Let \(X\) be a Calabi-Yau 3-fold. Assume that the nef cone \(\text{NEF}(X)\) is finite polyhedral and \(X\) does not have a small extremal ray. Then there exists a rational nef element \(h\) with \(h^3=0\) if \(\rho (X)>40\).
To prove these results on Calabi-Yau manifolds, we also use a result of the appendix by V. V. Shokurov on the length of divisorial extremal rays. Thus the above results on Calabi-Yau 3-folds are joint work with V. V. Shokurov. We also discuss the generalization of the above results to so-called \(\mathbb{Q}\)-factorial models of Calabi-Yau 3-folds. This sometimes enables us to play the same game even when the Mori cone is not polyhedral, or when there are small extremal rays. effective 1-cycles; length of extremal ray curve; Kähler cone; Calabi-Yau 3-folds; Picard number; nef cone; length of divisorial extremal rays Viacheslav V. Nikulin, The diagram method for 3-folds and its application to the Kähler cone and Picard number of Calabi-Yau 3-folds. I, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 261 -- 328. With an appendix by Vyacheslav V. Shokurov. Calabi-Yau manifolds (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), \(3\)-folds The diagram method for 3-folds and its application to the Kähler cone and Picard number of Calabi-Yau 3-folds. I. -- With an appendix by Vyacheslav V. Shokurov: Anticanonical boundedness for curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A polynomial map \(F:\mathbb{R}^2\to\mathbb{R}^2\) is said to satisfy the Jacobian condition if \(\forall (X,Y)\in \mathbb{R}^2\), \(J(F)(X,Y)\neq 0\). The real Jacobian conjecture was the assertion that such a map is a global diffeomorphism. Recently the conjecture was shown to be false by S. Pinchuk. According to a theorem of J. Hadamard any counterexample to the conjecture must have asymptotic values. We give the structure of the variety of all the asymptotic values of a polynomial map \(F:\mathbb{R}^2\to \mathbb{R}^2\) that satisfies the Jacobian condition. We prove that the study of the asymptotic values of such maps can be reduced to those maps that have only \(X\)- or \(Y\)-finite asymptotic values. We prove that a \(Y\)-finite asymptotic value can be realized by \(F\) along a rational curve of the type \((X^{-k},A_0+A_1X+\cdots+ A_{N-1}X^{N-1}+ YX^N)\), where \(X\to 0\), \(Y\) is fixed and \(K,N>0\) are integers. More precisely we prove that the coordinate polynomials \(P(U,V)\) of \(F(U,V)\) satisfy finitely many asymptotic identities, namely, identities of the following type, \(P(X^{-k},A_0+ A_1X+\cdots+ A_{N-1} X^{N-1}+ YX^N)= A(X,Y)\in R[X,Y]\), which `capture' the whole set of asymptotic values of \(F\). asymptotic values of a real polynomial; Jacobian condition; real Jacobian conjecture Peretz, Ronen: The variety of asymptotic values of a real polynomial étale map, J. pure appl. Algebra 106, 102-112 (1996) Topology of real algebraic varieties, Polynomial rings and ideals; rings of integer-valued polynomials The variety of the asymptotic values of a real polynomial étale map | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper the syzygies of a canonical curve given by a system of equations of Petri's type are studied. Curves, which allow such a system of equations, are precisely those curves, which are canonical and have a simple (g-2)-secant. This includes for example all irreducible reduced nonstrange canonical curves, but also various reducible curves. Petri's non-minimal presentation of the homogeneous ideal is generalized to a non-minimal resolution, and a simplified proof of Petri's theorem about the generators of the homogeneous ideal of a smooth canonical curve is given. Petri's theorem uses the irreducibility of the curve crucially. I analyze what happens in case of reducible canonical curves.
The main result is a proof of Green's conjecture about syzygies of canonical curves for the special case of the second syzygy module.
An appendix contains a short introduction to standard (or Gröbner) basis, including a quick proof of Hilbert's syzygy theorem. standard basis; Gröbner basis; syzygies of a canonical curve; equations of Petri's type; non-minimal resolution; reducible canonical curves; Green's conjecture; second syzygy module; Hilbert's syzygy theorem Milnor, J.: On the 3-dimensional Brieskorn manifolds \textit{M(p, q, r)}. In: Neuwirth, L.P. (ed.) Knots, Groups, and 3-Manifolds (Papers Dedicated to the Memory of R. H. Fox), pp. 175-225. Princeton Univ. Press, Princeton, N. J. (1975) Singularities of curves, local rings, Computational aspects of algebraic curves, Syzygies, resolutions, complexes and commutative rings, Global theory and resolution of singularities (algebro-geometric aspects), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) A standard basis approach to syzygies of canonical curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A conjecture of R. Hartshorne, stating that for every \(d\), \(g\) such that \(d\leq g+3\) and \(H_{d,g}\) (the Hilbert scheme of smooth and connected curves of \({\mathbb{P}}^ 3\) of degree \(d\) and genus \(g\)) is not empty, then \(H_{d,g}\) contains a linearly normal curve, is proved. For \(d\), \(g\) as above, this implies, by semicontinuity, the existence of an irreducible component of \(H_{d,g}\) consisting of generically linearly normal curves. Moreover, we give a range for \(d\), \(g\) where \(H_{d,g}\) is reducible, showing the existence of components consisting of non linearly normal curves. These constructions give further counter-examples to conjectures of R. Hartshorne and J. Kleppe. Finally, a problem about components of \(H_{d,g}\) containing families of curves lying on a smooth cubic surface of \({\mathbb{P}}^ 3\) is posed. linearly normal curve; families of curves lying on a smooth cubic surface; reducibility of Hilbert scheme Dolcetti, A; Pareschi, G, On linearly normal space curves, Math. Z., 198, 73-82, (1988) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) On linearly normal space curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is devoted to canonical maps of general hypersurfaces in abelian varieties. The main result obtained is that, for a general pair \((A,X)\) of an (ample) hypersurface \(X\) in an abelian variety \(A\), the canonical map \(\Phi_X\) of \(X\) is birational onto its image if the polarization given by \(X\) is not principal (i.e., its Pfaffian \(d\) is not equal to \(1\)). This work was motivated by a theorem obtained by the first author in a joint work with \textit{F.-O. Schreyer} [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 79--116 (2002; Zbl 1053.14048)] on canonical surfaces: if you have a polarization of type \((1, 1, 2)\) then the image \(\Sigma\) of the canonical map \(\Phi_X\) is in general a surface of degree \(12\) in \(\mathbb{P}^3\), birational to \(X\), while for the special case where \(X\) is the pull-back of the theta divisor of a curve of genus \(3\), then the canonical map has degree \(2\), and \(\Sigma\) has degree \(6\). The authors also show that, setting \(g=\dim(A)\), and letting \(d\) be the Pfaffian of the polarization given by \(X\), then if \(X\) is smooth and \(\Phi_X:X\longrightarrow\mathbb{P}^{N:=g+d-2}\) is an embedding, then necessarily they have the inequality \(d\geq g+1\), equivalent to \(N:=g+d-2\geq2\dim(X)+1\). Hence the authors formulate the following interesting conjecture, motivated by work of the second author: if \(d\geq g+1\), then, for a general pair \((A,X)\), \(\Phi_X\) is an embedding. This paper is organized as follows: Section 1 is an introduction to the subject and a description of the results. Section 2 is devoted to the proof of the main result. Section 3 deals with embedding obstruction. In section 4, the authors conclude this paper with some remarks on the conjecture. hypersurfaces; abelian varieties; canonical maps; Gauss maps; theta divisors; automorphisms of a covering; monodromy groups; generic coverings Rational and birational maps, Embeddings in algebraic geometry, Theta functions and abelian varieties, Algebraic theory of abelian varieties, Structure of families (Picard-Lefschetz, monodromy, etc.), Jacobians, Prym varieties Canonical maps of general hypersurfaces in abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0721.00006.]
\(k\) denotes a non-archimedean local field of characteristic zero and \(\chi: k^*\to\mathbb Q_ p\) denotes a continuous character. Let \(J\) be the Jacobian variety of a curve \(X\) over \(k\) (having a \(k\)-rational point). The aim of the paper is to construct a \(p\)-adic height pairing on \(J\).
In the case that the residue characteristic of \(k\) is different from \(p\), arithmetic intersection theory is used to produce a unique pairing \(\langle a,b\rangle\), with values in \(\mathbb Q_ p\), defined on relatively prime divisors \(a\) and \(b\) on \(X\) (defined over \(k)\) and satisfying: continuous, symmetric, bi-additive and \(\langle(f),b\rangle=\chi(f(b))\) for \(f\in k(X)^*\). In the case \(k\supset\mathbb Q_ p\), a rigid analytic analysis of differentials of the third kind and the de Rham cohomology is made to arrive at a definition of the pairing. The pairing which is constructed depends on a suitable choice of a direct sum decomposition \(H^ 1_{DR}(X/k)=H^ 0(X,\Omega_ X)\oplus W\). In case \(X\) has a good ordinary reduction one can take the unit root space as a choice for \(W\). With this choice the pairing coincides with the canonical \(p\)-adic height pairings constructed by \textit{P. Schneider} [Invent. Math. 69, 401--409 (1982; Zbl 0509.14048 and 79, 329--374 (1985; Zbl 0571.14021)] and by \textit{B. Mazur} and \textit{J. Tate} in Arithmetic and geometry, Pap. dedic. Shafarevich, Vol. I. Arithmetic, Prog. Math. 35, 195--237 (1983; Zbl 0574.14036)]. A proof of the last statement is given in the sequel of this paper [\textit{R. F. Coleman}, ``The universal vectorial bi-extension and \(p\)-adic heights'', Invent. Math. 103, No. 3, 631--650 (1991; Zbl 0763.14009)]. \(p\)-adic height pairing; Jacobian variety; arithmetic intersection theory; differentials of the third kind; de Rham cohomology R. F. Coleman and B. H. Gross, ''\(p\)-adic heights on curves,'' in Algebraic Number Theory, Adv. Stud. Pure Math. 17, Academic Press, Boston, 1989, 73--81. Local ground fields in algebraic geometry, Arithmetic varieties and schemes; Arakelov theory; heights, Arithmetic ground fields for curves \(p\)-adic heights on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For any curve of genus two with a rational Weierstrass point defined over a field of characteristic other \(than\quad 2,\) we find equations defining a projective embedding for its Jacobian, and determine its group law. When the curve is defined over a ring complete under a non-archimedean valuation, this allows us to find explicit parameters for the formal group on the kernel of reduction of the Jacobian modulo the maximal ideal of the ring. The calculations are facilitated by first considering curves defined over the complex numbers and employing theta functions. curve of genus two; rational Weierstrass point; embedding; Jacobian; formal group; theta functions Grant D.: Formal groups in genus 2. J. Reine. Angew. Math. 411, 96--121 (1990) Special algebraic curves and curves of low genus, Theta functions and abelian varieties, Jacobians, Prym varieties, Riemann surfaces; Weierstrass points; gap sequences, Formal groups, \(p\)-divisible groups Formal groups in genus 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study the one dimensional regular rigid analytic spaces of finite genus over a complete valued field \(k\). We show that such a space \(X\) has prestable reduction. If \(k\) is maximally complete, \(X\) is isomorphic to an analytic open set of an algebraic curve (analytified). Finally, we characterize all analytic spaces which are the complement of a compact set in an algebraic curve. regular rigid analytic spaces of finite genus; analytic spaces which are the complement of a compact set in an algebraic curve Liu, Q, Ouverts analytiques d'une courbe algébrique en géométrie rigide, Annales de l'Institut Fourier, 37, 39-64, (1987) Local ground fields in algebraic geometry, Arithmetic ground fields for curves, Analytic subsets of affine space Open analytic sets of an algebraic curve in rigid geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a normal affine variety acted on by a connected reductive group \(G\) with Borel subgroup \(B\). Suppose that the complexity of the action (that is, the codimension of a generic \(B\)-orbit in \(X)\) is one, and suppose that the categorical quotient of \(X\) by \(G\) is one-dimensional. In this paper it is shown that the closures of all \(G\)-orbits in \(X\) are normal (as was already known for actions of complexity zero). normal variety; orbit closure; complexity of the action of a reductive group Arzhantsev, I.V.: On the normality of the closures of spherical orbits. Func. Anal. Prilozh. 31(4), 66--69 (1997) (in Russian), English transl.: Func. Anal. Appl. 31, No. 4 (1997), 278--280 Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations On the normality of the closures of spherical orbits | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0614.00007.]
In this paper the author studies the behaviour of the singularities of a 3-dimensional variety under a sequence of quadratic transformations. In particular, under certain hypothesis of ``good position'' and with a suitable choice of the local parameters, he gives the construction of a ``Newton polygon'' which extends the classical one. singularities of a 3-dimensional variety; quadratic transformations Global theory and resolution of singularities (algebro-geometric aspects), \(3\)-folds, Singularities in algebraic geometry, Rational and birational maps Techniques pour la désingularisation des champs de vecteurs. (Techniques for the desingularization of vector fields) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is known that the Jacobian variety of the Klein curve is isomorphic to \(E\times E\times E\), where \(E\) is an elliptic curve. In this paper we compute the defining equation of \(E\) in Weierstrass normal form and show that the Klein curve covers \(E\) doubly and triply. orbit space; theta matrix; coverings; Klein curve; Jacobian variety Jacobians, Prym varieties, Elliptic curves, Coverings in algebraic geometry, Coverings of curves, fundamental group A note on Klein curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper studies the performance of the index calculus attack to the discrete logarithm problem (DLP) in the Jacobian of algebraic curves \(\mathcal{C}\), both hyperelliptic and non-hyperelliptic, curves defined over finite fields \(\mathbb{F}_q\) and composite extensions \(\mathbb{F}_{q^n}\) of these fields. In the paper \(n\) and the genus \(g\) of \(\mathcal{C}\) are supposed fixed while \(q\) grows to infinity.
The Introduction exposes the state of this problem today: \textit{P. Gaudry} et al. [Math. Comput. 76, No. 257, 475--492 (2007; Zbl 1179.94062)] gave a method solving the DLP for a hyperelliptic curve defined over \(\mathbb{F}_q\) with asymptotic complexity \(\tilde{O}(q^{2-2/g})\), while for \(\mathcal{C}\) non-hyperelliptic, \textit{C. Diem} [Lect. Notes Comput. Sci. 4076, 543--557 (2006; Zbl 1143.11361)] proposed an algorithm that solves the DLP in \(\tilde{O}(q^{2-2/(g-1)})\). In the case \(n>1\), \textit{K. Nagao} [Lect. Notes Comput. Sci. 6197, 285--300 (2010; Zbl 1260.11045)] gave, for hyperelliptic curves, an index calculus method with complexity \(\tilde{O}(q^{2-2/ng})\).
The aim of the present paper is then to provide an answer to the following question: ``Is it possible to combine both the non-hyperellipticity and the extensions?''
Section 2 gives a framework of the index calculus in \(J(\mathcal{C})\) using the notion of linear systems of a divisor and the Weil descent method and Section 3 summarizes the methods of Gaudry, Diem and Nagao.
Finally, in order to answer the above question, Section 4 considers the combination of different techniques both in the case \(n=1\) and in the case \(n>1\). The conclusion is that the optimal method is Nagao's algorithm. discrete logarithm problem; hyperelliptic curve cryptography; index calculus; divisor class group; Jacobian variety Algebraic coding theory; cryptography (number-theoretic aspects), Arithmetic ground fields for curves, Jacobians, Prym varieties, Cryptography Field extensions and index calculus on algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The bounded derived category of coherent sheaves $\mathsf{D}^{\text b}(\text{coh}X)$ for a smooth, projective variety $X$ is an important, and sometimes classifying, invariant of the variety. In this work, $\mathsf{D}^{\text b}(\text{coh}X)$ is broken down into simpler pieces of building blocks that can be analysed to give information about $X$. The tool used in this article for this procedure is called semi-orthogonal decomposition. There exist many examples of such decompositions, but no general algorithm to determine what semi-orthogonal decompositions that exist of a given bounded derived category $\mathsf{D}^{\text b}(\text{coh}X).$ In this article, the authors give a method for finding semi-orthogonal decompositions of bounded derived categories of coherent sheaves together with a complete description of all the involved components. \par The approach in this article is based on birational methods as one expects that the derived categories of sheaves on birational varieties should be related. There is no known semi-orthogonal decomposition for a sufficiently general class of birational transformations, where by sufficiently general means that the birational class should at least include blow-ups at smooth centres. In this article, the class of birational transformations comes from Geometric Invariant Theory (GIT). \par The article makes the link between GIT and birational geometry clear. There is no canonical choice of linearisation of a group action on a variety, and this establishes a feature for constructing new birational models of a GIT quotient. The meaning of this sentence is explained by treating GIT thoroughly through a separate preliminary section: Changing the linearisation leads to birational transformations between the GIT quotients. This is what is called variations of GIT structures (VGIT). Conversely, any birational map between smooth projective varieties can be obtained through such GIT variations, and one then says that two different GIT quotients are related by wall-crossing. The last is because there is a natural fan structure on the set of linearisations. \par The present methods focused on semi-orthogonal decompositions coming from wall-crossing in VGIT give a new perspective on the relationship between birational geometry and derived categories, leading to new and important results. \par The main results on derived categories of sheaves are stated as follows in the article: Let $X$ be a smooth, projective variety with an action of a reductive linear algebraic group $G$. Assume that $X$ has two $G$-equivariant ample line bundles $\mathcal L_-$ and $\mathcal L_+$ satisfying: \begin{itemize} \item[i)] For $t\in[-1,1],$ let $\mathcal L_t=\mathcal L_-^{\frac{1-t}{2}}\otimes\mathcal L_+^{\frac{1+t}{2}}.$ \end{itemize} Then the semi-stable locus should be constant for $-1\leq t<0$ and for $0<t\leq 1.$ Now name $X^{\text{ss}}(-):=X^{\text{ss}}(\mathcal L_t)$ for $-1\leq t<0,$ $X^{\text{ss}}(0):=X^{\text{ss}}(\mathcal L_0),$ $X^{\text{ss}}(+):=X^{\text{ss}}(\mathcal L_t)$ for $0<t\leq 1.$ Then: \begin{itemize} \item[ii)] The set $X^{\text{ss}}(0)\setminus(X^{\text{ss}}(-)\cup V(+))$ is connected, \item[iii)] For any point $x\in X^{\text{ss}}(0)\setminus(X^{\text{ss}}(-)\cup X^{\text{ss}}(+)),$ the stabilizer $G_x$ is isomorphic to $\mathbb G_m.$ \end{itemize}
When these conditions satisfied, work of \textit{M. Thaddeus} [J. Am. Math. Soc. 9, No. 3, 691--723 (1996; Zbl 0874.14042)] and \textit{I. V. Dolgachev} and \textit{Y. Hu} [Publ. Math., Inst. Hautes Étud. Sci. 87, 5--56 (1998; Zbl 1001.14018)] show that there is a one-parameter subgroup $\lambda:\mathbb G_m\rightarrow G,$ a connected component $Z^0_\lambda$ on the fixed locus of $\lambda$ in $X^{\text{ss}}(0),$ and disjoint decompositions $X^{\text{ss}}(0)=X^{\text{ss}}(+)\sqcup S_\lambda\text{ and }X^{\text{ss}}(0)=X^{\text{ss}}(-)\sqcup S_{-\lambda},$ where $S_\lambda$ is the $G$-orbit of all points in $X$ that flow to $Z^0_\lambda$ as $\alpha\rightarrow 0$ in $\mathbb G_m$ and $S_{-\lambda}$ is the $G$-orbit of all points in $X$ that flow to $Z^0_\lambda$ as $\alpha\rightarrow\infty$ in $\mathbb G_m.$ \par To state the article's first main statement: Let $C(\lambda)$ be the centralizer of $\lambda$ and $G_\lambda=C(\lambda)/\lambda.$ Let $X/\!/ +:=[X^{\text{ss}}(+)/G]\text{ and }X/\!/ - :=[X^{\text{ss}}(-)/G]$ be the global quotient stacks of the $(+)$ and $(-)$ semi-stable loci by $G$. Let $\mu$ be the weight of $\lambda$ on the anti-canonical bundle of $X$ along $Z^0_\lambda.$ The authors assume for simplicity that there exists a splitting $C(\lambda)\cong\lambda\times G_\lambda,$ and they put $X^\lambda/\!/_0 G_\lambda$ the GIT quotient stack $[(X^\lambda)^{\text{ss}}(\mathcal L_0)/G_\lambda]$ of the fixed locus $X_\lambda$ by $G_\lambda$ using the equivariant bundle $\mathcal L_0.$ We state theorem more or less verbatim: \par Theorem 1. Fix $d\in\mathbb Z.$ (a) If $\mu>0,$ then there are fully-faithful functors $\Phi_d^+:\mathsf{D}^{\text b}(\text{coh}X/\!/-)\rightarrow \mathsf{D}^{\text b}(\text{coh}X/\!/ +),$ and, for $d\leq j\leq\mu+d-1,\Upsilon_j^+:\mathsf{D}^{\text b}(\text{coh}X^\lambda/\!/_0 G_\lambda)\rightarrow \mathsf{D}^{\text b}(\text{coh}X/\!/ +)$ and a semi-orthogonal decomposition $\mathsf{D}^{\text b}(\text{coh}X/\!/ +)=\langle \Upsilon^+_d,\dots,\Upsilon^+_{\mu+d-1},\Phi^+_d\rangle.$ (b) If $\mu=0,$ then there is an exact equivalence $\Phi^+_d:\mathsf{D}^{\text b}(\text{coh}X/\!/ -)\rightarrow\mathsf{D}^{\text b}(\text{coh}X/\!/ +).$ (c) If $\mu<0,$ then there are fully-faithful functors $\Phi_d^-:\mathsf{D}^{\text b}(\text{coh}X/\!/ +)\rightarrow\mathsf{D}^{\text b}(\text{coh}X/\!/-)$ and, for $\mu+d+1\leq j\leq d,\;\Upsilon_j^-:\mathsf{D}^{\text b}(\text{coh}X^\lambda/\!/_0 G_\lambda)\rightarrow\mathsf{D}^{\text b}(\text{coh}X/\!/ -)$ and a semi-orthogonal decomposition $\mathsf{D}^{\text b}(\text{coh}X/\!/-)=\langle\Upsilon^-_{\mu+d+1},\dots,\Upsilon^-_d,\Phi^-_d\rangle.$
The above theorem provides a framework to view some exsting results. For a particular choice of wall-crossing, \textit{D. O. Orlov}'s description [Russ. Acad. Sci., Izv., Math. 41, No. 1, 1 (1992; Zbl 0798.14007); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 4, 852--862 (1992)] of the derived category of a blow up with smooth center, can be recovered. Also, it can be used to prove the $D$-domination and $K$-domination for such variations, and to give a streamlined proof of a result of \textit{Y. Kawamata} [J. Differ. Geom. 61, No. 1, 147--171 (2002; Zbl 1056.14021)] stating that for $X$ a smooth projective toric variety, the derived category $\mathsf{D}^{\text b}(\text{coh}X)$ possesses a full exceptional collection.
Now, \textit{M. M. Kapranov} [J. Algebr. Geom. 2, No. 2, 239--262 (1993; Zbl 0790.14020)] presented $\overline{M}_{0,n}$ as an iterated blow up of $\mathbb P^{n-3}$ along strict transforms of linear spaces and so the existence of a full exceptional collection was known. The article generalizes this result by establishing the corresponding result for \textit{B. Hassett}'s moduli spaces [Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)] of stable symmetrically-weighted rational curves $\overline{M}_{0,n\times\epsilon}.$
\textit{D. Orlov} [Prog. Math. 270, 503--531 (2009; Zbl 1200.18007)] has given a result relating the derived categories of projective complete intersections and singularity categories of affine cones. As a final main result of the present work, it is proved how to recover this result using VGIT.
In addition to give the results mentioned above, the article recall the GIT in a relative elementary way. Thus the ideas and the extracted definitions from GIT are as important as the results themselves.
The ideas in the paper appeared first by \textit{Y. Kawamata} [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 197--215 (2002; Zbl 1092.14023)] in his work on derived categories treating $\mathbb G_m$ actions. \par Independently, \textit{M. van den Bergh} [in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3--8, 2002. Berlin: Springer. 749--770 (2004; Zbl 1082.14005)] also studied these actions on affine space via windows, giving the fully-faithful functors and the criterion for equivalence. The present article makes manifest that windows and VGIT are an essential framework for Orlov's work. The authors mention Segal's, Orlov's and others important work, using the framework developed by Orlov and highlighted in this article.
I would like to end the review with the author's own words verbatim: ``Neither of these works provides descriptions of the full semi-orthogonal decompositions arising from wall-crossing. Consequently, applications, outside of those to construction of equivalences, are more limited in these works than here. This includes all applications mentioned.'' projective toric variety; one-parameter subgroup; wall-crossing; semi-orthogonal decomposition; singularity categories; full exceptional collection; GIT; windows; geometric invariant theory; fan structure; bounded derived category of coherent sheaves; equivariant ample line bundle; VGIT; reductive linear algebraic group; variations of GIT structures; birational methods; blow-ups at smooth centres; linearisation of a group action Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Geometric invariant theory, Variation of Hodge structures (algebro-geometric aspects), Fine and coarse moduli spaces, Derived categories, triangulated categories Variation of geometric invariant theory quotients and derived categories | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0745.00034.]
Let \(F\) be an algebraic curve in \(\mathbb{C}^ n\) defined by a system of polynomial equations \(f_ i(X)=0\), \(i=1,\dots,n-1\). Let \(X=(x_ 1,\dots,x_ n)=0\) be a singular point of \(F\), and let \(x_ i=\sum^ \infty_{k=1}b_{ik}t^{p_{ik}}\), \(i=1,\dots,n\) (where \(p_{ik}\) are integers, \(0>p_{ik}>p_{i,k+1}\), \(b_{ik}\) complex numbers, and the series converge for large \(| t|)\), be a local uniformization of a branch of \(F\) passing through the point \(X=0\). The authors give an algorithm for finding any initial parts of the above series, with the aid of Newton polyhedra and power transformations. local uniformization of a branch of an algebraic curve; Newton polyhedra Plane and space curves, Modifications; resolution of singularities (complex-analytic aspects), Computational aspects of algebraic curves The local uniformization of branches of an algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We say that a curve \(C\subset {\mathbb{P}}^ n\) has maximal rank if for every \(k\geq 0\) the restriction map \(H^ 0({\mathbb{P}}^ n,{\mathcal O}_{{\mathbb{P}}^ n}(k))\to H^ 0(C,{\mathcal O}_ C(k))\) is either injective or surjective. The maximal rank conjecture states that a general embedding of a general smooth curve of genus g has maximal rank. In this paper we prove this conjecture for non special embeddings in \({\mathbb{P}}^ 4\). The proof uses the method of \textit{R. Hartshorne} and \textit{A. Hirschowitz} [``Droites en position generale dans \({\mathbb{P}}^ n\)'' in Algebraic Geometry, Proc. int. Conf., La Rábida/Spain 1981, Lect. Notes Math. 961, 169-188 (1982; Zbl 0555.14011)] and \textit{A. Hirschowitz} [Acta Math. 146, 209-230 (1981; Zbl 0475.14027)] to show inductively the existence of a reducible curve with maximal rank. hyperplane section; genus; moduli; semicontinuity; non special embeddings in projective 4-space; maximal rank conjecture; embedding of a general smooth curve Ballico, E; Ellia, P, On postulation of curves in \(\mathbb{P}^4\), Math. Z., 188, 215-223, (1985) Families, moduli of curves (algebraic), Projective techniques in algebraic geometry, Embeddings in algebraic geometry, Algebraic moduli problems, moduli of vector bundles On postulation of curves in \({\mathbb{P}}^4\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0745.00062.]
In this paper sequences of curves \(X\) over a fixed finite field \(\mathbb{F}_ q\) whose genus tends to infinity are studied. First the author derives a formula for the asymptotic behavior of \(| X (\mathbb{F}_ q) |\), the number of \(\mathbb{F}_ q\)-rational points on \(X\), or rather of \(B_ m\), the number of points of degree \(m\). This formula generalizes the Drinfeld- Vladut inequality \(\limsup_{g \to \infty} | X (\mathbb{F}_ q) |/g \leq \sqrt q - 1\) where \(g\) is the genus.
Of special interest are sequences of curves for which \(\lim_{g \to \infty} B_ m/g\) exists for every \(m\): asymptotically exact families of curves. For the number of rational points on the Jacobian, denoted by \(h = | J_ X (\mathbb{F}_ q) |\) the author studies \(\liminf_{g \to \infty} \log h/g\) and \(\limsup_{g \to \infty} \log h/g\). These quantities coincide for asymptotically exact families. -- The last part of the paper treats the asymptotic behavior of the number of effective divisors of degree \(m\). curves over a finite field; number of rational points; number of points of degree \(m\); genus; asymptotic behavior; number of effective divisors M. A. Tsfasman, ''Some Remarks on the Asymptotic Number of Points, Coding Theory and Algebraic Geometry,'' in Lecture Notes in Math. (Springer-Verlag, Berlin, 1992), Vol. 1518, pp. 178--192. Arithmetic ground fields for curves, Enumerative problems (combinatorial problems) in algebraic geometry, Finite ground fields in algebraic geometry, Curves over finite and local fields, Computational aspects of algebraic curves Some remarks on the asymptotic number of points | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a nonsingular irreducible curve of degree \(d\) over a finite field \(\mathbb{F}_q\) and let \(K\), \(\overline K\) be the corresponding function fields over \(\mathbb{F}_q\) resp. over its algebraic closure. Let \(n\) be an integer and assume that \(r \in K\) is a rational function of degree \(m\) which for no divisor \(k\) of \(q^n - 1\) is the \(k\)-th power of a rational function in \(\overline K\). Then the number of prime divisors of \(C\) of degree \(n\) with respect to which \(r\) is a primitive root equals \(\varphi (q^n - 1)/n +\) error term, which in an explicit way depends on \(m,n,d,q\) and the number of distinct prime divisors of \(q^n - 1\).
The novelty of this result lies in the error term, since asymptotics in this problem have been found already by \textit{H. Bilharz} [Math. Ann. 114, 476-492 (1937; Zbl 0016.34301)] who did not restrict himself to the case of rational \(r\). The proof is easily reduced to the count of primitive roots of \(\mathbb{F}_{q^n}\) having a special form and this is done using a bound for character sums obtained by \textit{G. I. Perelmuter} [Mat. Zametki, 5, 373-380 (1969; Zbl 0179.49903)] as a consequence of Riemann's hypothesis for function fields. irreducible curve; rational function; number of prime divisors Pappalardi, F.; Shparlinski, I.: On Artin's conjecture over function fields. Finite fields appl. 1, 399-404 (1995) Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Curves over finite and local fields On Artin's conjecture over function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [\textit{W. Zhao}, J. Algebra 324, No.~2, 231--247 (2010)] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [\textit{O. H. Keller}, Monatsh. Math. Phys. 47, 299--306 (1939; Zbl 0021.15303; JFM 65.0713.02)] is reduced to an open problem on this deformation of polynomial algebras. the generalized Laguerre polynomials; total symbols of differential operators; the image conjecture; the Jacobian conjecture Wenhua Zhao, A deformation of commutative polynomial algebras in even numbers of variables, Cent. Eur. J. Math. 8 (2010), no. 1, 73 -- 97. Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Differential operators in several variables, Jacobian problem A deformation of commutative polynomial algebras in even numbers of variables | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For an elliptic curve C over a number field K, i.e. a curve of genus \(g=1\) having a rational point over K, the group \(T_ K\) of rational torsion points on C over K is finite by the Mordell-Weil theorem, and the (still unproved) boundedness conjecture implies that \(T_ K\) is bounded by a constant depending only on K. For (smooth) curves C of higher genus \(g\geq 2\) in an abelian variety J, the generalized Manin-Mumford conjecture states the finiteness of the set \(T_{\bar K}\) of torsion points of J arizing from C and being defined over the algebraic closure \(\bar K\) of K. This latter conjecture was proved by S. Lang for abelian varieties J admitting complex multiplication (CM) and by Raynaud for arbitrary abelian varieties. Assuming that the abelian variety J has potential complex multiplication, the author proves the following boundedness theorem: \(\#T_{\bar K}\leq pg,\) where p denotes the smallest prime of \({\mathbb{Q}}\) divisible by a prime in the set of primes \({\mathfrak p}\) of the number field K such that (i) \({\mathfrak p}\) does not divide 2 or 3, (ii) K is unramified at \({\mathfrak p}\), (iii) C has good ordinary reduction over K at \({\mathfrak p}.\)
Generalizations of this theorem are also obtained. Moreover, the theorem can be applied to completely determining the torsion points on the Fermat curves \(F(m):X^ m+Y^ m+Z^ m=0\) provided that \(m+1\) is a prime and \(m\geq 10.\)
As a tool for proving his theorem, the author develops a theory of p-adic abelian integrals based on Tate's rigid analysis and Monsky-Washnitzer's dagger analysis. In particular, the p-adic integrals of the first kind on an abelian variety turn out to satisfy an addition law as a result of which the torsion points on a curve C in its Jacobian J can be identified as the common zeros of these integrals. This establishes the connection between torsion points and p-adic abelian integrals.
Some interesting examples of torsion points on genus \(g=2\) curves are given at the end of the paper showing, among other things, that the primes 2 and 3 play a special role and that the CM-hypothesis is indispensable in the theorem. boundedness conjecture for group of rational torsion points; elliptic curve; Mordell-Weil theorem; abelian variety; potential complex multiplication; torsion points on the Fermat curves; p-adic abelian integrals Coleman, Robert F., Torsion points on curves and \textit{p}-adic abelian integrals, Ann. of Math. (2), 121, 1, 111-168, (1985), MR782557 Analytic theory of abelian varieties; abelian integrals and differentials, Local ground fields in algebraic geometry, Complex multiplication and abelian varieties, Rational points, Special algebraic curves and curves of low genus, Elliptic curves, Algebraic functions and function fields in algebraic geometry, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) Torsion points on curves and p-adic abelian integrals | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study when an ordering \(\alpha\) in a real variety V can be described by a half branch. That means there exists \(\gamma: (0,\epsilon)\to V,\) analytic, such that for every \(f\in R[V]:\) \(sgn_{\alpha}f=sgn f(\gamma (t))\) for t small enough. It was proved by \textit{M. E. Alonso, J. M. Gamboa} and \textit{J. M. Ruiz} [J. Pure Appl. Algebra 36, 1-14 (1985; Zbl 0559.14015)] that this is true for real surfaces. Here we prove that the result is also true for any dimension if the valuation associated to the ordering has maximum rank, even more if the ordering is centered at a regular point we show that the curve can be extended \(C^{\infty}\) to \(t=0\). C\({}^{\infty }\) curve germs; ordering in a real variety Real algebraic and real-analytic geometry A note on orderings on algebraic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors discuss the classical formula for the genus of a plane algebraic curve applied to intersections of surface patches used in computer graphics and conclude that in almost all cases the genus is \(>1\) and therefore the curve is neither rational nor given by square roots of rational functions.
\{The correct formula to apply would be the genus of a space curve, intersection of two surfaces of degrees m and \(n: \pi =mn(m+n-4)+1-\Sigma r_ i(r_ i-1)\) where \(r_ i\) is the order of contact when it is \(>0\), cf. \textit{G. Salmon}, Cambr. and Dublin Math. J. 5, 24 ff. (1849) who also treated the case of base points and curves. In any case, the curve can always be given by a Puiseux series which can be approximated for computer use by a finite asymptotic expression.\} genus of a plane algebraic curve; intersections of surface patches; computer graphics; genus of a space curve; Puiseux series Katz S, Sederberg T. Genus of the intersection curve of two rational surface patches. CAGD, 1988, 5(3): 253--258. Algorithms for approximation of functions, Descriptive geometry, Graphical methods in numerical analysis, Projective techniques in algebraic geometry Genus of the intersection curve of two rational surface patches | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\Gamma\) be a discrete subgroup of \(\text{SL}_ 2(\mathbb R)\) such that the corresponding modular curve \(X=X(\Gamma)\) has finite volume. It is of interest to study the subgroup \(C(\Gamma)\) of \(J=\text{Jac}(X)\) generated by the divisors of degree \( 0\) supported on the cusps of \(X\). If \(\Gamma\) is a congruence subgroup, then Manin and Drinfeld used the theory of Hecke operators to prove that \(C(\Gamma)\) is finite. A second proof can be given by explicitly constructing modular functions with the appropriate zeros and poles [see \textit{D. S. Kubert} and \textit{S. Lang}, ``Modular units'' (1981; Zbl 0492.12002)].
In this paper the authors give a third proof, based on ideas of \textit{B. Schoeneberg} [``Elliptic modular functions'' (1974; Zbl 0285.10016)], \textit{G. Stevens} [``Arithmetic on modular curves'', Prog. Math. 20 (1982; Zbl 0529.10028)], and \textit{A. J. Scholl} [Math. Proc. Camb. Philos. Soc. 99, 11--17 (1986; Zbl 0564.10023)]. Associated to a cuspidal divisor is a differential of the third kind, which in turn is given by an Eisenstein series of weight 2; and the divisor has finite order in \(J\) if and only if the Eisenstein series has algebraic Fourier coefficients. The authors review this material, and then use Ramanujan sums to obtain an explicit expression for the Fourier coefficients. Since this expression is visibly algebraic, they conclude that \(C(\Gamma)\) is finite.
In case \(\Gamma\) is not a congruence subgroup, it is possible for \(C(\Gamma)\) to be infinite. The authors next consider the well-known realization of the Fermat curve \(F_ N: X^ N+Y^ N=1\) as \(X(\Gamma)\) for a non-congruence subgroup, where the cusps are the \(N\) points ``at infinity''. They find that the finiteness of \(C(\Gamma)\) is equivalent to the algebraicity of a complicated (but very explicit) expression involving generalized Ramanujan sums. Since \textit{D. E. Rohrlich} [Invent. Math. 39, 95--127 (1977; Zbl 0357.14010)] has shown in this case that \(C(\Gamma)\) is finite, the authors conclude that their expression is algebraic.
In the final section the authors look at the (unramified) correspondence between \(F_ N\) and \(X(2N)\) over \(X(2)\) considered by Kubert and Lang (op. cit.). This gives a divisor on the surface \(F_ N\times X(2N)\), and they show that this divisor has finite order in the relative Néron-Severi group \(\text{NS}(F_ N\times X(2N))/(\text{NS}(F_ N)\oplus \text{NS}(X(2N)))\). number of divisors; modular curve; cuspidal divisor; Ramanujan sums; Fourier coefficients; Fermat curve [MR]V.K. Murty andD. Ramakrishnan, The Manin-Drinfeld Theorem and Ramanujan Sums, Proc. Indian Acad. Sci. 97 (1987), 251--262. Arithmetic ground fields for curves, Global ground fields in algebraic geometry, Holomorphic modular forms of integral weight The Manin-Drinfeld theorem and Ramanujan sums | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0563.00006.]
Let G be a semisimple connected linear algebraic group over an algebraically closed field k. Let \(\sigma\) be an involution on G. Let \(K=G^{\sigma}\) and H \(= normalizer\) of K in G. C. De Concini and C. Procesi have constructed a canonical projective G-equivariant compactification X of G/H. This X is referred to as a complete symmetric variety. In this paper, the authors determine the Betti numbers of a complete symmetric variety, by writing down its Poincaré polynomial in an explicit way. In fact, the authors exhibit two methods for writing the Poincaré polynomial; the first method uses a cellular decomposition of X, while the second method uses reduction modulo p and the counting of F- rational points, F being a finite field. This paper is an important contribution to the study of complete symmetric varieties. Betti numbers of a complete symmetric variety; Poincaré polynomial; cellular decomposition; rational points Corrado De Concini and Tony A. Springer, \(Betti numbers of complete symmetric varieties. \)Geometry today (Rome, 1984), 87--107, Progr. Math., 60, Birkhauser Boston, Boston, MA, 1985. Homogeneous spaces and generalizations, Enumerative problems (combinatorial problems) in algebraic geometry, Classical real and complex (co)homology in algebraic geometry Betti numbers of complete symmetric varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study the behaviour of invariants like the Euler characteristic defined for (possibly singular) algebraic varieties under (projective) morphisms. In particular they are interested in the question if an invariant has what they call the SMP (stratified multiplicative property), which means the following. If we have a surjective morphism \(f : X \to Y\) with general fibre \(F\), then an invariant \(e\) is said to have the SMP, if \(e(X) = e(F) e(Y) + C\), where \(C\) is a correction term of a very specific form, namely \(C = \sum \widehat e (\overline V) [e(P_{V,f}) - e(F) e(P_{V,Y})]\). Here the sum is over strata \(V\) over which the map is locally trivial. \(P_{V,Y} \) is the projective normal cone of \(V\) in \(Y\), \(P_{V,f}\) its general fibre, and \(\widehat e\) an inductively defined corrected version of \(e\): \(\widehat e (\overline V) : = e (\overline V) - \sum \widehat e (\overline W) e(P_{W,V})\), where the sum runs over all strata \(W\) of \(\overline V \backslash V\).
The authors state that the genus \(\chi_y (X)\) and a similarly defined genus in intersection homology \(I\chi_y (X)\) has the SMP. Furthermore, the authors give an explicit formula for the Todd-class of a (simplicial) toric variety \(X_\Sigma\) for which they used the SMP of the genus \(\chi_y\). As an application a rather complicated formula for the \(r\)-th coefficient of the Ehrhart polynomial for a simplex is given. stratified multiplicative property; Todd-class of a toric variety; Euler characteristic; Ehrhart polynomial S. E. Cappell and J. L. Shaneson, ''Genera of algebraic varieties and counting of lattice points,'' \textit{Bulletin (New Series) Amer. Math. Soc.}, \textbf{30}, No. 1 (1994). Topological properties in algebraic geometry, Birational geometry Genera of algebraic varieties and counting of lattice points | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper we classify the subcanonical (smooth connected) space curves \(C\) (i.e. smooth connected curves \(C\subseteq \mathbb{P}^3\) such that \(\omega_C \simeq {\mathcal O}_C (\alpha)\), for some integer \(\alpha)\) on certain smooth surfaces \(S\). Such classification is complete when \(S\) is a smooth cubic (see theorem 2.3) or a smooth quartic containing a line and with \(\text{Pic} (S) \simeq\mathbb{Z}^2\) (see proposition 4.8 and remark 4.9). We also give the list of all subcanonical curves \(C\) on smooth surfaces \(S\) of degree at least 4, provided that \(S\) contains a line and the class of \(C\) in \(\text{Pic} (S)\) is in the subgroup generated by the line and by the plane section (see theorem 3.4): from results of \textit{A. F. Lopez} [``Noether-Lefschetz theory and the Picard group of projective surfaces,'' Mem. Am. Math. Soc. 438, 100 p. (1991; Zbl 0736.14012)] this suffices to get the complete classification when \(S\) is general among the surfaces containing a line (see remark 3.6). -- All results are applications of theorem 1.2 where we translate the existence of non-complete intersection, \(\alpha\)-subcanonical curves with \(\alpha\geq \deg (S)-3\) in terms of the existence of effective divisors \(Y\) on \(S\) with \(h^0 ({\mathcal O}_Y) =1\) (fixed divisors).
In order to work, this approach needs a good description of the Picard group of the considered surfaces, but unfortunately this is only available in a few cases. -- Some extensions of the previous results are obtained in case of a smooth surface \(S\) containing a given complete intersection curve provided that \(\deg (S)\) is big enough in the paper by the reviewer and \textit{Ph. Ellia} [``Curves on generic surfaces of high degree through a complete intersection in \(\mathbb{P}^3\)'', Geom. Dedicata 65, No. 2, 203--213 (1997; Zbl 0880.14015)]. subcanonical curves on smooth surfaces; space curves; non-complete intersection; existence of effective divisors; Picard group Plane and space curves, Special surfaces, Projective techniques in algebraic geometry On subcanonical curves lying on smooth surfaces in \(\mathbb{P}^ 3\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(m\) be a fixed positive integer, and let \(F_ m\) denote the complete plane curve over the complex field with projective equation \(X^ m+Y^ m+Z^ m=0\); \(F_ m\) is called the Fermat curve of exponent \(m\) over \(\mathbb{C}\). Let \(J_ m\) denote the Jacobian of \(F_ m\). The object of this paper is to give a characterization of the endomorphism ring \(\text{End}(J_ m)\) of \(J_ m\) when \(m\) is relatively prime to 6. The author gives necessary and sufficient conditions for an element of \(\text{End}^ 0(J_ m)=\text{End}(J_ m)\otimes\mathbb{Q}\) to be element of \(\text{End}(J_ m)\). In particular, he finds examples of endomorphisms of \(J_ m\) which are not induced from elements of the integral group ring \(\mathbb{Z}[\text{Aut}(F_ m)]\). Jacobian of Fermat curve; endomorphism ring Lim, C. H.: Endomorphisms of Jacobian varieties of Fermat curves. Compositio math. 80, 85-110 (1991) Jacobians, Prym varieties, Algebraic theory of abelian varieties, Automorphisms of curves, Special algebraic curves and curves of low genus Endomorphisms of Jacobian varieties of Fermat curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Plane rational algebraic curves can be described either implicitly as loci of bivariate polynomials, or parametrically as the Zariski closure of a set of points derived from evaluating a pair of rational functions. Both descriptions have their advantages, and transformation algorithms for parametrization and implicitization have been described in the literature. In this paper the author takes a parametric description of a rational curve and gives formulae for certain interesting characteristica of the curve, such as the degree and the multiplicity of singularities. Singularities can even be classified into ordinary and non-ordinary ones without resorting to the implicit defining polynomial of the curve. rational curve parametrization; algebraic curve; degree of an algebraic curve; singularities of an algebraic curve; multiplicity of a point Pérez-Díaz, S., Computation of the singularities of parametric plane curves, J. Symb. Comput., 42, 8, 835-857, (2007) Computational aspects of algebraic curves, Singularities of curves, local rings Computation of the singularities of parametric plane curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author gives a description of the space of projective unitary representations of the orbifold fundamental group of a compact holomorphic orbifold \(X\) of dimension 2 in terms of the moduli space of stable parabolic bundles over \(X\). This allows the author to compute the cohomology of the \(SU(2)\)-character variety \({\mathcal R}(\Sigma)\) of any Seifert-fibered homology sphere. In particular, he shows that \(H^ k({\mathcal R}(\Sigma))=0\), for \(k\) odd. This extends to a complete computation of the cohomology of the character variety for Seifert fibrations which are torsion-free and of genus one and a partial computation for genus greater than one. Related results have been independently obtained by \textit{M. Furuta} and \textit{B. Steer} [Adv. Math. 96, No. 1, 38-102 (1992)] and \textit{S. Bauer} [Math. Ann. 290, No. 3, 509- 526 (1991; Zbl 0752.14035)]. \(SU(2)\)-character variety of a Seifert-fibered homology sphere; compact holomorphic orbifold of dimension 2; space of projective unitary representations of the orbifold fundamental group; moduli space of stable parabolic bundles Boden Hans U, Representations of orbifold groups and parabolic bundles, Comment. Math. Helv. 66(3) (1991) 389--447 Topology of general 3-manifolds, Algebraic moduli problems, moduli of vector bundles, Yang-Mills and other gauge theories in quantum field theory, Topology of Euclidean 2-space, 2-manifolds, Fundamental group, presentations, free differential calculus Representations of orbifold groups and parabolic bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the review below of the author's same titled paper in C. R. Acad. Sci., Paris, Sér. I 312, No. 11, 853-856 (1991). product theorem of Nadel; bound for Chern class; Fano variety; movable rational curve Campana, F. Une version géométrique généralisée du théorème du produit de Nadel,Bull. Soc. Math. France 119(4), 479--493 (1991). Fano varieties, Characteristic classes and numbers in differential topology A generalized geometric version of the product theorem of Nadel | 0 |
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