text stringlengths 571 40.6k | label int64 0 1 |
|---|---|
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 0624.00007.]
For rigid analytic varieties (analytic) covering maps are defined with respect to the Grothendieck topology of affinoid subspaces [see \textit{M. van der Put}, Ann. Inst. Fourier 33, No.1, 29-52 (1983; Zbl 0495.14017)]. This leads to corresponding notions of fundamental group and universal covering in rigid analysis. The author studies the basic properties of these concepts and gives a number of (mostly well-known) examples: The punctured affine line \(k^*\) is simply connected, as is any affinoid subspace of the projective line, and the Schottky uniformization of a Mumford curve is the universal covering. It is an open question whether the n-dimensional unit ball is simply connected for \(n\geq 2\). rigid analytic varieties; covering maps; Grothendieck topology; fundamental group; Schottky uniformization of a Mumford curve P. Ullrich, Rigid analytic covering maps , Proceedings of the conference on \(p\)-adic analysis (Houthalen, 1987), Vrije Univ. Brussel, Brussels, 1986, pp. 159-171. Local ground fields in algebraic geometry, Coverings in algebraic geometry, Non-Archimedean analysis Rigid analytic covering maps | 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 \(U(n,d)\) denote the moduli space of stable vector bundles of rank \(n\) and degree \(d\) on a generic curve \(C\) of genus \(g\geq 2\) over an algebraically closed field. Consider the set of subbundles \(E'\) of rank \(n'\) of an element \(E\in U(n,d)\) and define \(s_{n'}(E)= n'd-n\max (\deg E')\). It is well known that \(0<s_{n'} (E)<n'(n-n')g\). Hence one can stratify \(U(n,d)\) according to the value of \(s_{n'}\). Define \(U_{s,n}(n,d): =\{E\in U(n,d)\mid s_{n'} (E)=s\}\) for any \(0<s\leq n'(n-n')g\). These loci are locally closed, the function \(s_{n'}\) being upper semicontinuous. The question is whether these sets are non-empty. This has been shown for rank 2 by \textit{H. Lange} and \textit{M. S. Narasimhan} [Math. Ann. 266, 55-72 (1983; Zbl 0507.14005)], for arbitrary rank and \(0<s< \min(n',(n-n') (g-1))\) by \textit{E. Ballico}, \textit{L. Brambila-Paz} and \textit{B. Russo} [Math. Machr. 194, 5-11 (1998; Zbl 0938.14013)], for arbitrary rank by \textit{L. Brambila-Paz} and \textit{H. Lange} under the assumption \(g\geq (n+1)/2\) [J. Reine Angew. Math. 494, 173-187 (1998; Zbl 0919.14016)] and in general by \textit{B. Russo} and \textit{M. Teixidor} i \textit{Bigas} [J. Algebr. Geom. 8, No. 3, 483-496 (1999; Zbl 0942.14013)]. Here the author gives another proof applying a degeneration technique. It gives slightly more information, namely
(i) If \(0<s\leq n'(n-n')(g-1)\) then a generic \(E\in U_{s,n'} (n,d)\) admits only finitely many maximal subbundles of rank \(n'\).
(ii) If \(s\geq n'(n-n')(g-1)\) the set of maximal subbundles of a generic \(E\in U_{s,n'}(n,d)\) is of dimension \(s-n'(n-n')(g-1)\).
(iii) The results hold roughly speaking for any semistable curve of genus \(g\), and
(iv) the loci \(U_{n',S}(n,d)\) are irreducible.
The proof is by reduction to a curve of genus \(g-1\) with an elliptic tail. It uses heavily Atiyah's classical results on vector bundles on elliptic curves. extensions of vector bundles; moduli space of stable vector bundles on a generic curve; subbundles; elliptic tail Teixidor i Bigas, J. Reine Angew. Math. 528 pp 81-- (2000) Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles On Lange's conjecture | 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 Translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No.5, 1135-1150 (Russian) (1983; Zbl 0514.14022). topology of real algebraic manifolds; number of ovals; complex orientations of a real curve; non-existence of curves of isotopy types; M-curve O. Ya. Viro,Real plane curves of degrees 7 and 8: new prohibitions, Mathematics of the USSR Izvestia23 (1984), 409--422. Enumerative problems (combinatorial problems) in algebraic geometry, Topological properties in algebraic geometry, Special algebraic curves and curves of low genus, Real algebraic and real-analytic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects) Real plane curves of degrees 7 and 8: New prohibitions | 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 one opens ``Formal Groups and Applications'', one realizes immediately that it has an enthusiastic author who wants to bring the subject across to the reader. The book begins with a table of contents, has a preface, a ``Leitfaden and Indicien'' and an introduction. After reading all this, one knows that further aids are in the extensive bibliography (of 511 titles), a detailed index and an alphabetical collection of all used notations, subdivided into ``standard notations'', ``generic notations'', and ``incidental notations''. On top of that, each of the seven chapters has its own bibliographical, historical and other notes. Furthermore certain sidelines of the theory appear in extensive remarks within the chapters. The most helpful achievement is -- for the reviewer's taste -- that each chapter has a first paragraph of a dozen pages with the title ``Definitions and survey of the results'', after which one knows precisely the mathematical objects, their problems, the important properties, theorems, and examples, which will be worked out in the rest of the chapter. No doubt, this book will not only be the standard reference for formal groups but also an excellent textbook for everybody who wants to learn more about formal groups.
The book studies formal groups in the sense of power series, bialgebras are only introduced in the last chapter, the functorial approach is completely omitted.
Chapter I: This chapter gives definitions of the fundamental concepts for one-dimensional formal groups and the most important theorems about them. A one-dimensional formal group law over a ring \(A\) is a formal power series \(F(X,Y)\in A[[X,Y]]\) of the form \(F(X,Y) = X+Y+\) terms of higher degree in \(X\) and \(Y\) such that \(F(X,F(Y,Z)) = F(F(X,Y),Z)\). It is commutative, if \(F(X,Y) = F(Y,X)\). There exists always an inverse \(\iota(X)\in A[[X]]\) such that \(F(X,\iota(X)) = 0\). A homomorphism \(F(X,Y)\rightarrow G(X,Y)\) between two formal group laws over \(A\) is a power series \(\alpha(X)\in A[[X]]^+\) without constant term, such that \(\alpha(F(X,Y)) = G(\alpha(X),\alpha(Y))\). Isomorphisms are homomorphisms which possess an inverse homomorphism.
The fundamental example of an isomorphism consists of the additive (formal) group (law) \(\widehat G_a = X + Y\), the multiplicative group \(\widehat G_m(X,Y)= X + Y + XY\) and the isomorphisms \(E(X) = \displaystyle \sum_{n=1}^\infty \frac{X^n}{n!}\) resp. \(\log (1 + x) = \displaystyle \sum_{n=1}^\infty (-1)^{n+1}\frac{X^n}{n}\) over the ring \(A = \mathbb Q\) and also over every \(\mathbb Q\)-algebra \(A\). For fields \(A\) of characteristic \(p\), however, \(\widehat G_a\) and \(\widehat G_m\) are not isomorphic.
The first important theorem is, that every one dimensional, commutative formal group law \(G\) (not only \(\widehat G_m)\) over a \(\mathbb Q\)-algebra \(A\) is isomorphic to \(\widehat G_a\) by a generalized logisomorphism \(f(X) \in A[[X]^+\). Thus \(G(X,Y) = f^{-1}(f(X) + f(Y))\) and every \(f\) induces a group law over \(A\). If \(A\) is a \(\mathbb Z\)-algebra, \(A\rightarrow A\otimes \mathbb Q\) injective, and \(f(X)\in A\otimes \mathbb Q[[x]]^+\) satisfies a certain functional equation, then \(G(X,Y) = f^{-1}(f(X) + f(Y))\) has coefficients in \(A\). So \(f\) defines a commutative formal group law over \(A\), which is not necessarily isomorphic to \(\widehat G_a\).
Another important theorem states, that all one-dimensional formal group laws over \(A\) are commutative, iff \(A\) contains no nonzero nilpotent torsion elements.
The main aim of this chapter is, however, to construct a universal commutative one-dimensional formal group law . It consists of a formal group law \(F_U(X,Y)\) over a ring \(L\), such that for every one-dimensional commutative formal group law \(G(X,Y)\) over a ring \(A\) there is a unique ring homomorphism \(\varphi\colon L\rightarrow A\) such that \(\varphi_*F_U(X,Y) = G(X,Y)\). It turns out that \(L = \mathbb Z[U_2,U_3,\ldots]\) is a polynomial ring. Thus every one-dimensional commutative formal group law is of the form \(\varphi_*F_U(X,Y)\) and every ring homomorphism \(\varphi\colon \mathbb Z[U_2,U_3,\ldots] \rightarrow A\) induces such a law. Observe that not all group laws over \(L\) are isomorphic, nor that every group law isomorphic to \(F_U(X,Y)\) over \(L\) is universal. The central technique is given by the functional equation lemma here proved in a very general and powerful form, which is too involved to be quoted here. This is the first time that this lemma is published in such a general form. Examples of formal groups are given with constructions due to Honda resp. Lubin-Tate.
Chapter II: Here one finds the fundamental theory of higher dimensional formal group laws \(F(X,Y)\). The difference to the one-dimensional formal group laws is, that one needs \(n\) formal power series in \(2n\) variables for the \(n\)-dimensional case. The additive group is then \(\widehat G_a^n\), the multiplicative group \(\widehat G_m\) (the automorphism group of a 1-dimensional space) changes to a noncommutative \(n^2\)-dimensional general linear group. Part of the 1-dimensional theory can be extended, in particular, the functional equation lemma. As one important consequence one obtains, that every commutative formal group law over a \(\mathbb Q\)-algebra \(A\) is isomorphic to (a power of) the additive group \(\widehat G_a^n\). Again there is a universal \(n\)-dimensional formal group law with corresponding ring \(\mathbb Z[U_1,U_2,\ldots]\). But here the differences from the 1-dimensional case begin. It turns out that this universal formal group law does not generalize the 1-dimensional case. Curvilinearity, i.e. the vanishing of certain mixed terms in \(F(X,Y)\), is one of the new phenomena here, although every commutative \(n\)-dimensional formal group law is isomorphic to a curvilinear one.
After the construction of generalized Honda and Lubin-Tate group laws, the characteristic zero Lie theory is done. Here the main theorem is, that the category of finite-dimensional formal group laws and the category of (free) finite rank Lie algebras are equivalent.
Chapter III studies groups of curves in commutative formal group laws \(F(X,Y)\). The elements are \(n\)-tuples of power series \(\gamma(t)\in A[[t]]^+\). The ``addition'' is \(\gamma_1(t) + \gamma_2(t) = F(\gamma_1(t), \gamma_2(t))\). The curves in \(F(X,Y)\) form a group with additional operators, the Verschiebung, Frobenius and homothety operators. The Frobenius operators are introduced by primitive roots of unity as well as by elementary symmetric polynomials. Curves in \(F(X,Y)\) are called \(p\)-typical (for a prime \(p)\) if the \(q\)-th Frobenius operators for all primes \(q\ne p\) vanish on them. \(F(X,Y)\) is \(p\)-typical if the curves which consist just of \(p^i\)-th powers of \(t\) are \(p\)-typical (in characteristic zero). Again there is a universal \(p\)-typical group law. Over \(\mathbb Z_{(p)}\) it turns out that every formal group law is isomorphic to a \(p\)-typical formal group law.
The main aim of this chapter is the introduction of generalized Witt polynomials using the coefficients of the logarithm of a one-dimensional formal group law \(F(X,Y)\) in characteristic zero. Using these Witt polynomials, rings (groups) of infinite Witt vectors and an Artin-Hasse exponential mapping over \(F(X,Y)\) are constructed and studied. Special cases are the ordinary infinite Witt vectors coming from \(\widehat G_m^-(X,Y) =X +Y - XY\), the Witt vectors with respect to a prime \(p\) and ``ramified Witt vectors'' over certain Lubin-Tate formal groups.
In chapter IV homomorphisms and isomorphisms of commutative formal group laws are studied in more detail. In particular cases a classification of all isomorphisms by Eisenstein polynomials is given. Furthermore in characteristic \(p\) the concept of height of a homomorphism is introduced, the height of a formal group law corresponds to the height of the endomorphism induced by \(p\). The concept of height allows the definition of a topology on the set of homomorphisms between certain formal group laws, such that this set is complete. This leads to the classification of the one-dimensional formal group laws by their heights and Galois descent. Also the moduli problem is attacked, the problem of classifying liftings of one-dimensional formal group laws from a residue field to a complete noetherian ring. A large part of this chapter is filled by the theory of formal \(A\)-modules, i.e. formal group laws on which an algebra \(A\) operates by endomorphisms. It turns out that most of the theory developed up to here, especially in the one-dimensional case, generalizes to formal \(A\)-modules.
In chapter V it is shown that each commutative formal group law defines a set of curves. These curves form an abelian complete Hausdorff topological group with certain continuous operators, a so-called reduced Cartier-Dieudonné module. This construction is a functorial equivalence, and the functor from commutative formal group laws to reduced Cartier-Dieudonné modules is representable by the formal group law of Witt vectors. Similar results hold for \(A\)-modules. Over an algebraically closed field, the finite dimensional \(A\)-modules may be decomposed ``up to isogeny'' into very simple components. A paragraph on the ``tapis de Cartier'' closes this chapter.
Chapter VI introduces various applications of formal group laws, to name a few: complex oriented cohomology theories, Brown-Peterson cohomology, Tate modules, local class field theory, and zeta functions of elliptic curves.
The last, fairly short, chapter VII gives an introduction on how to study for formal group laws their co- and contravariant bialgebras (or hyperalgebras), their connection with the Lie algebra of a formal group law, and how curves can be considered as sequences of divided powers. Furthermore, a noncommutative analogue of the ring of Witt vectors is constructed.
In two appendices a brief introduction to power series rings is given for the convenience of the reader and some more examples where the whole theory might be applied, e.g. global class field theory, \(p\)-divisible groups, lifting abelian varieties, arithmetic algebraic geometry, \(L\)-functions on elliptic curves, extraordinary \(K\)-theories, and Hopf rings. Honda group law; additive group; universal n-dimensional formal group law; Lubin-Tate group laws; finite rank Lie algebras; generalized Witt polynomials; Eisenstein polynomials; height; formal A-modules; reduced Cartier-Dieudonné module; Brown-Peterson cohomology; hyperalgebras M. Hazewinkel, \textit{Formal Groups and Applications}, Acad. Press, New York (1978). Formal groups, \(p\)-divisible groups, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Affine algebraic groups, hyperalgebra constructions, Witt vectors and related rings, Formal power series rings Formal groups and applications | 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 \(R\) be a strictly henselian discrete valuation ring, \(\overline R\) the integral closure of \(R\) in an algebraic closure of \(K : = \text{Fr}R\), and let \({\mathcal X}\) be a normal, with geometrically connected fibres, semistable \(R\)-curve. Let \({\mathcal U}\) be the open subscheme of \({\mathcal X}\) obtained by deleting the double points and let \(\overline {\mathcal U} : = {\mathcal U} \times_R \overline R\). We give a description of the fundamental group \(\pi_1 (\overline {\mathcal U})\) in terms of the fundamental group of a certain graph of groups. As an application we determine a quotient of the fundamental group of a generic curve. graph of groups; fundamental group of a generic curve Saı\ddot{}di, M.: Revêtements étales et groupe fondamental de graphe de groupes. C. R. Acad. sci. Paris sér. I 318, 1115-1118 (1994) Coverings of curves, fundamental group, Homotopy theory and fundamental groups in algebraic geometry Etale coverings and fundamental group of graphs of 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 Some results about the analytic branches of an algebraic affine variety along a singular subvariety are proved, using the theory of henselian rings. Precisely, let \(X=Spec(A)\), with A a noetherian domain, and Y a closed irreducible subvariety of X corresponding to the prime \({\mathfrak p}\) of A. The first result is that the global branches of X along Y, which by definition are the minimal primes of the henselization of the couple (A,\({\mathfrak p})\), correspond to the connected components of \(p^{-1}(Y)\), where \(p: X'\to X\) is the normalization morphism. Moreover, there is an open subset U of X such that there is a natural canonical correspondence between the global branches of U along \(U\cap Y\) and the branches of X at the generic point y of Y. A similar result is then proved for the geometric global branches of X along Y, i.e. the minimal primes of the strict henselization of the couple (A,\({\mathfrak p})\), replacing the branches of X in y with the geometric branches of X in y. Furthermore it is shown how to reconstruct the local rings of the branches at each point of a dense open subset of Y knowing the branches at the generic point y, under some conditions for the behaviour of X along Y. This result is finally extended to the general case, passing to a suitable étale covering of X.
For closely related results proved with completely different topological techniques see the paper by \textit{G. Tedeschi}, Boll. Unione Mat. Ital., VI. Ser., D, Algebra Geom. 4, No.1, 17-27 (1985; see the preceding review)]. analytic branches of an algebraic affine variety along a singular subvariety; henselian rings; geometric global branches Ramification problems in algebraic geometry, Henselian rings Global branches and parametrization | 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 \(n,q>0\) be integers, and let \(IMM_q^n\) be the polynomial on \(n\)-tuples of \(q\times q\) matrices whose value on \((X_1,\dots,X_n)\) is \(\text{tr}(X_n\cdots X_1)\). Then \(IMM_q^n\) is a homogeneous polynomial in \(nq^2\) variables; if we let \(V\) be the complex vector space of \(n\)-tuples of \(q\times q\) matrices, then we write \(IMM_q^n\in S^{n}V\). In the work under review, some geometric properties of \(IMM_q^n\) are given. The tools used are representation theory and algebraic geometry.
Geometric Complexity Theory seeks to study polynomials which are characterized by their symmetry group. Recall that \(\text{GL}(V)\) acts on \(S^nV\); the symmetry group \(\mathcal{S}\) of \(IMM_q^n\in S^nV\) is the stabilizer of \(IMM_q^n\) under this action. A main result in this paper is that \(\mathcal{S}\cong \mathcal{S}_0\rtimes D_n\), where \(\mathcal{S}_0\) is the connected component of the identity of \(\mathcal{S}\) and \(D_n\) is the group of symmetries of a regular \(n\)-gon. Alternatively, any polynomial in \(S^nV\) which is stabilized by \(\mathcal{S}_0\) is shown to be a complex multiple of \(IMM_q^n\), hence \(IMM_q^n\) is characterized by its stabilizer.
The remainder of the results concerns the hypersurface determined by the polynomial. Let \(\mathcal{I}mm_q^n\) be the hypersurface \(V(IMM_q^n)\subseteq\mathbb{P}V^*\). It is shown that its dual variety \((\mathcal{I}mm_q^n)^{\vee}\) is also a hypersurface in \(\mathbb{P}V^*\). Let \(\mathcal{S}ing_q^n\) denote the singular locus of the affine cone in \(V^*\) over \(V(IMM_q^n)\subseteq\mathbb{P}V^*\). Then a description of the irreducible components of \(\mathcal{S}ing_q^n\) in terms of certain nilpotent representations of the Euclidean equioriented quiver and dimensions are computed; examples are provided for \(n\leq 3\). Finally, let \(\mathcal{W}=V(J)\) where \(J\) is the ideal of \(IMM_q^n\) generated by all \(n-2\) order partial derivatives. The irreducible components of \(\mathcal{W}\) are given, and it is shown that \(\text{dim} \mathcal{W}=\lfloor (5/4)q^2 \rfloor\). iterated matrix multiplication; symmetry group of a polynomial; Jacobian loci of hypersurfaces F. Gesmundo, \textit{Geometric aspects of iterated matrix multiplication}, J. Algebra, 461 (2016), pp. 42--64. Other algebraic groups (geometric aspects), Linear preserver problems, Representations of quivers and partially ordered sets Geometric aspects of iterated matrix multiplication | 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 unipolar \(\mathbb{Q}\)-Fano variety \(X\) is a \(n\)-dimensional complex projective variety such that \(X\) is normal and \(\mathbb{Q}\)-factorial and {Weil divisors}/(modulo numerical equivalence) equal \(\mathbb{Z} [D]\).
Theorem. (1) Suppose that \(X\) is log-terminal and 1-canonical (i.e., for any resolution \(f : Y \rightarrow X\) the differential \(df : f^*(\Omega_X)\rightarrow \Omega_Y\) factors through a map \(f^*(\Omega_X^{\vee \vee})\rightarrow \Omega_Y\), or equivalently 1-forms on the smooth locus of \(X\) lift to holomorphic forms on \(Y\)). Then \((-K_X)^n\) is bounded from above by the \(n\)-th power of the maximum of \(((n+1)\)(Cartier index of \(D\))) and \(-2(C . K_X)/\mu_{\min}(T_X)\), where \(C = H^{n-1}\) with a very ample divisor \(H\) and \(\mu_{\min}(T_X)\) is the minimum slope of the subquotients in a Harder-Narasimhan filtration of the tangent bundle \(T_X\) of \(X\) with respect to \(H\).
(2) Suppose that \(T_X\) is semi-stable then \((-K_X)^n\) is bounded from above by the \(n\)-th power of the maximum of \(2n\) and \(((n+1)\)(Cartier index of \(D\))).
The upper bound for \((-K_X)^n\) exists for smooth Fano varieties due to Nadel, Kollar-Miyaoka-Mori. The technique used here is different from known ones and is based on the study of positivity properties of sheaves of differential operators on ample line bundles. There are some related theorems in math.AG preprints by \textit{H. Tsuji} (\url{http://front.math.ucdavis.edu/}). singularity; Fano variety; positivity properties of sheaves of differential operators; line bundles; canonical divisor; Weil divisors Fano varieties, Divisors, linear systems, invertible sheaves A new method in Fano 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 The author considers projective curves \(C\) of degree \(d\) in \(\mathbb P^4\) subject to a flag condition of type \((s,t)\), which means that \(C\) does not lie on surfaces of degree \(< s\), nor on hypersurfaces of degree \(< t\). A bound \(G(d,s,t)\) for the arithmetic genus \(g_C\) of such curves was known in the range \(s>t^2-t\) and \(d\gg s\). The author proves that the bound also holds when \(s \leq t^2-t\), provided that \(d\) is sufficiently large with respect to \(t\). A precise inequality on \(d\) which implies that the bound \(g_C\leq G(d,s,t)\) holds is given in the paper in terms of the arithmetics of the three numbers \(d,s,t\). The author also determines several properties of curves for which the bound is attained. Consequent bounds on the speciality index of curves satisfying flag conditions are listed. genus of a complex projective curve; Castelnuovo-Halphen theory; flag condition Projective techniques in algebraic geometry, Special algebraic curves and curves of low genus A remark on the genus of curves in \(\mathbf{P}^4\) | 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 Für eine Raumcurve mit der Gleichung \(\varrho =\psi (t)\) ist der Bogen
\[
s =\int T \varrho' dt,
\]
wo \(\varrho'\) die Tangente an die Curve im Punkte \(\varrho\) bedeutet. Für eine Fläche \(\varrho =\chi (t,u),\) ist das Flächenstück
\[
S =\int TV \varrho_1' \varrho_2' dt du,
\]
wo \(\varrho_1'\) und \(\varrho_2'\) zwei in ihrem gemeinsamen Berührungspunkte sich schneidende Tangenten an die Fläche sind. Aehnliche Ausdrücke ergeben sich bei der Cubatur der Körper und für die Schwerpunkte der vorher betrachteten Gebilde. Auch die speciellen Fälle der Ebene und des Umdrehungskörpers sind berücksichtigt. Den Schluss bilden Anwendungen auf das Ellipsoid. arc of a curve; tangent of a curve; tangential surface; barycentre; quaternion Analytic theory of abelian varieties; abelian integrals and differentials The quaternion formulae for quantification of curves, surfaces and solids, and for barycentres. | 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 purpose of this paper is to study \(K_1\)-groups of algebraic curves, in particular the Block conjecture on it. For \(C\) a nonsingular projective curve over the complex number field \(\mathbb{C}\), the author studies \(K_1\) of the curve by using arithmetic Hodge structure.
First, the author reviews admissible variation of mixed Hodge structure and calculates the Yoneda extension groups. Then he constructs the regulator map
\[
\rho^2_C: V(C)\to \text{Ext}^2_{{\mathcal M}(C)}(Z(0),\;H^1(C,Z(2))
\]
[a supplement of the construction of \textit{M. Asakura}, CRM Proc. Lect. Notes 24, 133-154 (2000; Zbl 0966.14014)].
It is also given a survey of the theory of generalized Jacobian rings with the theorem on Mori's connectivity for open curves. This result and the Mordell-Weil theorem over function fields are used to prove the main theorems in this paper:
Theorem 1.2: Let \(C\subset\mathbb{P}^2_{\mathbb{C}}\) be a generic smooth plane curve of degree \(d\geq 4\), and \(\{P_1,\dots, P_n\}\) a set of closed points of \(C\) in a generic position. Then the map
\[
\bigoplus^{n-1}_{i=1} \mathbb{C}^*\otimes [P_i- P_{i+1}]\to \text{Ext}^2_{{\mathcal M}(C)}(Z(0), H^1(C,Z(2)))
\]
induced from \(\rho^2_C\) is injective modulo torsion.
Theorem 1.4: Let \(C\subset\mathbb{P}^2_\mathbb{C}\) be a generic plane curve of degree \(d\geq 4\). Then the map
\[
\overline \mathbb{Q}^*\otimes \text{Pic}^0(\mathbb{C})\to \text{Ext}^2_{{\mathcal M}(C)}(Z(0), H^1(C,Z(2)))
\]
induced from \(\rho^2_C\) is injective. \(K_1\)-groups of algebraic curves; arithmetic Hodge structure; generalized Jacobian rings Relations of \(K\)-theory with cohomology theories, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects) On the \(K_1\)-groups of 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 (Siehe auch JFM 09.0488.01) 1. In der Theorie der algebraischen Curven sind gewisse Punktgruppen zu betrachten, einerseits solche, welche Paaren conjugirter Punkte bei Kegelschnitten, andererseits soche, welche conjugirten Tripeln entsprechen. In Bezug auf die erstere Art verweist der Herr Verfasser auf die Arbeit von Battaglini: Sulle forme ternarie Differentialgleichung grado qualunque (Atti di Napoli IV.), Untersuchungen Bezug auf die andere auf Reye's Arbeit: Trägheits- und höhere Momente etc. (Borchardt J. LXXII.).
In der vorliegenden Arbeit werden solche Punktgruppen bei Curven dritten Grades betrachtet, doch lassen sich die erhaltenen Sätze, wie der Herr Verfasser bemerkt, nicht auch auf Curven beliebigen Grades übertragen. Drei Punkte werden in Beaug auf eine Curve dritten Grades conjugirt gennant, wenn die gemischte Polare in Bezug auf je zwei dieser Punkte durch den dritten geht.
(Unter der gemischten Polare einer Curve in Bezug auf zwei Punkte \(A\) und \(B\) versteht man bekanntlich die erste Polare in Bezug auf den einen von der ersten Polare der Curve selbst in Bezug auf den andern).
Vier Punkte heissen conjugirt in Bezug auf eine Curve dritter Ordnung, wenn je drei derselben conjugirt sind, oder wenn die Gerade, welche irgend zwei derselben verbindet, die gemischte Polare der beiden andern ist. Es wird nun in der Arbeit eine grosse Anzahl von Sätzen über derartige Punktgruppen aufgestellt.
2. Die Zweite Note betrifft Covarianten, welche bei der Betrachtung von Büschenln von Curven dritter Ordnung auftreten, insbesondere die Enveloppe der Hesse'schen Curven, die zu den Gliedern eines solchen Büschels gehören. Wenn ein Büschel von Curven dritter Ordnung gegeben ist, so bilden die ersten Polaren in Bezug auf einen beliebigen Punkt ebenfalls ein Büschel; und wenn der Pol eine Gerade beschreibt, beschreiben die vier Basispunkte des Büschels eine Curve \(4^{\text{ter}}\) Ordnung. Allen Geraden der Ebene entspricht demnach ein Netz von Curven \(4^{\text{ter}}\) Ordnung. Die Jacobische Curve \(J\) dieses Netzes ist der Ort der Punkte, welche zugleich Doppelpunkte des Büschels \(3^{\text{ter}}\) Ordnung sind. Durchläuft ein Punkt die Curve \(J\), so beschreiben die zusammenfallenden Basispunkte des Polarenbüschels eine Curve \(H\), die beiden nicht zusammenfallenden eine Curve \(K\). Die Curve \(H\) ist die Enveloppe der Hesse'schen Curven; sie ist von der Ordnung 12, vom Geschlecht 16 und von der Klasse 27. Die Curve \(K\) ist von der \(30^{\text{ten}}\) Ordnung und hat die Punkte \(D\) zu achtfachen Punkten.
Hieran knüpfen sich noch einige weitere Untersuchungen ähnlicher Art, in Bezug auf welche auf die Note selbst verwiesen werden muss. Third order curves; conjugated points; Hesse's curve; pencil of curves; envelope; Jacobian curve; covariant; nets Pencils, nets, webs in algebraic geometry, Curves in algebraic geometry Theorems about pencils of third order 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 thesis studies the classification of indecomposable vector bundles on a complete regular absolutely irreducible curve X defined over a perfect ground field K, as well as of those among them that are selfdual. This is done by ground field extension to the algebraic closure \(\bar K.\) A given indecomposable vector bundle (locally free \({\mathcal O}_ X\)- module) \({\mathcal M}\) splits over \(\bar K\) into Krull-Schmidt components \({\mathcal N}\). In particular, the classification is obtained for an elliptic curve X possessing a rational point. (The latter condition guarantees that \(D({\mathcal M})=End_ K{\mathcal M}/Rad\) is a commutative field K' and then the components \({\mathcal N}\) are already defined over the various imbeddings of K' in \(\bar K;\) in general one takes a maximal commutative field extension YK' contained in D(\({\mathcal M})\) over K.) The result generalizes that of Atiyah for K algebraically closed: for fixed rank n and degree the isomorphism classes of indecomposable \({\mathcal M}\) correspond to the places of the elliptic curve X/K with residue degrees that divide n. By definition, \({\mathcal M}\) is selfdual if there is an isomorphism \(b:{\mathcal M}\to {\mathcal M}^*\) (''bilinear form'') and, depending on M, this may then be taken either selfdual \((b=b^*)\) or anti-selfdual \((b=-b^*)\). They correspond to algebras D(\({\mathcal M})\) with an involution. One remarks that there may be selfdual \({\mathcal M}\) such that the corresponding \({\mathcal N}\) are not selfdual; this namely is the case when the involutions on D(\({\mathcal M})\) induced by the bilinear forms are all of the second kind (here char \(K\neq 2\) assumed). splitting of vector bundle; classification of indecomposable vector bundles on a complete regular absolutely irreducible curve A. Tillmann, Unzerlegbare Vektorbündel über algbraischen Kurven, Dissertation. Fern Universität, Hagen, 1983. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special algebraic curves and curves of low genus Unzerlegbare Vektorbündel über algebraischen Kurven | 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 \textit{R. F. Coleman} [``Effective Chabauty'', Duke Math. J. 52, 765-770 (1985; Zbl 0588.14015)], following Chabauty, has shown how to obtain good bounds on the cardinality of \(K\)-rational points \((K\) number field) on a curve of genus \(g \geq 2\), if the rank of the Mordell-Weil group \(J(K)\) of the corresponding Jacobian \(J\) is less than \(g\). -- The author applies these methods to the Fermat curve \(x^ p + y^ p = 1\). And he shows under certain assumptions that the second case of Fermat's Last Theorem is true.
In the first part it is shown that the conditions on the rank of \(J(K)\) in the Coleman-Chabauty paper (loc. cit.) are satisfied. Here the author has to assume the existence of suitable cocycles in the Selmer group. -- Then the second part contains a detailed applications of Coleman's method to the Fermat curve and other related curves. In the case that the prime number \(p\) is regular the existence of these cocycles can be guaranteed. This provides an independent proof of Kummer's result, also independent of the recent work of \textit{A. Wiles} [cf. Ann. Math., II. Ser. 141, No. 3, 443-551 (1995; Zbl 0823.11029)]. The paper shows in a good way how Coleman's method can be fashioned into a quite precise tool for bounding rational points on curves. effective Chabauty; cardinality of rational points on a curve; Fermat's Last Theorem; rank of the Mordell-Weil group; Fermat curve; rational points on curves W. G. McCallum, ''On the method of Coleman and Chabauty,'' Math. Ann., vol. 299, iss. 3, pp. 565-596, 1994. Rational points, Arithmetic ground fields for curves, Higher degree equations; Fermat's equation On the method of Coleman and 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 [For the entire collection see Zbl 0676.00006.]
F. W. Long has introduced a Brauer group BD(R,G) of a commutative ring R, and a group G, which consists of equivalence classes of dimodule algebras, on which the grading group G acts as a group of automorphisms; the multiplication is induced from the smash product. The author discusses the main result of his joint paper with \textit{T. Ford} [J. Pure Appl. Algebra 54, No.2/3, 197-208 (1988; Zbl 0662.16007)], which describes BD(R,\({\mathbb{Z}}/2{\mathbb{Z}})\), in case R is a commutative ring in which 2 is a unit and which has only trivial idempotents, by mapping it bijectively to \(MF(R)=:{\mathbb{Z}}/2{\mathbb{Z}}\times H^ 1(R,{\mathbb{Z}}/2{\mathbb{Z}})\times H^ 1(R,{\mathbb{Z}},2{\mathbb{Z}})\times B(R)\). Here B(R) is the Brauer group of R and \(H^ 1(R,{\mathbb{Z}}/2{\mathbb{Z}})\) is the étale cohomology group which classifies the Galois extensions of R with Galois group isomorphic to \({\mathbb{Z}}/2{\mathbb{Z}}\). The multiplication in BD(R,\({\mathbb{Z}}/2{\mathbb{Z}})\) is described in MF(R) as a twisted product involving the cup product.
This result and the description of various subgroups of BD(R,\({\mathbb{Z}}/2{\mathbb{Z}})\) are then applied in cases R is the coordinate ring of a nonsingular affine real curve. coordinate ring of a nonsingular affine real curve Brauer groups of schemes, Algebraic functions and function fields in algebraic geometry, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) The Brauer Long group of \({\mathbb{Z}}/2\) dimodule algebras | 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 a semisimple adjoint algebraic group \(G\) and a Borel subgroup \(B\), consider the double classes \(BwB\) in \(G\) and their closures in the canonical compactification of \(G\); we call these closures large Schubert varieties. We show that these varieties are normal and Cohen-Macaulay; we describe their Picard group and the spaces of sections of their line bundles. As an application, we construct geometrically a filtration à la van der Kallen of the algebra of regular functions on \(B\). We also construct a degeneration of the flag variety \(G/B\) embedded diagonally in \(G/B\times G/B\), into a union of Schubert varieties. This yields formulae for the class of the diagonal of \(G/B\times G/B\) in \(T\)-equivariant \(K\)-theory, where \(T\) is a maximal torus of \(B\). semisimple adjoint algebraic group; large Schubert varieties; Picard group; filtration; algebra of regular functions; flag variety; equivariant \(K\)-theory Brion, M; Polo, P, Large Schubert varieties, Represent. Theory, 4, 97-126, (2000) Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Representation theory for linear algebraic groups, \(K\)-theory of schemes, Group actions on varieties or schemes (quotients) Large Schubert 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 \(K\) be an algebraically closed field of characteristic zero and \(C\) an integal, nodal curve of arithmetic genus \(g\) over \(K\). Assume \(d\geq 2g-1\), \(d \geq 0\) and \(n \geq 2\) for certain integers \(d\) and \(n\). In this article, the author constructs a smooth compactification \(\bar{M}\) of the moduli space \(M\) of morphisms to projective space \(\mathbb{P}^n\) of a fixed degree \(d\) and computes the top intersection number \(c_1(\mathcal{O}_{\bar{M}}(1))^{(n+1)(d+1-g)+g-1}[\bar{M}]\) of a certain Cartier divisor \(X\) in \(M\) to be \((n+1)^g\). where \(g\) is the arithmetic genus of the curve \(C\).
The computation of the above result -- which was known in the case of a smooth curve (see \textit{A. Betram, G. Dasklopoulos} and \textit{R. Wentworth} [J. Am. Math. Soc. 9, No. 2, 529--571 (1996; Zbl 0865.14017)]) -- is made possible by the computation of the Chern character of the degree \(d\) Picard bundle on the natural desingularization \(\tilde{J}^d(C)\) of the compactified Jacobian \(\bar{J}^d(C)\) (i.e. the moduli space of degree \(d\) torsion-free sheaves of rank one on \(C\)). Further applications of this result to Brill-Noether loci are also given. nodal curves; torsion-free sheaves; generalized Jacobian; Picard bundle Bhosle Usha, N, Maps into projective spaces, Proc. Indian Acad. Sci. (Math. Sci.), 123, 331-344, (2013) Vector bundles on curves and their moduli Maps into projective spaces | 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 0747.00028.]
The curves of this article are locally Cohen-Macaulay in \(\mathbb{P}^ 3\) over an algebraically closed field \(k\). Associated to each curve \(C\) is a graded module \(M(C) = \displaystyle\oplus_ n H^ 1(\mathbb{P}^ 3), {\mathcal I}_ C(n))\) of finite length over \(R=k [x_ 0, x_ 1, x_ 2, x_ 3]\), called the Hartshorne-Rao module. It is well known that the even linkage classes of these curves can be classified by isomorphism classes of their Hartshorne-Rao modules, up to shift in grading. For each class there are minimal curves, namely those whose Hartshorne-Rao module is shifted furthest to the left. Given a nonzero graded module \(M\) of finite length over \(R\), some sufficiently large shift of \(M\) is the Hartshorne- Rao module of a smooth curve \(C\); the goal of this article is to explicitly construct a minimal curve for the class of \(C\) and to prove its uniqueness up to deformation. Moreover, this minimal curve has the property of Lazarsfeld-Rao, namely that all other curves in its even linkage class are obtainable from it by a sequence of basic double links, and a deformation. The minimal curve is constructed from two ``canonical resolutions'' of the ideal sheaf \({\mathcal I}_ C\): \(0 \to {\mathcal E} \to {\mathcal F} \to {\mathcal I}_ C \to 0\) and \(0 \to {\mathcal P} \to {\mathcal N} \to {\mathcal I}_ C \to 0\), where \({\mathcal F}\) and \({\mathcal P}\) are direct sums of invertible \({\mathcal O}_{\mathbb{P}^ 3}\)-modules and \({\mathcal N}\) and \({\mathcal E}\) are, up to addition of direct sums of invertible \({\mathcal O}_{\mathbb{P}^ 3}\)-modules, sheafifications of syzygies of the module \(M\) (minimality corresponds to the elimination of the extra summands). In general minimal curves of a class are not smooth, but it follows from the Lazarsfeld-Rao property that the minimal curves constructed here have minimal genus and degree for their class. The article concludes with specific illustrations of the construction in which \(M\) is first taken to be \(R\) modulo a regular sequence, and then Buchsbaum. biliaison; minimal curve of a linkage class; Hartshorne-Rao module; even linkage class Linkage, Plane and space curves, Syzygies, resolutions, complexes and commutative rings Minimal curves in even linkage classes | 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 \textit{R. Pandharipande} [Proc. Am. Math. Soc. 125, No.12, 3471--3479 (1997; Zbl 0883.14031)] counted algebraic elliptic curves of given degree, with given \(j\)-invariant, and passing through given generic points in the plane -- the answer is independent of the value of \(j\)-invariant, unless the latter equals 0 or 1728. The authors prove that this answer coincides with the number of tropical elliptic curves of given degree, with given \(j\)-invariant, and passing through given generic points (the tropical curves should be counted with natural multiplicities, coming from the moduli space of tropical elliptic curves, constructed in the paper). In particular, this weighted number of tropical curves is independent of the value of \(j\)-invariant and position of points. This gives a generalization of \textit{G. Mikhalkin}'s tropical enumeration [J. Am. Math. Soc. 18, 313--377 (2005; Zbl 1092.14068)] to elliptic curves with given \(j\)-invariant (although no explicit correspondence between tropical and classical elliptic curves is established). Immediate applications include a simplification of Mikhalkin's lattice path count for curves in the projective plane. elliptic curve; tropical curve; tropical variety; intersection product; moduli space of curves Kerber, M.; Markwig, H., Counting tropical elliptic plane curves with fixed \textit{j}-invariant, Comment. Math. Helv., 84, 2, 387-427, (2009) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), , Polyhedra and polytopes; regular figures, division of spaces, Enumerative problems (combinatorial problems) in algebraic geometry Counting tropical elliptic plane curves with fixed \(j\)-invariant | 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 set of \(\mathbb{F}_q\)-rational points of the Plücker embedding of the Grassmann variety \(G(l,m)\) as a projective system in \(\mathbb{P}^{{m \choose l}-1}\). The parameters of such systems and thereby of corresponding linear codes are calculated. We also computed some generalized weights of these codes and their spectra in the case \(l=2\). Plücker embedding of Grassmann variety; generalized weights Nogin, D.Yu., Codes Associated to Grassmannians, Arithmetic, Geometry and Coding Theory, Pellikaan, R., Perret, M., and Vladut, S., Eds., Berlin: Gruyter, 1996, pp. 145--154. Geometric methods (including applications of algebraic geometry) applied to coding theory, Grassmannians, Schubert varieties, flag manifolds Codes associated to Grassmannians | 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 Wir bezeichnen \(f(y)=a_m y^m+a_{m-1} y^{m-1} + \cdots +a_1 y+a_0.\; \varXi\) sei die Discriminante von \(f\); die \(a\) sind Function von \(x\). Ferner sei
\[
\varXi' =\frac{\partial\varXi}{\partial x};\quad \varXi_k =\frac {\partial\varXi} {\partial a_k},\quad \varXi_{ki}= \frac{\partial^2 \varXi} {\partial \alpha_k\partial \alpha_i}\,\dots;\quad (s_i t_k \dots)= \sum_{i,k\dots} s_i t_k \ldots
\]
Wenn \(x=\xi\) eine Wurzel von \(\varXi=0\), und \(y=\eta\) die entsprechende Doppelwurzel von \(f(y)=0\) ist, gilt die Formel
\[
\frac{\partial f(\eta)}{\partial a_k} \varXi' (\xi)=\varXi_k \frac {\partial f(\eta)}{\partial \xi}.
\]
Man kann \(\frac{\partial f(\eta)}{\partial a_k}\) stets von 0 verschieden annehmen. Verschwindet also \(\varXi\) von höherem, als dem ersten Grade, so ist entweder \(\varXi_k=0\) oder \(\frac{\partial f(\eta)}{\partial \xi}=0.\) Sei \(1)\;\varXi_k=0, \frac{\partial f(\eta)}{\partial \xi}\) nicht=0. Verschwindet \(\varXi\) von der Ordnung \(\mu\) für \(x=\xi\), so hat \(f \; \mu\) Paare Doppelwurzeln \(\eta_1, \eta_2 \ldots \eta_{\mu}\), wobei das Zusammenfallen von 2 Paaren Doppelwurzeln \(\eta_1, \eta_2\) eine dreifache Wurzel anzeigt, Hierfür hat man die Formel
\[
\left(\frac{\partial f(\eta_1)}{\partial a_{i_1}} \frac{\partial f(\eta_2)}{\partial a_{i_2}}\,\cdots\right)=\mu!\quad \varXi_{i_1 i_2} \ldots \frac {\partial f(\eta_1)}{\partial \xi}\;\cdots\;\frac{\partial f(\eta_\mu)}{\partial\xi}\,.
\]
Ist \(2)\quad \frac{\partial f(\eta)}{\partial \xi}=0, \varXi_k\) nicht \(=0\), so erhält man ähnliche Resultate
\[
\text{z.B.}\quad \frac{\partial f(\eta)}{\partial a_k}\;\varXi'' (\xi) =\varXi_k \frac{d^2 f(\eta)}{\partial \xi^2}.
\]
Bildet man die Discriminante \(D\) von \(\varXi\), so tritt diese unter der Form auf \(D=\varDelta B^2.C^3\), wobei (vorausgesetzt dass \(a_k =a_n^{(k)} x^n +\cdots +\alpha_0^{(k)}\) \(B=0\) von der Ordnung \(4n(m-2)(m-3)\) in den \(\alpha_{i}^{(k)}\) ist und die Bedingung anzeigt, dass \(f(y)\) für einen bestimmten Werth von \(x\) 2 Doppelwurzeln hat; wobei \(C=0\) von der Ordnung \(6n(m-2)\) die Bedingung für eine dreifache Wurzel; und \(\varDelta=0\) von der Ordung \(6mn-4(m+n)+4\) die Bedingung ist, dass für eine Doppelwurzel \(\eta\) von \(f(y)\) der Differenzialcoefficient \(\frac{\partial f(\eta)}{\partial \xi}\) verschwindet. Der Satz bleibt auch dann bestehen, wenn \(f\) eine Function mehrerer Veränderlichen ist.
Ist für homogene Coordinaten \(v=\varSigma a_{k \lambda \mu} x_1^k x_2^{\lambda} x_3^{\mu}\), \(k+\lambda +\mu =r\) gegeben, und die \(a\) sind Functionen \(m^{\text{ter}}\) Ordnung von \(y_1\), \(y_2\), \(y_3,\) so möge der Punkt \(y_1 y_2 y_3\) in erweiterter Bedeutung der Pol der entsprechenden Curve \(v\) heissen. Die Pole \((y)\) der Curven \(v\) mit Doppelpunkten bilden eine Curve \(\varDelta\) (Discriminante von \(v\)) der Ordung \(3m(r-1)^2\). Der Ort der Doppelpunkte von \(v\) ist eine Curve \(k\) der \(3m^2(r-1)^{\text{ten}}\) Ordung. Die Pole \((y)\), für welche \(v\) 2 Doppelpunkte oder eine Spitze hat, sind Doppelpunkte bez. Spitzen von \(\varDelta\). Es ist \(\varDelta\) von der Classe \(3m(r-1)(2mr-m-r+1).\)
Betrachtet man nun eine Curve \(u\) von der Ordnung \(n=m+r,\) so bilden die ersten, zweiten u. s. w. Polarcurven \textit{aller} Punkte der Ebene Systeme von Curven \(v\). Die Curve \(\varDelta\) für die \(m^{\text{ten}}\) Polarcurven ist gleich der Curve \(k\) für die \((n-m-1)^{\text{ten}}\) Polarcurven und umgekehrt. Steiner nennt diese Curven, ``conjugirte Kerncurven.'' Sie mögen \(s^{(\lambda)}\) bezeichnet werden, wo \(\lambda\) den Grad der Polarcurven angiebt, für die \(s^{(\lambda)}\) die Curve \(\varDelta\) ist. \(S^{(m)}\) und \(S^{(n-m-1)}\) entsprechen sich also punktweise. Die Tangenten von \(S^{(m)}\) sind die Linearpolaren der Punkte von \(S^{(n-m-1)}\) in Bezug auf die \((m-1)^{\text{ten}}\) Polaren der entsprechenden Punkte von \(S^{(m)}\). Die \((n-m)^{\text{ten}}\) Polaren der Punkte von \(S^{(n-m-1)}\) berühren \(S^{(m)}\) in den entsprechenden Punkten. Die \((m+1)^{\text{ten}}\) Polaren der Punkte von \(S^{(m)}\) berühren \(S^{(n-m-1)}\). Discriminants; polar curves; to vanish; double root; triple root; differential coefficient; function of several variables; homogeneous coordinates; pole of a curve; order; double points; locus; peak; tangent Algebraic functions and function fields in algebraic geometry, Plane and space curves, Real polynomials: location of zeros, Continuity and differentiation questions, Curves in Euclidean and related spaces, Steiner systems in finite geometry On certain formulae concerning the theory of discriminants; with applications to discriminants of discriminants, and to the theory of polar 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 arbitrary algebraic curve, we define an infinite sequence of invariants (meromorphic forms \(W^{(g)}_k(p_1,\dots,p_k)\) on the curve where \(k,g \in {\mathbb N}\) and \(p_1,\dots,p_k\) are points of the curve -- \textit{the reviewer}). We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition, we find that they can be used to defines a formal series, which satisfies formally a Hirota equation, and we thus obtain a new way of onstructing a \(\tau\)-function attached to an algebraic curve.
These invariants are constructed in order to coincide with the topological expansion of a matrix model integral, when the algebraic curve is chosen as the large \(N\) limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in paerticular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e., the \((p,q)\) minimal models of conformal field theory.
As an example to illustrate the efficiency of our method, we apply it to the Kontsevich integral, and give a new and extremely easy proof that the Kontsevich integral depends only on odd times, and that it is a KdV \(\tau\)-function. algebraic curve; spectral curve; matrix model; topological expansion of a matrix integral; tau-function; Hirota equation B. Eynard and N. Orantin, \textit{Invariants of algebraic curves and topological expansion}, \textit{Commun. Num. Theor. Phys.}\textbf{1} (2007) 347 [math-ph/0702045] [INSPIRE]. Relationships between algebraic curves and physics, Groups and algebras in quantum theory and relations with integrable systems, String and superstring theories in gravitational theory, KdV equations (Korteweg-de Vries equations), Quantization in field theory; cohomological methods Invariants of algebraic curves and topological expansion | 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 volume of a Cartier divisor \(D\) on a projective variety \(X\) measures the asymptotic growth of the space of sections of multiples of that divisor \(H^0(X, \mathcal O_X(nD))\) as \(n\) tends to infinity. What real algebraic numbers are volumes of divisors is not known. The set of such numbers is countable and is a multiplicative semigroup. \textit{S.~D. Cutkosky} gave examples of divisors with irrational volumes in [Duke Math. J. 53, 149--156 (1986; Zbl 0604.14002)]. This article is concerned with the realization of certain irrational numbers as algebraic volumes of divisors.
The authors show that a primitive element of every totally real Galois number field can be realised as the algebraic volume of a divisor on a smooth variety. Their construction relies on studying the volume of divisors on projective bundles over abelian varieties with real multiplication (an extension of Cutkosky's construction). The authors also show \(\pi\) can be realised as the volume of a divisor by exhibiting a divisor whose volume is a multiple of \(\pi\) on a projective bundle over \(E\times E\), where \(E\) is an elliptic curve without complex multiplication. volumes of Cartier divisors; geometric realizations of algebraic numbers; geometric realizations of \(\pi \) Divisors, linear systems, invertible sheaves, Algebraic numbers; rings of algebraic integers, Complex multiplication and abelian varieties, Diophantine approximation, transcendental number theory Algebraic volumes of divisors | 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 explicit method for constructing a finite dihedral Galois covering of a smooth projective variety \(Y\) with prescribed branch locus \(B\), i.e. a finite morphism \(\pi:X\to Y\) with \(X\) normal variety such that \(\mathbb{C}(X)\) is a Galois extension of \(\mathbb{C}(Y)\) having dihedral group \({\mathcal D}_{2n}\) as its Galois group. The branch locus \(\Delta=\Delta (X/Y)\) is defined by \(\Delta= \{y\in Y\mid \#^{-1}(y)< \deg\pi\}\).
If we choose generators \(\sigma\) and \(\tau\) of \({\mathcal D}_{2n}\) such that \(\sigma^2 = \tau^n= (\sigma \tau)^2 = \text{id}\), the invariant subfield \(\mathbb{C}(X)^\tau\) of \(\mathbb{C}(X)\) by \(\tau\) is a quadratic extension of \(\mathbb{C}(Y)\). Then if we note \(D(X/Y)=D\) the \(\mathbb{C}(X)^\tau\)-normalisation of \(Y\) we get a factorization of the covering \(\pi:X\to Y\) by \(D\), where \(\beta_2: X\to D\) and \(\beta_1: D\to Y\) are cyclic coverings respectively of degree \(n\) and 2. We can reduce the study of dihedral covering to that of these two coverings \(\beta_1\) and \(\beta_2\). We have to distinguish the cases \(n\) odd and \(n\) even, and we get the following:
Let \(f:Z\to Y\) be a smooth finite double covering of \(Y\) and \(\sigma\) be the involution on \(Z\) determined by \(f\). If we get three effective divisors \(D_1\), \(D_2\) and \(D_3\) in the \(n\)-odd case, or four effective divisors \(D_1\), \(D_2\), \(D_3\) and \(D_4\) in the \(n\)-even case, satisfying some convenient properties, then there exists a dihedral \({\mathcal D}_{2n}\) covering \(\pi: X\to Y\) such that: (i) \(D(X/Y)=Z\) and (ii) \(\Delta(X/Y) = \Delta(Z/Y)\cup f(\text{Supp} (D_1))\) in the \(n\)-odd case, and \(\Delta(X/Y) = \Delta(Z/Y) \cup f(\text{Supp} (D_1+D_2))\) in the \(n\)-even case.
This construction is universal and we get a converse of this result:
Let \(\pi: X\to Y\) be a dihedral \({\mathcal D}_{2n}\) covering such that \(D(X/Y)\) is smooth, and let \(\sigma\) be the involution on \(D(X/Y)\) determined by \(\beta_1\), then there exist three effective divisors \(D_1\), \(D_2\) and \(D_3\) in the \(n\)-odd case, or four effective divisors \(D_1\), \(D_2\), \(D_3\) and \(D_4\) in the \(n\)-even case, on \(D(X/Y)\) which satisfy the previous conditions and such that \(\text{Supp} (D_1+ \sigma^*D_1)\) in the \(n\)-odd case, or \(\text{Supp} (D_1+ \sigma^*D_1+D_2)\) in the \(n\)-even case, is the branch locus of \(\beta_2\).
In the last part of the article the author deduces from these propositions a result about dihedral covering of the projective plane \(\mathbb{P}^2\): Let \(\pi: S\to\mathbb{P}^2\) be a dihedral \({\mathcal D}_{2p}\) (with \(p\) odd prime) coverings of \(\mathbb{P}^2\) and let \(\Delta(S/ \mathbb{P}^2)\) be the branch locus of \(\pi\). Then \(\deg\Delta(S/ \mathbb{P}^2) \geq 3\). Furthermore if \(\deg\Delta(S/ \mathbb{P}^2)\leq 4\), the author gives the list of the all possibilities for the branch locus \(\Delta(S/ \mathbb{P}^2)\). dihedral group as Galois group; dihedral Galois covering of a smooth projective variety; prescribed branch locus Tokunaga H. (1994) On dihedral Galois coverings. Can. J. Math. 46: 1299--1317 Coverings in algebraic geometry, Ramification problems in algebraic geometry, Low codimension problems in algebraic geometry, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Inverse Galois theory On dihedral Galois coverings | 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 Prym map associates to every étale double covering \(\pi\) of some curve of genus g a principally polarized abelian variety P, the Prym variety of \(\pi\). The Torelli problem for Prym varieties is to determine the preimages of the Prym map. Differently from the usual Torelli problem for curves the Prym map is not always injective. However it is shown by \textit{V. I. Kanev} [Math. USSR, Izv. 20, 235-257 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.2, 244-268 (1982; Zbl 0566.14014)] for \(g\geq 9\) and \textit{R. Friedman} and \textit{R. Smith} [Invent. Math. 67, 473-490 (1982; Zbl 0506.14042)] for \(g\leq 7\) that a double covering of a general curve of genus g is determined by its Prym variety. A constructive proof of this result for \(g\geq 17\) was given by \textit{G. E. Welters} [Am. J. Math. 109, 165-182 (1987; Zbl 0639.14026)]. The present paper gives a constructive proof for \(g\geq 8.\)
The proof is analogous to Green's proof of the usual Torelli theorem for non-hyperelliptic and non-trigonal curves of genus \(\geq 7:\) Let P be a general Prym variety of dimension \(g-1\) and \(\Theta\) its theta divisor. The tangent cones of \(\Theta\) at singular points x of multiplicity 2 are quadrics of rank \(\leq 6\) in the projectivized tangent space of P at x which contain the base curve \(C\) of the double covering. The first step in the proof consists in showing that for a general covering these quadrics generate the vector space of all quadrics containing C. - The second step, analogue to Petri's theorem is to show that the ideal of the curve C is generated by quadrics. As a byproduct one gets that generically the singular locus of \(\Theta\) is not empty of dimension \(g- 7\) for \(g\geq 7\), reduced for \(g\equiv 7\), and irreducible and reduced for \(g\geq 8\) with cohomology class \(16\frac{1}{6!}[\Theta]^ 6\). Prym map; Torelli problem for Prym varieties; double covering of a general curve; theta divisor; Petri's theorem Debarre, O., Sur le problème de Torelli pour LES variétés de Prym, Amer. J. Math., 111, 1, 111-134, (1989) Algebraic moduli of abelian varieties, classification, Picard schemes, higher Jacobians Sur le problème de Torelli pour les variétés de Prym. (On the Torelli problem for Prym 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 \(X\) be a modular curve defined over \(\mathbb{Q}\) of genus greater than \(0\), and let \(J_X\) be its Jacobian variety defined over \(\mathbb{Q}\) with an embedding of \(X\) into \(J_X\) sending the cusp at infinity to the distinguished rational point \(O\). The cuspidal group of \(J_X\), denoted by \(C(J_X)\), is the subgroup of \(J_X\) generated by all cusps of \(X\), and the \(\mathbb{Q}\)-rational cuspidal group of \(J_X\), denoted by \(C(J_X)_\mathbb{Q}\), is the subgroup of \(J_X\) consisting of \(\mathbb{Q}\)-rational points generated by cusps of \(X\). It is a classical result that \(C(J_X)\) is a subgroup of the torsion subgroup of \(J_X\), denoted by \(T(J_X)\), and this relation holds for the \(\mathbb{Q}\)-rational subgroups, i.e., \(C(J_X)_\mathbb{Q}\) is a subgroup of \(T(J_X)_\mathbb{Q}\). The author proves that if \(X=X_1(2p)\) and it has genus \(\geq 1\), then the order of \(C(J_X)_\mathbb{Q}\) is equal to
\[
\frac{p^2-1}{24}\;\cdot\;p\,\prod_{\psi} \frac{B_{2,\psi}^2}{16}(4-\psi(2)),
\]
where \(\psi\) runs over all even, primitive Dirichlet characters modulo \(p\), and \(B_{2,\psi}\) denotes the generalized Bernoulli number defined by
\[
B_{2,\psi}=p\sum_{a=1}^{p-1} \psi(a)\left\{ \left(\frac ap\right)^2 -\frac ap + \frac16\right\}.
\]
The author also proves a general result about explicit structures of \(C(J_X)_\mathbb{Q}\), and uses it to compute explicit structures for the primes \(7\leq p \leq 127\), and for primes \(\leq 4001\); more precisely, he determines the structures of the Sylow \(p\)-subgroups of \(C(J_X)_\mathbb{Q}\).
For general modular curves, it is believed that \(C(J_X)_\mathbb{Q}\) is in fact equal to \(T(J_X)_\mathbb{Q}\), and the author introduces a nice survey of results related to this conjecture. modular curve; Jacobian variety; rational point; torsion subgroup; cuspidal class number; modular unit Arithmetic aspects of modular and Shimura varieties, Modular and automorphic functions, Rational points, Modular and Shimura varieties, Jacobians, Prym varieties The \(\mathbb{Q}\)-rational cuspidal group of \(J_{1}(2p)\) | 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 finite extension of the field of \(p\)-adic numbers \(\mathbb{Q}_p\) and \(O_K\) be the ring of integers in \(K\). Let \(C\) be a geometrically connected smooth projective curve over \(K\). The index \(I(C)\) of \(C\) is defined to be the greatest common divisor of the divisors on \(C\). In this paper, which is the sequel of [\textit{J. Van Geel} and \textit{V. I. Yanchevskii}, Manuscr. Math. 96, 317--333 (1998; Zbl 1015.11023) and Ann. Fac. Sci. Toulouse, VI. Sér., Math. 8, 155--172 (1999; Zbl 0986.11036)], the authors deal with the index of curves \(C\) defined by an affine equation of the form \(Y^2= \pi f(X)\), where \(\pi\) is a uniformizing element and \(f(X)\) a monic irreducible polynomial of \(O_K[X]\). They give necessary and sufficient conditions for \(I(C)\) to be 1 and 2 in cases where \(K\) is dyadic and non-dyadic, respectively. hyperelliptic curves; \(p\)-adic field; index of a curve Varieties over finite and local fields, Local ground fields in algebraic geometry, Arithmetic ground fields for curves The index of certain hyperelliptic curves over \(p\)-adic 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 Pairings over abelian varieties have been widely used in cryptography, specially the Weil pairing \(e\) and the Tate-Lichtenbaum pairing \(t\) defined on a elliptic curve \(E\) over the finite field \(K:=\mathbb F_q\) of order \(q\). In this paper a fundamental difference between these pairings is studied: While \(e\) only admits trivial self pairings, \(t\) might do not; see for instance [\textit{I. F. Blake} (ed.) et al., Advances in elliptic curve cryptography. Lond. Math. Soc. Lect. Note Ser. 317. Cambridge: Cambridge University Press (2005; Zbl 1089.94018)]
Among other interesting results, the author shows here that \(t\) only allows trivial self pairings if and only if the Frobenius morphism \(\Phi\) over \(K\) acts on \(E\) like an integer on \(E[n^2]\), the set of \(n^2\)-torsion points of \(E\), whenever \(E[n]\subseteq E(K)\) and \(\gcd(n,q)=1\). The condition on \(\Phi\) might be not easy to check in practice; the author then restate it by using complex multiplication. Finally, the author generalizes the results obtained in her thesis [PhD Thesis, University of Maryland, College Park, Maryland (2007)] to Jacobian of curves of arbitrary genus over finite fields. finite field; elliptic curve; Jacobian of curves Schmoyer, S. L., The triviality and nontriviality of Tate-lichtenbaum self-pairings on Jacobians of curves, (2006) Curves over finite and local fields, Elliptic curves The triviality and nontriviality of Tate-Lichtenbaum self pairings on Jacobians 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 We prove that, if C is a smooth irreducible curve on a Del Pezzo surface S such that \(K^ 2_ S\geq 2\), then - with one exception, involving curves of genus 3 - the gonality of the smooth curves in the linear system \(| C|\) is constant, and that, if the genus of C is at least 4, also the Clifford index of the smooth curves in \(| C|\) is constant. Both statements are not true for curves on Del Pezzo surfaces S such that \(K^ 2_ S=1\). Such results, especially the ones concerning the gonality, are predicted - and clarified - by Green and Lazarsfeld's conjectures on syzygies of curves embedded by line bundles of high degree. Kodaira divisor; smooth irreducible curve on a Del Pezzo surface; gonality; linear system; Clifford index; syzygies of curves Pareschi, G., Exceptional linear systems on curves on del Pezzo surfaces, Math. Ann., 291, 17-38, (1991) Divisors, linear systems, invertible sheaves, Vector bundles on curves and their moduli, Fano varieties Exceptional linear systems on curves on Del Pezzo 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 The author proves Picard-type theorems about the degeneracy of holomorphic mappings at a point. He essentially uses the relation with logarithmic differentials.
Let W be a compact complex algebraic variety, D a divisor on W with normal intersection, and \(H^ 0(T^*(\log D))\) the space of D- logarithmically meromorphic 1-forms on W [see \textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 40, 5-57 (1971; Zbl 0219.14007)].
Theorem 1. Let W and D be as above and \(f: {\mathbb{C}}\to W\setminus D\) be a nonconstant holomorphic mapping. Then there exists a nonconstant holomorphic mapping \(\check f: {\mathbb{C}}\to W,\) on which all differentials \(\omega_ j\in H^ 0(T^*(\log D))\) take the constant values \((\omega_ j,\check f')=c_ j\), where \(c_ j\) are constants and \(\check f': {\mathbb{C}}\to T(W)\) is the derivative of the curve \v{f}.
As an application of Theorem 1, the author proves the following theorem.
Theorem 2. Let \(D_ j\) be hypersurfaces of certain degrees in \(CP_ n\) and \(D=\sum^{2n+1}_{j=1}D_ j\) be a divisor with normal intersection. If \(f: C\to CP_ n\setminus D\) is a holomorphic mapping, then f is equal to a constant. Picard-type theorems; degeneracy of holomorphic mappings at a point; logarithmic differentials; divisor with normal intersection Babets, V. A.: Picard-type theorems for holomorphic mappings, Siberian math. J. 25, 195-200 (1984) Picard-type theorems and generalizations for several complex variables, Divisors, linear systems, invertible sheaves, Curves in algebraic geometry, Holomorphic mappings and correspondences, Entire functions of one complex variable (general theory) Picard-type theorems for holomorphic mappings | 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 0596.00007.]
The aim of this paper is to introduce the reader to Arakelov intersection theory (with its subsequent developments due to Faltings and Hriljac). The basic definition and results of this theory are presented. The details are in general not included, although the author sketches the proofs of some results.
In the first section the author defines the Arakelov intersection pairing of two divisors on a regular curve over the spectrum of the ring of integers of a number field K. - In the second section one discusses the metrized line bundles, and in the third, one sketches the proof of a result of Faltings allowing to assign to each admissible hermitian line bundle L on X a hermitian metric satisfying certain properties. The fourth section deals with the Riemann-Roch theorem and the adjunction formula, and in the last section the Hodge index theorem is proved. Arakelov intersection theory; Arakelov intersection pairing; divisors on a regular curve; metrized line bundles; Riemann-Roch theorem; adjunction formula; Hodge index theorem Chinburg, T.: An introduction to Arakelov intersection theory. In: Cornell, G., Silverman, J.H. (eds.) Arithmetic Geometry, pp. 289--307. Springer, New York (1986) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Divisors, linear systems, invertible sheaves, Arithmetic ground fields for abelian varieties An introduction to Arakelov intersection theory | 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 From the introduction: ``We describe a Prym construction which associates abelian varieties to certain graphs. More precisely, given the adjacency matrix \(A = (a_{ij} )_{i,j=1}^d\) of a vertex-transitive strongly regular graph \(\mathcal G\) along with a covering of curves \(p :C \to{\mathbb P}^1\) of degree \(D\) and a labeling \(\{x_1,...,x_d \}\) of an unramified fiber such that the induced monodromy group of \(p\) is represented as a subgroup of the automorphism group of \(G\), we construct a symmetric divisor correspondence \(D\) on \(C\times C \) which then serves to define complementary subvarieties \(P_+\) and \(P_-\) of the Jacobian \(J(C)\). The correspondence \(D\) is defined in such a way that the point \((x_i ,x_j )\) appears in \(D\) with multiplicity \(a_{ij}\), analogous to \textit{V. Kanev}'s construction [in: Theta functions, Proc. 35th Summer Res. Inst. Bowdoin Coll., Brunswick/ME 1987, Proc. Symp. Pure Math. 49, part 1, 627--645 (1989; Zbl 0707.14041)]. The varieties \(P_{\pm}\) are given by \(P_{\pm} = \text{ker}(\gamma - r_{\mp}\text{id}_{J(C)})_0\), where \(r_{\pm}\) are special eigenvalues of \(A\) and \(\gamma\) is the endomorphism on \(J(C)\) canonically associated to \(D\) (i.e., sending the divisor class \([x - x_0]\) to the class \([D(x)-D(x_0)]\)). It is easy to show that
\[
(\gamma -r_+ \text{id}_{J(C)})(\gamma - r_- \text{id}_{J(C)}) = 0
\]
and \(P_{\pm} = \text{im}(\gamma - r_{\pm}\text{id}_{J(C)})\). In particular, if \(D\) is fixed point free and \(r_+ = 1\), then \(P_+\) is a Prym--Tyurin variety of exponent \(1 - r_-\) for \(C\). Given the ramification of \(p\) it is not hard to compute the dimension of \(P_{\pm}\). For a thorough definition of \(D\) we consider the Galois closure \(p :X\to {\mathbb P}^1\) of \(p\) and use the induced representation \(\text{Gal}(p)\to \text{Aut}(G)\) to construct symmetric correspondences \(D_+\) and \(D_-\) on \(X \times X\) (much the way \textit{J.-Y. Mérindol} did in [J. Reine Angew. Math. 461, 49--61 (1995; Zbl 0814.14043)], or \textit{R. Donagi} in [Astérisque 218, 145--175 (1993; Zbl 0820.14031)]). With \(C\) being a quotient curve of \(X\), the correspondence \(D\) is derived from \(D_{\pm}\) taking quotients and adding \(r_{\pm}\Delta_C\), where \(\Delta_C\) is the diagonal of \(C \times C\); see Section 4. Given the endomorphisms \(\gamma_{\pm}\) on \(J(X)\) canonically associated to \(D_{\pm}\), we show that \(\text{im}\gamma_{\pm}\) and \(P_{\pm}\) are isogenous. The lattice graphs \(L_2(n), n\geq 3\), and their complements \(\overline{L_2(n)}\) offer important examples. For instance, applying the method to \(\overline{L_2(n)}\) and appropriate coverings \(C\to {\mathbb P}^1\) of degree \(n\) with branch loci of cardinality \(2(l+2n-2)\) for \(l\geq 1\), we obtain \(l\)-dimensional Prym--Tyurin varieties of exponent \(n\) for the curves \(C\); see Section 7. We give a characterization of these varieties and show that for \(n = 3\) they coincide with the non-trivial Prym--Tyurin varieties of exponent \(3\) described by [\textit{H. Lange, S. Recillas} and \textit{A. M. Rojas}, J. Algebra 289, 594--613 (2005; Zbl 1089.14005)].'' Jacobian variety; Prym variety; coverings of curves DOI: 10.1016/j.jalgebra.2007.01.001 Jacobians, Prym varieties, Algebraic theory of abelian varieties Prym varieties associated to 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 This paper announces results on the variations of constants in Faltings' theorem on the Mordell conjecture, using the proof of the reviewer [Ann. Math. 133, 509--548 (1991; Zbl 0774.14019)] as further simplified by \textit{E. Bombieri} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, 615--640 (1990; Zbl 0722.14010); errata 18, 473 (1991)]. To state the result precisely, let \(k\) be a number field, let \(T\) be a projective variety over \(k\), and let \({\mathcal C}\) be a flat family over \(T\), whose generic fiber is smooth and projective over \(k(T)\), of genus \(g>1\). Let \(T^0\) be the (nonempty) open subset of \(T\) over which the family is smooth. Then there exists a constant \(\gamma_1 ({\mathcal C})\), depending only on \({\mathcal C}\) and on the choice of a height function \(h\) on \(T\) relative to an ample divisor, and a constant \(\gamma_2 (g)\), depending only on \(g\), such that for every closed point \(t\in T^0\) and for every finite extension \(k'\) of \(k(t)\), one has
\[
\text{Card} \biggl\{x\in {\mathcal C}_t(k') \mid |x|\geq\gamma_1 ({\mathcal C}) \bigl(\sqrt {h(t)} + 1\bigr) \biggr\} \leq\gamma_2 (g) \cdot 7^{\text{rank}I_t(k')},
\]
where \({\mathcal J}\) denotes the relative Jacobian of \({\mathcal C}\) over \(T^0\). If, in addition, \(T\) is a curve and if the relative Jacobian is isogenous to a product of non-isotrivial families of elliptic curves, then she removes the constant \(\gamma_1\) at the cost of increasing the 7 in the bound:
\[
\text{Card} {\mathcal C}_t(k') \leq c_7 ({\mathcal C}, \sigma)^{\text{rank} {\mathcal J}_t(k')},
\]
where \(c_7({\mathcal C}, \sigma)\) is a constant depending only on \([k': \mathbb Q]\), on \({\mathcal C}\), and on the Szpiro constants of the elliptic curve factors of \({\mathcal J}_t\). If the Szpiro conjecture holds, then the dependence on \(\sigma\) can be removed. The proof is sketched (details will appear elsewhere); it depends on careful attention to the constants in Bombieri's proof. The second result also uses work of Merel. algebraic families of curves; Faltings' theorem; Mordell conjecture; projective variety; Szpiro constants; elliptic curve De Diego, Teresa: Théorème de faltings (conjecture de Mordell) pour LES familles algébriques de courbes, C. R. Acad. sci. Paris sér. I math. 323, No. 2, 175-178 (1996) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Families, moduli of curves (algebraic), Rational points, Arithmetic ground fields for curves Faltings' theorem (Mordell's conjecture) for algebraic families 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 Let \(X\) be a curve defined over an algebraically closed field \(k\), let \( x \in X \) be a singular point of \(X\), let \( R = {\mathcal O}_{X, x} \) be the local ring of \(x\) on \(X\), and let \( \Omega_{R/k} \) be the \(R\)-module of differentials of the \(k\)-algebra \(R\).
\textit{R. Berger} [Math. Z. 81, 326-354 (1963; Zbl 0113.26302)] was the first to study the torsion submodule \(T\) of \( \Omega_{R/k} \) which is an \(R\)-module of finite length. After Berger, many authors investigated \(T\) [cf., e.g., \textit{E. Kunz}, Arch. Math. 19, 47-58 (1968; Zbl 0162.05502), \textit{S. Susuki}, J. Math. Kyoto Univ. 4, 471-475 (1965; Zbl 0127.26102), \textit{B. Ulrich}, Arch. Math. 36, 510-523 (1981; Zbl 0458.13008), \textit{R. Waldi}, Math. Ann. 242, 201-208 (1979; Zbl 0426.14004) and \textit{O. Zariski}, Proc. Natl. Acad. Sci. U.S.A. 56, 781-786 (1966; Zbl 0144.20201)].
This interesting paper gives further results on \(T\). In the first part, the author studies, more generally, the torsion submodule of an \(R\)-module \( R^n/a R \) where \(n\) is a positive integer and \( a \in R^n\), \(R\) being a one-dimensional local ring. Let \( a = a_1 e_1 + \cdots + a_n e_n \), \( \{e_1, \ldots, e_n \} \) being the canonical basis of \( R^n\), and let \( J = a_1 R + \cdots + a_n R \). If \(J\) contains a regular element, then \( \ell_R(T) = \ell_R(J^{-1} / R) \) is finite, and if \( n = 2 \), then we have \( \ell_R(T) = \ell_R(R/J)\). Now let \(R\) be an arbitrary domain, let \(M\) be a finitely generated \(R\)-module of rank \(n\), and let \(J\) be the \(n\)-th Fitting ideal of \(M\). Assume that \(M\) admits a finite representation
\[
\to R^p \to R^q \to M \to 0.
\]
Let \(T\) be the torsion submodule of \(M\). Then (theorem 3) there exists an exact sequence
\[
0 \to T \to (J^{-1} / R)^p \to (J^{-1}/R)^q
\]
which implies that \( T = 0 \) is equivalent with \( J^{-1} = R \) (corollary 4). Now consider the particular case where \(R\) is the local ring of a point \(x\) of an algebraic variety defined over \(k\), and \( M = \Omega_{R/k}\); then \(J\) is the jacobian ideal. In theorem 2 the author shows that \(R\) is normal if \( J^{-1} = R \) , and if \(X\) is locally a complete intersection in \(x\), then \( \Omega_{R/k} \) is reflexive iff the height of \(J\) is \( \geq 3 \). In section 3 the author studies quadratic transformations of a complete local one-dimensional \(k\)-algebra \(R\) which is a complete intersection, and he shows the inequality \( \ell_{R_1}(T_1)+ \cdots+ \ell_{R_n}(T_1) \leq\ell_R(T)\), where \( R_1, \ldots, R_n \) are the rings in the first neighbourhood of \(R\) and \(T_1,\ldots,T_n\) are the corresponding torsion modules [with respect to the construction of \( R_1, \ldots, R_n\); the author should have included in his bibliography also the paper by \textit{J. Lipman}, Am. J. Math. 93, 649-685 (1971; Zbl 0228.13008)].
From now on let \(R\) be the ring of a plane irreducible algebroid curve, i.e., \( R= \mathbb{C} [[X, Y ]] /(F)\) where \(F\) is an irreducible power series. Then the integral closure \( \overline{R}\) of \(R\) has the form \( \overline{R}=\mathbb{C}[[t ]] \); let \(\nu\) be the canonical valuation of \( \overline{R}\), \( \Gamma= \{\nu(z) \mid z \in R \} \) the semigroup of values, and \( \Lambda= \{\nu(t \omega) \mid \omega \in x' R+ y' R \}\) where \( x, y \) are the images of \( X, Y \) in \(R\), and \( x'= d x/dt \), \( y'= dy/dt \). In section 4 the author shows that \( \ell_R(T)= c - \#(\Lambda)\), \(c\) being the conductor of \( \Gamma\). Some generalizations are contained in theorem 7.
Zariski [cf. the paper referred to above] has characterized those curves with \( \ell_R(T)= c \); they are isomorphic to \( Y^m - X^n= 0 \), i.e., to curves having one Puiseux pair. In section 6 the author calculates \( \Lambda \) for the case of curves having two Puiseux pairs. Finally, taking Ebey's list [cf. \textit{S. Ebey}, Trans. Am. Math. Soc. 118, 454-471 (1965; Zbl 0132.41602)], the author calculates \( \ell(T)\) for certain curves having only one Puiseux pair.
The following two works could also have been included in the bibliography: \textit{A. Campillo}, ``Algebroid curves in positive characteristic'', Lect. Notes Math. 813 (1980; Zbl 0451.14010); \textit{P. Russell}, Manuscr. Math. 31, 25-95 (1980; Zbl 0455.14018). algebroid curves; Puiseux pairs; differential module of a curve; one-dimensional local rings; local ring of singular point of a curve; torsion submodule; Fitting ideal Carbonne, P.: Sur LES différentielles de torsion, J. algebra 202, 367-403 (1998) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Singularities of curves, local rings, Regular local rings, Commutative rings of differential operators and their modules On the torsion differentials | 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 Consider a discrete subgroup \(\Gamma \subset U((2,1),{\mathbb{C}})\), acting on the two-dimensional unit ball \({\mathcal B}\) through its linear action on projective coordinates, and, for \(\gamma\in \Gamma\), denote by \(j_{\gamma}\) the jacobian determinant of this action. Let \(\chi: \Gamma\to {\mathbb{C}}^*\) be a character of finite order, and let \(S(\Gamma,\chi)\) be the graded algebra over \({\mathbb{C}}\) of modular forms on \({\mathcal B}\) satisfying \(f=\chi (\gamma)\cdot j^ k_{\gamma}\cdot \gamma^*(f)\), \(\gamma\in \Gamma\). The principal groups studied in this paper are \(\Gamma =U((2,1)\), \({\mathcal O}_{{\mathbb{Q}}(\sqrt{-3})})\) and its congruence subgroup \(\Gamma(\sqrt{-3})\). The author gives an explicit description by generators and relations of \(S(\Gamma,1)\) and \(S(\Gamma(\sqrt{-3}),1)\) (the latter has 11 generators with 28 relations).
The method is to start with modular forms that arise naturally as the coefficients and ramification points of the normal form \(y^ 3=x^ 4+G_ 2x^ 2+G_ 3x+G_ 4=\prod^{4}_{i=1}(x-\xi_ i) \) for a Picard curve, just as the classical modular forms \(g_ 2\) and \(g_ 3\) on the upper half-plane arise as coefficients for the normal form of an elliptic curve. It turns out that there is a certain character \(\chi\) of order 6 such that \(S(\Gamma,\chi)={\mathbb{C}}[G_ 2,G_ 3,G_ 4]\) and S(\(\Gamma\) (\(\sqrt{-3}))={\mathbb{C}}[\xi_ 1,\xi_ 2,\xi_ 3]\). The key result of the paper under review is the explicit determination of the character \(\chi\), which enables the author to derive the structure of \(S(\Gamma,1)\) and \(S(\Gamma(\sqrt{-3}),1)\). (The description of \(\chi\) and the structure of \(S(\Gamma,1)\) were previously obtained by J.-M. Feustel using theta constants.) Finally, the author examines the algebra of modular forms satisfying \(f(z_ 1,z_ 2)=(c_ 0+c_ 1z_ 1+c_ 2z_ 2)^{-3k} \gamma^*(f)(z_ 1,z_ 2),\) where \((c_ 0,c_ 1,c_ 2)\) is the bottom row of \(\gamma\), by identifying it with \(S(\Gamma,\psi)\), where \(\psi\) is a certain character of order 6, not equal to \(\chi\). generators; relations; Picard curve; S(\(\Gamma \) ,1); S(\(\Gamma \) (\(\sqrt{-3}),1)\); algebra of modular forms Holzapfel, R.-P. , On the nebentypus of Picard modular forms . To appear in Math. Nach. Theta series; Weil representation; theta correspondences, Special algebraic curves and curves of low genus, Special varieties On the nebentypus of Picard modular forms | 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 \(f: D\to {\mathbb{P}}^{g-1}\) be the rational Gauss map for a principally polarized abelian variety (X,D) of \(\dim enson\quad g.\) Provided that D is symmetric and \(\dim (D_{\sin g})\leq g-3\), the author shows that (X,D) is a jacobian variety of a non-hyperelliptic curve if and only if there exists a curve C in the dual space \(({\mathbb{P}}^{g-1})^*\) such that for each point \(P\in C\), \(f^{-1}(H_ P)\)- where \(H_ p\) means the hyperplane in \(({\mathbb{P}}^{g-1})^*\) defined for each \(P\in {\mathbb{P}}^{g-1}\) by incidence - breaks into two different conjugated components. The author gives almost the same result for hyperelliptic curves. Moreover he discusses a characterization of (X,D) being a jacobian variety by using a curve C as above and the surface \(S^ 2C.\)
The crucial point of these characterizations of jacobian varieties is that the existence of a curve is assumed. jacobianness of abelian variety; rational Gauss map for a principally polarized abelian variety Muñoz Porras, J.M.: Characterization of Jacobian varieties in arbitrary characteristic. Compos. Math. (1987) Jacobians, Prym varieties, Picard schemes, higher Jacobians Characterization of Jacobian varieties in arbitrary characteristic | 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 \(C \subset \mathbb P^r\), \(r\geq 3\), be a smooth curve and \(N \subset \mathbb P^2\) the set of all nodes of a general linear projection of \(C\) into \(\mathbb P^2\). The authors give strict lower bounds for the postulation of \(N\) and show connections with Castelnuovo's and Halphen's theory. They prove some sharp lower bounds for the difference function \(\partial h_N\) of \(N\). Set \(d:= \deg (C)\), \(n:= h^0(C,O_C(1))-1\) and \(\alpha := \min \{j:h^0(I_N(j)) \neq 0\}\). Then they prove
\[
\partial h_N(j+1) \leq \partial h_N(j)-n+2 \leq \partial h_N(j)-r+2
\]
and hence \(\partial h_N\) is strictly decreasing in the interval \([\alpha ,d-2]\). space curve; nodal plane curve; general projection; postulation of a finite subset L. Chiantini - N. Chiarli - S. Greco, Halphen conditions and postulation of nodes, Adv. Geom. 5 (2005), 237--264 Plane and space curves, Special divisors on curves (gonality, Brill-Noether theory) Halphen conditions and postulation of nodes | 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 0659.00006.]
Fermat's Last ``Theorem'' says that the equation \(A^ n+B^ n+C^ n=0\) has no solutions in coprime, non-zero integers \(A, B, C\), whenever \(n\geq 3\). This problem has attracted the attention of many mathematicians over the past 300 years, with the result, for example, that the theorem is known for all \(n\leq 150 000\). In this well-written and extremely informative survey, the author describes some new approaches to the Fermat problem which have recently attracted considerable interest.
The initial observation, due to Frey, is that if \((A,B,C)\) is a solution to Fermat's equation with prime exponent \(p\), then one should look at the elliptic curve
\[
E: Y^ 2=X(X-A^ p)(X+B^ p).
\]
This is an elliptic curve, defined over \(\mathbb Q\), which has such amazing properties that one can hope to show that \(E\) does not exist.
For example, the discriminant of \(E\) is essentially \((ABC)^{2p}\), while its conductor is at most \(2ABC\). This would contradict a conjecture of Szpiro, and also the ``abc-conjecture'' of Masser and the author. The author describes the relationships between these conjectures, and shows how they would imply Fermat's Theorem.
Next comes a description of Serre's conjecture, which asserts that every continuous representation \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to \text{GL}_ 2(\overline{\mathbb F}_ p)\) which is absolutely irreducible and odd, is obtained from a cusp form for \(\Gamma_ 0(N)\) by a process described by Deligne. Further, Serre (conjecturally) describes the level, weight, and character of the cusp form in terms of the conductor, weight, and character of the representation. The author explains how the representation on the \(p\)- torsion subgroup of \(E\) and Serre's conjecture would imply the existence of a cusp form of weight 2 for \(\Gamma_ 0(2)\). Since there are no such modular forms, Fermat's Theorem follows. The author also describes some other consequences of Serre's conjecture, including the conjecture of Taniyama-Weil that every elliptic curve over \(\mathbb Q\) is modular (i.e. is parametrized by modular functions).
Since both Fermat's Theorem and the Taniyama-Weil conjecture are consequences of Serre's conjecture, this suggests a relationship between the two. The final part of the paper describes theorems of Mazur and Ribet which give conditions under which a modular representation has lower level than one would expect. The more difficult case, which was recently resolved by Ribet, has as a consequence that the conjecture of Taniyama-Weil implies Fermat's Theorem. The author gives a brief sketch of the proof of Mazur's and Ribet's result which should be helpful for anyone planning to study the details of these difficult theorems. Fermat last theorem; survey; Fermat's equation; elliptic curve; conjecture of Szpiro; abc-conjecture; Serre's conjecture; existence of a cusp form of weight 2 for \(\Gamma _ 0(2)\); Taniyama-Weil conjecture; theorems of Mazur and Ribet; modular representation Oesterlé ( J. ) .- Nouvelles approches du ''théorème'' de Fermat , Séminaire Bourbaki n^\circ 694 (1987-88). Astérisque 161 -162, 165 - 186 ( 1988 ) Numdam | MR 992208 | Zbl 0668.10024 Higher degree equations; Fermat's equation, Elliptic curves over global fields, Galois representations, Special surfaces New approaches to Fermat's Last Theorem | 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 curve on the weighted projective plane \(\mathbb{P}(1, 1, 4)\) which is of Fermat type or almost Fermat type. We construct \(K3\) surfaces which are double covers of \(\mathbb{P}(1, 1, 4)\) and which contain pointed curves with symmetric Weierstrass semigroups which are double covers of \(C\). Weierstrass semigroup of a point; \(K3\) surface; weighted projective plane; numerical semigroup; double cover of a curve \(K3\) surfaces and Enriques surfaces, Coverings of curves, fundamental group, Plane and space curves, Riemann surfaces; Weierstrass points; gap sequences, Commutative semigroups Curves on weighted \(K3\) surfaces of degree two with symmetric Weierstrass semigroups | 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 linear codes associated to Schubert varieties in Grassmannians. A formula for the minimum distance of these codes was conjectured in 2000 and after having been established in various special cases, it was proved in [\textit{X. Xiang}, IEEE Trans. Inf. Theory 54, No. 1, 486--488 (2008; Zbl 1308.94115)]. We give an alternative proof of this formula. Further, we propose a characterization of the minimum weight codewords of Schubert codes by introducing the notion of Schubert decomposable elements of certain exterior powers. It is shown that codewords corresponding to Schubert decomposable elements are of minimum weight and also that the converse is true in many cases. A lower bound, and in some cases, an exact formula, for the number of minimum weight codewords of Schubert codes is also given. From a geometric point of view, these results correspond to determining the maximum number of \(\mathbb{F}_q\)-rational points that can lie on a hyperplane section of a Schubert variety in a Grassmannian with its nondegenerate embedding in a projective subspace of the Plücker projective space, and also the number of hyperplanes for which the maximum is attained. Grassmannian; Schubert variety; Grassmann code; Schubert code; minimum distance of a code; minimum weight codewords Ghorpade, SR; Singh, P., Minimum distance and the minimum weight codewords of Schubert codes, Finite Fields Appl., 49, 1-28, (2018) Linear codes (general theory), Geometric methods (including applications of algebraic geometry) applied to coding theory, Grassmannians, Schubert varieties, flag manifolds, Applications to coding theory and cryptography of arithmetic geometry Minimum distance and the minimum weight codewords of Schubert codes | 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 projective curve with bad reduction over a discretely- valued complete local field k. Using line bundles on the universal covering of C in the category of formal schemes over the valuation ring of k a uniformization of the Jacobian Jac(C) of C is constructed. This is an isomorphism G/\(\Lambda\simeq Jac(C)\) in the analytic category, where G is an algebraic group, an extension of an abelian variety with good reduction by a torus of rank h, and where \(\Lambda \simeq {\mathbb{Z}}^ h\) is a discrete subgroup of G. The group \(\Lambda\) reflects in some sense the bad reduction of the curve C. The uniformization of an arbitrary abelian variety Z over k deduces from this with the help of a surjective homomorphism of the Jacobian of a curve onto Z. Different constructions of uniformizations of abelian varieties where given by \textit{M. Raynaud} [cf. Actes Congr. internat. Math. 1970, Vol. 1, 473-477, (1971; Zbl 0223.14021)] and \textit{S. Bosch} and \textit{W. Lütkebohmert} [cf. Invent. Math. 78, 257-297 (1984; Zbl 0554.14015)]. curve with bad reduction; complete local field; uniformization of the Jacobian; uniformizations of abelian varieties Fresnel, J.; Van Der Put, M.: Uniformisation des variétés abéliennes. Ann. fac. Sci. univ. Toulouse 6, 7-42 (1994) Algebraic theory of abelian varieties, Jacobians, Prym varieties, Formal groups, \(p\)-divisible groups, Local ground fields in algebraic geometry Uniformization 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 We study the question of determining the number of conics tangent to a general plane curve at five unassigned points. This is related to the number of rational curves in a system of curves on the \(K3\)-surface obtained as a double cover of the plane ramified along a sextic. \(K3\)-surface; number of conics tangent to a general plane curve; number of rational curves Enumerative problems (combinatorial problems) in algebraic geometry, Special algebraic curves and curves of low genus, \(K3\) surfaces and Enriques surfaces Conics five-fold tangent to a plane 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 paper is a report on some known results about generic hyperplane sections of a curve in \({\mathbb{P}}^ n\). However some new results are contained. Of some interest is the following ``lifting lemma'': Let C be a curve (not necessarily reduced or irreducible) in \({\mathbb{P}}_ k^ 3\), \(k\) algebraically closed field of characteristic 0, and let \(\Gamma =C\cap H\) be its generic plane section. Assume that in the minimal resolution of J (J= the homogeneous ideal of \(\Gamma\) in \(k[x_ 1,x_ 2,x_ 3])\) there are no syzygies in degree \(\leq t+2\). Then the restriction map: \(H^ 0({\mathcal I}_ C(n))\to H^ 0({\mathcal I}_{\Gamma}(n))\) is surjective for all \(n\leq t\). - From this result the following corollaries are easily deduced:
(1) Suppose that C is not contained in a quadric, \(\deg (C)>4\), and that \(\Gamma\) is a complete intersection; then C is a complete intersection [see: the author, Proc. Am. Math. Soc. 104, No.3, 711-715 (1988; Zbl 0693.14020)].
(2) (Laudal's lemma:) If C is integral of degree \(d>\sigma^ 2+1\) and \(\Gamma\) is contained in a curve of H of degree \(\sigma,\) then C is contained in a surface of degree \(\sigma\) [see \textit{O. A. Laudal} in Algebraic Geometry, Proc., Tromsø Symp. 1977, Lect. Notes Math. 687, 112-149 (1978; Zbl 0392.14002) and \textit{L. Gruson} and \textit{C. Peskine} in Enumerative geometry and classical algebraic geometry, Prog. Math. 24, 33-35 (1982; Zbl 0526.14021)]. minimal resolution of homogeneous ideal; postulation; generic hyperplane sections of a curve; lifting lemma; complete intersection Strano, R.: On the hyperplane sections of curves. In: Proceedings of the Geometry Conference (Milan and Gargnano, 1987). Rend. Sem. Mat. Fis. Milano Vol. 57, pp. 125--134 (1989) Plane and space curves, Complete intersections, Projective techniques in algebraic geometry On hyperplane sections 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 abelian variety; elliptic curve; Hasse invariant; Artin effect; infiniteness of Tate-Shafarevich group; quasi-global field; potentially good reduction; Neron minimal models O.N. Vvedenskiĭ : The Artin effect in elliptic curves I . Izv. Akad. Nauk SSSR 43 (1979) = Math. USSR Izv. 15 (1980) 277-288. Elliptic curves, Arithmetic ground fields for abelian varieties, Arithmetic ground fields for curves, Special algebraic curves and curves of low genus The Artin effect in elliptic curves. I | 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 generalized étale cohomology theory is a representable cohomology theory for presheaves of spectra on an étale site of an algebraic variety. These cohomology theories simultaneously generalize the homotopy-theoretic cohomologies of algebraic topology and the algebraic theories (for example: étale and crystalline) of Grothendieck.
Consequently this volume, in developing the techniques of the subject, introduces the reader to the stable homotopy category of simplicial presheaves. This is an extremely delicate development, obstructed by the need for coherent constructions involving very ``large'' objects such as limits of Čech constructions involving presheaves of spectra. The development of an adequate theory, particularly in respect of its applications to algebraic \(K\)-theory, was held up by difficulties with smash-products of spectra and with transfer constructions. This book provides the user with the first complete account which is sensitive enough to be compatible with the sort of closed model category necessary in \(K\)-theory applications [i.e., the closed model structure of \textit{A. K. Bousfield} and \textit{E. M. Friedlander}, Lect. Notes Math. 658, 80-130 (1978; Zbl 0405.55021)].
As an application of the techniques the author gives proofs of the descent theorems of \textit{R. W. Thomason} and \textit{Y. A. Nisnevich}. In particular, this implies the celebrated result of \textit{R. W. Thomason}, ``Algebraic \(K\)-theory and étale cohomology'', Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012)] which identifies \(\text{mod }p\) \(K\)-theory, after being inflicted with Bott periodicity in the manner introduced by the reviewer, with \(\text{mod }p\) étale \(K\)-theory.
The book concludes with a discussion of the Lichtenbaum-Quillen conjecture (an approximation to Thomason's theorem without Bott periodicity). The recent proof of this conjecture, by \textit{V. Voevodsky}, when \(p=2\) for fields of characteristic zero makes this volume compulsory reading for all who want to be au fait with current trends in algebraic \(K\)-theory! étale site of algebraic variety; generalized étale cohomology; presheaves of spectra; closed model category; Lichtenbaum-Quillen conjecture Jardine, J. F., Generalized Étale Cohomology Theories, Progress in Mathematics, vol. 146, (1997), Birkhäuser Verlag: Birkhäuser Verlag Basel Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Stable homotopy theory, spectra, Research exposition (monographs, survey articles) pertaining to category theory, Research exposition (monographs, survey articles) pertaining to algebraic topology, Étale and other Grothendieck topologies and (co)homologies, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Generalized étale cohomology theories | 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 0653.00006.]
Let X be a projective nonsingular curve of genus g and let \(SU_ X(2)\) be the moduli space of semistable vector bundles of rank 2 having trivial determinant. Let J be the Jacobian of X and let \(\theta\) be the natural divisor on \(J^{g-1}\). In a former paper [cf. Ann. Math., II. Ser. 89, 14-51 (1969; Zbl 0186.549)], the authors showed: If \(g=2\) then \(SU_ X(2)\) is canonically isomorphic to the projective space of divisors on J, linearly equivalent to \(2\theta\) [and is smooth]; if \(g\geq 3\) then the Kummer variety \({\mathcal K}\)- which is canonically embedded in \(SU_ X(2)\)- is the singular locus of \(SU_ X(2)\). In this paper, the authors sketch a proof of the following theorem: If X is non-hyperelliptic of genus 3, then \(SU_ X(2)\) is isomorphic to a quartic hypersurface in \({\mathbb{P}}_ 7\), and the Kummer surface \({\mathcal K}\) can be defined by cubic polynomials. Details of the proof are to appear elsewhere.
[One should also compare a paper by \textit{A. Beauville} and the authors, cf. J. Reine Angew. Math. 398, 169-179 (1989; Zbl 0666.14015)]. 2\(\theta\)-linear system on an abelian variety; semistable vector bundles; of rank 2; projective nonsingular curve; genus M. S. Narasimhan and S. Ramanan, 2\?-linear systems on abelian varieties, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 415 -- 427. Algebraic moduli of abelian varieties, classification, Theta functions and abelian varieties, Elliptic curves 2\(\theta\)-linear systems on 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 This paper uses mixed Hodge theory to study the homotopy of a complex algebraic variety in the neighborhood of a singular point, or more generally, in the neighborhood of a subvariety. An important application is finding topological restrictions on the links of isolated singular points. One such application is the following theorem: Let L be the link of an isolated singularity of an n-dimensional variety. If s, t\(<n\) and \(s+t\geq n\), then the cup product \(H^ s(L;{\mathbb{Q}})\otimes H^ t(L;{\mathbb{Q}})\to H^{s+t}(L;{\mathbb{Q}})\quad vanishes\). (Recall that the link of an isolated singularity in an n-dimensional variety is a real (2n-1)-manifold.)
This theorem is a direct consequence of the following three facts:
(1) The cohomology \(H^ k(L)\) (which is isomorphic to \(H_ x^{k+1}(X))\) has a natural mixed Hodge structure.
(2) The weights of \(H^ k(L)\) are less than or equal to k for \(k<n\), and greater than or equal to \(k+1\) for \(k\geq n.\)
(3) The cup product in the cohomology of L is a morphism of mixed Hodge structures and therefore preserves weights.
The first assertion is due to Deligne. The second is a well-known consequence of Gabber's purity theorem and the decomposition theorem of intersection homology, both deep theorems. This paper establishes fact 3.
It is more convenient to work not just with isolated singularities, but in the following more general situation: Suppose that X is a projective variety and that Z and Z' are closed subvarieties. Set \(Y=Z\cup Z'\) and suppose that the singular locus of X is contained in Y. Let T be a sufficiently nice neighborhood of Z in X. Define the link L(X,Y,Z) of Z in X with Y removed by \(L(X,Y,Z)=T-Y\). This is well defined up to homotopy.
A de Rham mixed Hodge complex is constructed for L(X,Y,Z). The main result of this paper then follows from work of \textit{R. M. Hain} [''The de Rham homotopy theory of complex algebraic varieties. I'', J. K-Theory (to appear)] or \textit{J. W. Morgan} [Publ. Math., Inst. Hautes Etud. Sci. 48, 137-204 (1978; Zbl 0401.14003)] This result may be summarized as follows: If \(L=L(X,Y,Z)\) is as above, then: \((i)\quad For\) all k, \(H^ k(L)\) has a real mixed Hodge structure. \((ii)\quad The\) cup product of H(L) is a morphism of mixed Hodge structures. \((iii)\quad The\) real homotopy type of L has a mixed Hodge structure. The main idea behind the proof is the use of the machinery developed in the paper of Hain cited above. mixed Hodge theory; homotopy of a complex algebraic variety; neighborhood of a subvariety; links of isolated singular points; cup product; decomposition theorem of intersection homology Hain, R.M. and Durfee, A.: Mixed Hodge structures on the homotopy of links. Math. Ann.,280, 69--83 (1988) Homotopy theory and fundamental groups in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Rational homotopy theory, Algebraic topology on manifolds and differential topology Mixed Hodge structures on the homotopy of links | 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 calculate the first Betti number of an abelian covering of a \(CW\)-complex \(X\) as the number of points with cyclotomic coordinates (of orders determined by the Galois group) which belong to a certain subvariety of a torus constructed from the fundamental group of \(X\). This generalizes the classical formulas for the cyclic coverings due to Zariski and Fox. We also describe certain properties of these subvarieties of tori in the case when \(X\) is a complement to an algebraic curve in \(\mathbb{C}\mathbb{P}^ 2\) which are analogs of the Traldi-Torres relations from link theory and the divisibility theorem for Alexander polynomials of plane algebraic curves. Betti number of an abelian covering of a \(CW\)-complex; cyclotomic coordinates; fundamental group; complement to an algebraic curve; link; Alexander polynomials of plane algebraic curves Libgober A.: On the homology of finite abelian coverings. Topol. Appl. 43(2), 157--166 (1992) Coverings in algebraic geometry, Fundamental group, presentations, free differential calculus, Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group, Covering spaces and low-dimensional topology On the homology of finite abelian coverings | 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 quadric surface fibrations over curves, defined over algebraically closed and finite fields. Our goal is to understand, in geometric terms, spaces of sections for such fibrations. We analyze varieties of maximal isotropic subspaces in the fibers as \(\mathbb P^1\) -bundles over the discriminant double cover. When the \(\mathbb P^1\)-bundle is suitably stable, we deduce effective estimates for the heights of sections over finite fields satisfying various approximation conditions. We also discuss the behavior of the spaces of sections as the base of the fibration acquires singularities. quadric surface fibration over a curve; discriminant curve; space of sections; rank-2 vector bundles over curves Hassett B., Tschinkel Yu., Spaces of sections of quadric surface fibrations over curves, In: Compact Moduli Spaces and Vector Bundles, Contemp. Math., 564, American Mathematical Society, Providence, 2012, 227--249 Fibrations, degenerations in algebraic geometry, Rational points, Vector bundles on curves and their moduli Spaces of sections of quadric surface fibrations over 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 \textit{M. Oka} in Complex analytic singularities, Proc. Semin., Ibaraki/Jap. 1984, Adv. Stud. Pure Math. 8, 405-436 (1987; Zbl 0622.14012) gave a new method by which algebraic surfaces were constructed as exceptional divisors through the resolution of a toric variety. In the paper under review a procedure is presented based on the Oka theory to classify algebraic surfaces obtained from such a resolution. resolution of singularities; resolution of a toric variety Computational aspects of algebraic surfaces, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Numerical algebraic 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 Over a series of papers, the authors have developed a theory of patching that is suitable for studying algebraic objects defined over fields that have a rich `local' theory, e.g. in terms of valuations. In the present context, the authors consider as standing hypothesis a complete discrete valuation ring \(T\) with fraction field \(K\) and residue field \(k\), \(\hat{X}\) a projective normal \(T\)-curve with function field \(F\) and closed fibre \(X\). Using local data related to the geometry of \(X\), they now construct collections of certain locally defined fields, various products of which fit into a commutative diamond that has the original field \(F\) as base and such that this diagram has some additional desirable properties. One then shows that this allows to study algebraic objects defined over \(F\), such as quadratic forms or central simple algebras, in terms of such objects defined over the locally defined fields from above, giving rise to various local-global properties.
The present paper builds on and extends earlier work by the authors on the topic of patching and applications to quadratic forms and central simple algebras. This is achieved by introducing a new `refinement principle' for patching which, in some sense, can be considered as a patching within a patching. To state the main applications of this new method, recall that the \(u\)-invariant of a field \(F\) is the supremum of the dimensions of anisotropic quadratic forms over \(F\), and that the period \(\mathrm{per}(\alpha)\) of an element \(\alpha\) in the Brauer group \(\mathrm{Br}(F)\) is just its order as element in a group.
Let either \(F\) be the fraction field of a two-dimensional Noetherian complete local domain \(R\) with residue field \(k\) (e.g., \(k((x,t))\)), or a finite separable extension of the fraction field of the \(t\)-adic completion of \(T[x]\), where \(T\) is a complete discrete valuation ring with uniformizer \(t\) and residue field \(k\). Assume that \(k\) has Brauer dimension \(d\) away from \(p=\mathrm{char}(k)\) (this is a somewhat technical condition). If \(\alpha\in\mathrm{Br}(F)\) with \(\mathrm{per}(\alpha)\) not divisible by \(p\), then the index \(\mathrm{ind}(\alpha)\) divides \(\mathrm{per}(A)^{d+1}\). In particular, if \(F\) is a finite extension of the fraction field of either \(\mathbb{Z}_p[[x]]\) or the \(p\)-adic completion of \(\mathbb{Z}_p[x]\), then \(\mathrm{ind}(\alpha)\) divides \(\mathrm{per}(\alpha)^{2}\) for any \(\alpha\in\mathrm{Br}(F)\).
Let \(k\) be a field with \(\mathrm{char}(k)\neq 2\), and let \(F\) be a finite separable extension of the fraction field of either \(k[x][[t]]\) or \(k[[x,t]]\). Then \(u(F)=2^n\) with: \(n=2\) if \(k\) is algebraically closed; \(n=3\) if \(k\) is finite or \(k=k_0((z))\) with \(k_0\) algebraically closed; \(n=4\) if \(k=k_0((z))\) with \(k_0\) finite, or if \(k=\mathbb{Q}_p\) for some prime \(p\); \(n=5\) if \(k=\mathbb{Q}_p(z)\) or \(k=\mathbb{Q}_p((z))\) for some prime \(p\). patching; local-global principle; two-dimensional complete domain; function field of a curve; quadratic form; Witt ring; \(u\)-invariant; Brauer group; period-index problem Harbater, D.; Hartmann, J.; Krashen, D., \textit{refinements to patching and applications to field invariants}, Int. Math. Res. Not. IMRN, 2015, 10399-10450, (2015) Algebraic functions and function fields in algebraic geometry, Brauer groups of schemes, Quadratic forms over general fields, Algebraic theory of quadratic forms; Witt groups and rings, Formal power series rings, Arithmetic ground fields for curves, Finite-dimensional division rings, Brauer groups (algebraic aspects), Valued fields Refinements to patching and applications to field invariants | 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 ``By a cyclic layer of a finite Galois extension, \(E/K\), of fields one means a cyclic extension, \(L/F\), of fields where \(E\supseteq L\supset F\supseteq K\). Let \(C(E/K)\) denote the subgroup of the relative Brauer group, Br\((E/K)\), generated by the various subgroups \(\text{cor(Br}(L/F))\) as \(L/F\) ranges over all cyclic layers of \(E/K\) and where cor denotes the corestriction map into Br\((E/K)\). We show that for \(K\) global, [Br\((E/K): C(E/K)]< \infty\) and we produce examples where \(C(E/K)\neq\text{Br}(E/K)\)''. cyclic layer of a finite Galois extension; relative Brauer group Brauer groups of schemes, Separable extensions, Galois theory Sums of corestrictions of cyclic algebras | 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 idea of stable cohomology of an algebraic variety was introduced by the author in a preprint [``Bloch's conjecture for torsion cohomology of algebraic varieties'', Preprint R '85-20, Univ.' Amsterdam (1985)], but in essence it was considered by M. Atiyah and A. Grothendieck in the 1960s for cohomology with complex coefficients. It arises naturally in the attempt to isolate that part of the cohomology of a variety that depends on the field of rational functions of the variety, and not on any concrete geometric model. The stabilization process destroys those cohomology classes that can be concentrated on proper subvarieties, and as a result we obtain only the classes that are nontrivial in the cohomology of a generic point of the variety. The geometry of a generic point is rather well approximated by the structure of the Galois group of the algebraic closure of the field of rational functions on the variety in the case of characteristic zero. Thus, the passage from the cohomology of a variety to that of a profinite group arises.
The basic observation in the preprint mentioned above was that with each finite group \(G\) a family of algebraic varieties is associated in a natural way, as varieties obtained as quotient spaces of faithful linear representations of this group modulo its action. In the case of the étale topology this allows us to construct an analogue of the theory of universal Eilenberg-Mac Lane spaces for rational mappings for the corresponding stable cohomology mappings.
The role of these spaces is played by varieties of the form \(V^ L/G\), where \(V\) is a faithful representation of the finite group \(G\), and \(V^ L\) is an open subvariety of \(V\) on which the action of \(G\) is free. Accordingly, the universal cohomology classes belong to the stable cohomology of the given finite group, which is defined in a natural way with respect to the quotient spaces \(V^ L/G\). This result follows rather simply from known results of the theory of étale cohomology.
The fact that the system of varieties \(V^ L/G\) allows the stable cohomology of a finite group to be defined in a compatible way also permits us to obtain less trivial consequences. Namely, it turns out that we can extend the concept of stable cohomology to profinite groups and prove the theorem that the stable cohomology of the Galois group of the algebraic closure of a function field of characteristic zero coincides with the ordinary cohomology. In the stable cohomology of varieties it is natural to distinguish the subgroups of unramified and projective elements. The former have trivial residue for all divisorial valuations, while the latter come with smooth projective models of the field. The analogous concepts also arise in a natural way for profinite groups. The main result of the paper establishes an isomorphism between the unramified cohomology groups for an algebraic variety defined over an algebraic closed field of characteristic zero and for the Galois group of the algebraic closure of the field of rational functions on it. stable cohomology; geometry of a generic point; Galois group of the algebraic closure of the field of rational functions on the variety; étale cohomology Bogomolov F.A., Stable cohomology of groups and algebraic varieties, Russian Acad. Sci. Sb. Math., 1993, 76(1), 1--21 Algebraic functions and function fields in algebraic geometry, (Co)homology theory in algebraic geometry, Galois cohomology, Varieties over global fields Stable cohomology groups and 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 Cayley-Chow variety \(C_{n,k} (\mathbb{P})\), parametrizing cycles of dimension \(n\) and degree \(k\) in projective space \(\mathbb{P}\), comes out equipped with a natural embedding into a certain projective space \(\mathbb{F}_{n,k}\). The main result of the paper is a formula expressing the degree of a subvariety \(T \subset C_{n,k} (\mathbb{P})\), relative to the natural embedding, in terms of the type, the sequence of multiplicities of irreducible components of the general cycle of the family \(\Sigma (T)\) parametrized by \(T\), and a collection of indices, numbers of cycles in the family which satisfy certain incidence conditions with respect to linear spaces of codimension \(n + 1\) in \(\mathbb{P}\). The total index \(\overline \mu\), the sum of all indices, is the number of cycles which meet \(m\) general linear spaces, where \(m = \dim T\), so it is what in enumerative geometry is called a characteristic number of the family. These indices depend both intrinsically on the family of cycles and on the projective characters of the carrier \(X \subset \mathbb{P}\), the locus filled in with cycles of the family (this is seen in the case of divisors). In particular, if the family is of simple type, i.e. the general cycle is irreducible, then \(\deg T = \overline \mu\). If moreover it is a family of divisors in a variety \(X\), then \(\overline \mu = d^m \overline \nu\), where \(d = \deg X\) and \(\overline \nu\) is the number of divisors passing through \(m\) general points of \(X\). As an application, a characterization of linear systems of divisors is derived. Cayley-Chow variety; cycles; degree of a subvariety; type; indices; characteristic number L. Guerra , Degrees of Cayley-Chow varieties , Math. Nachr. , 171 ( 1995 ), pp. 165 - 176 . MR 1316357 | Zbl 0838.14003 Parametrization (Chow and Hilbert schemes), Algebraic cycles Degrees of Cayley-Chow 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 Extending known results of \textit{P. Candelas}, \textit{X. C. de la Ossa}, \textit{P. S. Green} and \textit{L. Parkes} [``A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory'' in: Essays on mirror manifolds, 31-95 (1992; Zbl 0826.32016), see also Nuclear Phys., Particle Physics, B 359, No. 10, 21-74 (1991)], \textit{D. R. Morrison} [``Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians'', J. Am. Math. Soc. 6, No. 1, 223-247 (1993; Zbl 0843.14005) and ``Picard Fuchs equations and mirror maps for hypersurfaces'', in: Essays on mirror manifolds, 241-264 (1992; Zbl 0841.32013)] and many others, the authors formulate interesting conjectures on the mirror symmetry and generalized hypergeometric series for Calabi-Yau complete intersections in toric varieties. To state these conjectures, we need to introduce necessary ingredients as follows: Let \(P_\Sigma\) be a \((d + r)\)-dimensional projective toric variety corresponding to a complete simplicial fan \(\Sigma\) for a free \(\mathbb{Z}\)-module \(N\) of \(\text{rank} d + r\). Denote by \(E = \{v_1, \dots, v_k\}\) the set of primitive generators of one-dimensional cones in the fan \(\Sigma\), and let \(D_j\) be the irreducible torus-invariant Weil divisor on \(P_\Sigma\) corresponding to the one-dimensional cone spanned by \(v_j \in E\). Split \(E\) into a disjoint union \(E = E_1 \cup E_2 \cup \cdots \cup E_r\) and denote also by \(E_i\) the set of indices \(\{j \mid v_j \in E_i\}\). Assume that \(\sum_{j \in E_i} D_j\) for each \(1 \leq i \leq r\) is numerically effective (or equivalently, base-point-free in the present context) and is linearly equivalent to a hypersurface \(V_i \subset P_\Sigma\). Then the complete intersection \(V : = V_1 \cap V_2 \cap \cdots \cap V_r\) is a \(d\)-dimensional Calabi-Yau variety possibly with Gorenstein toroidal singularities, since \(\sum^k_{j = 1} D_j = \sum_i (\sum_{j \in E_i} D_j)\) is an anticanonical divisor of \(P_\Sigma\). When \(P_\Sigma\) is smooth, the kernel of the surjective homomorphism \(Z^k \ni \lambda = (\lambda_1, \dots, \lambda_k) \mapsto \sum^k_{j = 1} \lambda_j v_j \in N\) is known to coincide with the \(\mathbb{Z}\)-module \(R(E)\) of algebraic 1-cycles on \(P_\Sigma\). \(\lambda_j = \langle D_j, \lambda \rangle\) is the intersection number of the algebraic 1-cycle \(\lambda \in R(E)\) with the divisor \(D_j\). Thus \(R^+ (E) : = R(E) \cap (\mathbb{Z}_{\geq 0})^k\) is the submonoid of nef 1-cycles, where \(\mathbb{Z}_{\geq 0}\) is the set of nonnegative integers. We can choose a \(\mathbb{Z}\)-basis \(\{\lambda^{(1)}, \dots, \lambda^{(t)}\}\) of \(R(E)\) so that effective algebraic 1-cycles on \(P_\Sigma\), hence elements in \(R^+ (E)\) in particular, are nonnegative linear combinations of \(\lambda^{(1)}, \dots, \lambda^{(t)}\). Let us introduce a generalized hypergeometric series in complex variables \(u_1, \dots, u_k\) by
\[
\Phi_0 (u) : = \sum_{\lambda \in R^+ (E)} \prod^r_{i = 1} \left( \sum_{j \in E_i} \lambda_j \right)! \left( \left.\prod_{j \in E_i} u_j^{ \lambda_j}\right/\lambda_j! \right).
\]
Let \(T : = \Hom_\mathbb{Z} (N,C^\times)\) be the \((d + r)\)-dimensional algebraic torus with the character group \(N\). Denote by \(X^v\) the Laurent monomial corresponding to \(v \in N\). Then in terms of the Laurent polynomials \(P_{E_i} (X) : = 1 - \sum_{j \in E_i} u_j X^{v_j}\), \(i = 1, 2, \dots, r\), we have an integral representation
\[
\Phi_0 (u) = {1 \over (2i \sqrt {-1})^{d + r}} \int_{|X_1 |= 1, \dots, |X_{d + r} |= 1} {1 \over P_{ E_1} (X) \cdots P_{E_r}(X)} {dX_1 \over X_1} \wedge \cdots \wedge {dX_r \over X_r},
\]
where \(X_1, \dots, X_{d + r}\) are suitable coordinates for \(T\).
In terms of a new set of complex variables \(z_1, \dots, z_t\) defined by \(z_s : = \prod^r_{i = 1} \prod_{j \in E_i} u^{\lambda_j^{(s)}}_j\), \(s = 1, \dots, t\), \(\Phi_0 (u)\) can be expressed as a power series
\[
\Phi_0 (z) = \sum_{\lambda \in R^+ (E)} (\langle V_1, \lambda \rangle! \cdots \langle V_r, \lambda \rangle!/ \langle D_1, \lambda \rangle! \cdots \langle V_k, \lambda \rangle!) z^\lambda,
\]
where \(z^\lambda = z_1^{c_1} \cdots z_t^{c_t}\) with \(\lambda = c_1 \lambda^{(1)} + \cdots + c_t \lambda^{(t)}\), and \(\langle V_i, \lambda \rangle\) is the intersection number of \(V_i\) with the nef 1-cycle \(\lambda\).
Assume further that \(V\) is smooth and that the restriction map \(\text{Pic} (\mathbb{P}_\Sigma) \leftarrow \text{Pic} (V)\) is injective. There exists a flat ``\(A\)-model connection'' \(\nabla_{AP}\) on \(H^* (\mathbb{P}_\Sigma, \mathbb{C})\) which defines a quantum variation of Hodge structures on \(H^* (\mathbb{P}_\Sigma, \mathbb{C})\). Likewise, there exists a flat ``\(A\)-model connection'' \(\nabla_{AV}\) on \(H^* (V,\mathbb{C})\) which defines a quantum variation of Hodge structures on \(H^* (V,\mathbb{C})\). The complex variables \(z_1, \dots, z_t\) can be identified with \(\nabla_{AP}\)-flat coordinates on the image \(\widetilde H^2\) of \(H^2 (\mathbb{P}_\Sigma, \mathbb{C})\) and \(H^2 (V,\mathbb{C})\). -- Here are some of the authors' conjectures in terms of these ingredients:
(1) The generalized hypergeometric series \(\Phi_0 (z)\) in terms of \(z_1, \dots, z_r\) is a solution of the differential system \({\mathcal D}\) defined by the restriction of \(\nabla_{ AV}\) to \(\widetilde H^2\).
(2) The differential system \({\mathcal D}\) has logarithmic solutions of the form \(\Phi_s (z) = (\log z_s) \Phi_0 (z) + \Psi_s (z)\), \(s = 1, \dots, t\), with \(\Psi_s (z)\) holomorphic at \(z = 0\) and \(\Psi_s (0) = 0\). We can then define \(\nabla_{AV}\)-flat coordinates on \(\widetilde H^2\) by \(q_s : = \exp (\Phi_s (z)/ \Phi_0 (z))\), \(s = 1, \dots, t\). The coefficients of \(q_1, \dots, q_s\) with respect to \(z_1, \dots, z_t\) are integers.
(3) The Calabi-Yau variety mirror symmetric to \(V\) is obtained as a Calabi-Yau compactification of the complete intersection \(\{P_{E_1} (X) = 0\} \cap \cdots \cap \{P_{E_r} (X) = 0\}\) of affine hypersurfaces in the \((d + r)\)-dimensional algebraic torus \(T\).
The authors go on to check these conjectures by dealing with many examples of Calabi-Yau threefolds obtained as complete intersections in products of projective spaces. mirror variety; mirror symmetry; generalized hypergeometric series; Calabi-Yau complete intersections; toric varieties; quantum variation of Hodge structures; differential system; Calabi-Yau compactification V. Batyrev and D. Van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Universität Essen report, in preparation Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Generalized hypergeometric series, \({}_pF_q\) Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric 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 We show that symmetry defect hypersurfaces for two generic members of the irreducible algebraic family of \(n\)-dimensional smooth irreducible subvarieties in general position in \(\mathbb{C}^{2n}\) are homeomorphic and they have homeomorphic sets of singular points.
In particular symmetry defect curves for two generic curves in \(\mathbb{C}^2\) of the same degree have the same number of singular points. center symmetry set; affine algebraic variety; family of algebraic sets; bifurcation set of a polynomial mapping Hypersurfaces and algebraic geometry, Semialgebraic sets and related spaces, Fibrations, degenerations in algebraic geometry, Effectivity, complexity and computational aspects of algebraic geometry On a generic symmetrc defect hypersurface | 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 correct a theorem on the conductor of elliptic curves over \(Q\) given in the introduction of [ibid. 64, No. 1, 23--85 (2012; Zbl 06020907)]. modular curve; modular unit; cuspidal class number; elliptic curve; Jacobian variety; torsion subgroup Arithmetic aspects of modular and Shimura varieties, Modular and automorphic functions, Elliptic curves over global fields, Rational points, Modular and Shimura varieties, Jacobians, Prym varieties, Elliptic curves Erratum to ``The cuspidal class number formula for the modular curves \(X_{1}(2p)\)'' | 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 explicit formula is given for the cohomology class of the Brill Noether loci in the Prym varieties. Let \(\pi : \widetilde C \to C\) be an étale double cover of a smooth algebraic curve \(C\) of genus \(g\). The Brill-Noether varieties associated with this situation were defined by \textit{G. E. Welters} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 671- 683 (1985; Zbl 0628.14036)] by
\[
V^r : = \{L \in\text{Nm}^{-1} (\omega_C) |h^0 (L) \geq r + 1\quad\text{and}\quad h^0 (L) \equiv r + 1 \pmod 2\},
\]
where \(Nm : \text{Pic}^{2g - 2} (\widetilde C) \to \text{Pic}^{2g - 2} (C)\) is the norm map associated with \(\pi\). Using a construction of Mumford, the computation of the class of \(V^r\) is reduced to the computation of the class of the locus where two isotropic subbundles of a bundle endowed with a quadratic form, intersect in dimension exceeding a given number. There is a formula evaluating the cohomology dual to the fundamental class of such a locus as a quadratic expression in the so-called \(P\)-polynomials applied to the subbundles. This result stems from the reviewer's paper in: Topics in invariant theory, Proc. Sémin. Algèbre Dubreil-Malliavin, Paris 1989-1990, Lect. Notes Math. 1478, 130-191 (1991; Zbl 0783.14031) and the preprint by the reviewer and \textit{J. Ratajski}: ``Formulas for Lagrangian and orthogonal degeneracy loci'' (Max-Planck Inst. Math. 1994). This formula, appropriately specialized, yields the class of \(V^r\). The obtained formula for the class of \(V^r\) implies the non-emptiness of those loci in the range \(g \geq \left( \begin{smallmatrix} r + 1\\ 2 \end{smallmatrix} \right) + 1\), thus giving another proof of the ``existence theorem'' for \(V^r\) first proved by \textit{A. Bertram} [in Invent. Math. 90, 669-671 (1987; Zbl 0646.14006)] using different methods. The locus \(V^r\) is endowed, in the paper, with a scheme structure which is reduced, Cohen-Macaulay and normal for a general curve and any irreducible double cover of it. This scheme structure coincides on \(V^r \backslash V^{r + 2}\) with the scheme structure defined by Welters in the paper quoted above. Brill Noether loci; Prym varieties; cover of a smooth algebraic curve; isotropic subbundles De Concini, C.; Pragacz, P.: On the class of brill--Noether loci for Prym varieties. Math. ann. 302, 687-697 (1995) Picard schemes, higher Jacobians, Jacobians, Prym varieties, Algebraic cycles, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the class of Brill-Noether loci for Prym 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 article we give a numerical criterion, valid in all characteristics, for the very ampleness of a line bundle \(H\) on a curve \(C\) (possibly reducible and non-reduced) lying on a smooth algebraic surface.
We show by the way that this criterion essentially implies the results of Bombieri and Ekedahl on pluricanonical embeddings of surfaces of general type. We use our numerical criterion to obtain the fine classification of non-special rational surfaces in \(\mathbb{P}^4\) (with one exception dealt with in a forthcoming article). We also describe in full detail the embeddings when \(C\) is a curve of genus 2 and \(H\) has degree 5. very ampleness of a line bundle on a curve Catanese, F., Franciosi, M.: Divisors of small genus on algebraic surfaces and projective embeddings. In: Proceedings of the conference ''Hirzebruch 65'', Tel Aviv 1993, Contemp. Math., A.M.S. (1994), subseries 'Israel Mathematical Conference Proceedings', vol. 9, pp. 109--140 (1996) Vector bundles on curves and their moduli, Divisors, linear systems, invertible sheaves, Embeddings in algebraic geometry, Rational and ruled surfaces Divisors of small genus on algebraic surfaces and projective embeddings | 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 and Y be algebraic varieties of dimensions n, n-1 over an algebraically closed field k. (1) X is said to be ruled over Y if it is birationally isomorphic to \(Y\times {\mathbb{P}}^ 1\), and the induced rational map \(\phi\) : \(X\to Y\) is called a ruling. (2) If there is a dominant separable rational map \(g: X\to Y\) whose generic fibre is an irreducible rational curve, X is called quasi-ruled and g a quasi-ruling. (3) X is said to be uniruled if there is a ruled variety W and a dominant rational map \(\psi\) : \(W\to X\). As it is clear, all these notions are birational in nature and generalize in higher dimensions that of ruled surface. \textit{K. Kodaira} and \textit{D. C. Spencer} [Ann. Math., II. Ser. 67, 328-466 (1958; Zbl 0128.169)] have shown that all small deformations of a ruled surface are ruled. In Am. J. Math. 103, 997-1020 (1981; Zbl 0494.14002), the author gave a counterexample to the straightforward higher dimensional version of the Kodaira-Spencer theorem; indeed he showed the following. Let \(n\geq 3\), then there is a family of n- dimensional algebraic varieties parametrized by a smooth curve C, such that: (i) \(V_ o\) (o\(\in C)\) is ruled over an abelian variety and \(h^ 2(V_ o,{\mathcal O}_{V_ o})\neq 0\), (ii) \(V_ x\) is not ruled for a generic \(x\in C\); moreover, if \(n\geq 4\) there is a family as above such that: (i) \(V_ o\) is ruled over a ruled variety, (ii) \(V_ x\) is not quasi-ruled. Both the author and Fujiki have shown that if Char k\(=0\) all small deformations of a uniruled variety are uniruled. In the paper under review the author gives another partial generalization of Kodaira-Spencer theorem; indeed he proves the following. Let \(p: V\to M\) be a family of n-dimensional varieties. If the fibre \(V_ o\) (o\(\in M)\) is quasi-ruled over \(Y_ o\), then all fibres \(V_ i\) for u near o are quasi-ruled if one of the following holds: (1) \(Y_ o\) is projective and the quasi- ruling is a morphism, (2) \(Y_ o\) is not uniruled and either Char k\(=0\) or Char k\(>5\) and \(n\leq 3\). Moreover, if \(V_ o\) is ruled and \(h^ 2(V_ o,{\mathcal O}_{V_ o})=0\), then \(V_ o\) is ruled for every \(u\in M\). deformations of a ruled surface; deformations of a uniruled variety; quasi-ruling Levine, M.: The stability of certain classes of uni-ruled varieties. Comp. Math. (to appear) Families, moduli, classification: algebraic theory, Formal methods and deformations in algebraic geometry The stability of certain classes of uni-ruled 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 author examines the Hilbert function of the intersection of a finite set \(X\subset P^ n\) with a hypersurface defined by a form \(D\). Let \(I\subset R:=k[X_ 0,...,X_ n]\) be the saturated ideal of \(X\) and \(\beta:=\min\{t\mid \hbox{height} \langle I_{\leq t}\rangle\geq 2\}\). His main result is that if \(D\) is the g.c.d. of the forms in \(I\) of degree \(<\beta\), and if \(\Delta H(X,\beta)=\Delta H(R/(I,D),\beta)\), then the saturated ideal of that intersection \(Y\) is \((I,D)\) and its Hilbert function satisfies \(H(Y,t)=H(X,t)-H(X\setminus Y,t-deg(D))\) for all \(t\). (This formula is misprinted on p. 90, 1.4.)
Furthermore, the author shows that if \(X\subset \mathbb{P}^ 2\) is the complete intersection of a curve of degree \(a\geq 2\) and a curve of degree \(b\geq 1\), then every curve of degree \(d\leq a\) containing at least \(a+(b- 1)(d-1)+1\) points of \(X\) passes through exactly \(bd\) points of \(X\). For \(d=a\), this is also an easy consequence of the classical Cayley-Bacharach theorem [\textit{E. D. Davis, A. V. Geramita}, and \textit{F. Orecchia}, Proc. Am. Math. Soc. 93, 593-597 (1985; Zbl 0575.14040)]. number of points of complete intersection of a curve; Hilbert function; intersection; Cayley-Bacharach theorem Sodhi, A.: On the intersection of a hypersurface with a finite set of points in pn. J. pure appl. Algebra 74, 85-94 (1991) Complete intersections, Projective techniques in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry On the intersection of a hypersurface with a finite set of points in \({\mathbb{P}}^ n\) | 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 A real hyperelliptic curve \(X\) is said to be Gaussian if there is an automorphism \(\alpha : X_{\mathbb C} \to X_{\mathbb C}\) such that \(\bar{\alpha } = [-1]_{\mathbb C}\circ \alpha\), where \([-1]\) denotes the hyperelliptic involution. Gaussian hyperelliptic curves occur in several contexts (e.g. in the study of real Jacobian). Here the authors prove that a real hyperelliptic curve is Gaussian if and only if it is uniquely determined by its branch locus. They describe their moduli spaces.For many properties of real hyperelliptic curves, see \textit{M. Lattarulo} [Commun. Algebra 31, 1679--1703 (2003; Zbl 1059.14070)]. real algebraic curve; real abelian variety; real Jacobian Families, moduli of curves (analytic), Automorphisms of curves, Real algebraic and real-analytic geometry, Klein surfaces Imaginary automorphisms on real hyperelliptic 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 \(C_n\) be the affine scheme of all associative commutative multiplications on an \(n\)-dimensional vector space \(V_n\), and \(A_n\) be the affine scheme of those multiplications on \(V_n\) which determine commutative nilpotent algebras of class 3, and let \(C_n^{\text{red}}\) and \(A_n^{\text{red}}\) be the corresponding reduced schemes.
From the article by \textit{I.~R.~Shafarevich} [Leningr. Math. J. 2, No. 6, 1335--1351 (1991; Zbl 0743.13009)], it is known that an arbitrary component \(A_{n,r}\) of \(A_n^{\text{red}}\) is determined by the number \(r=\dim N^2\) for a generic algebra \(N\) in \(A_{n,r}\).
The main problem is to determine a condition on \(r\) for the component \(A_{n,r}\) to coincide with a proper component of \(C_n\).
The author proves some ``descent'' theorem. It follows that, under certain conditions, for sufficiently small number \(r\), the component \(A_{n,r}\) coincides with a proper component of \(C_n\). affine scheme; commutative nilpotent algebra of class 3; component of a variety General commutative ring theory, Schemes and morphisms, General and miscellaneous Stable nilpotent algebras of class 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 [For the entire collection see Zbl 0683.00009.]
This is a very nice down-to-earth introduction to Grothendieck's theory of cotangent complex applied to a concrete situation of deformations of a hyperplane section. It is applied to the problem of explicit description of graded canonical (or subcanonical) rings of surfaces and curves in terms of generators and relations.
The most interesting part of this article is a collection of concrete examples. For instance the author works out in detail the deformation theory in degree \(\leq 0\) of the half-canonical ring of a hyperelliptic curve of genus \( 6.\) This can be applied to some Horikawa surfaces (numerical quintics) whose canonical system contains hyperelliptic curves of genus 6. This example serves a warm-up example to the real job: to compute the canonical ring of surfaces with \(p_ g=0\) with torsion \({\mathbb{Z}}/2\). In this example the ring needs 8 generators, 20 relations, 64 syzygies and 90 second syzygies. An attempt to implement the computations using the Maple program has been made; its success could eventually decide the irreducibility of the moduli space of such surfaces.
Among concrete results proved in this paper we cite only one: Let X be a canonical surface of general type with \(q=0\), \(p_ g\geq 2\) and \(K^ 2\geq 3\). Assume X has an irreducible curve in its canonical linear system. Then the canonical ring is generated in degrees \(\leq 3\) and related in degrees \(\leq 6\). cotangent complex; deformations of a hyperplane section; half-canonical ring of a hyperelliptic curve; Horikawa surfaces; canonical ring of surfaces; Maple program; canonical surface of general type Miles Reid, Infinitesimal view of extending a hyperplane section --- deformation theory and computer algebra, Algebraic geometry (L'Aquila, 1988) Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 214 -- 286. Formal methods and deformations in algebraic geometry, Computational aspects of algebraic curves, Computational aspects of algebraic surfaces, Surfaces of general type, Local deformation theory, Artin approximation, etc. Infinitesimal view of extending a hyperplane section - deformation theory and computer algebra | 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 projective variety \(X \subset {\mathbb P}^N\) satisfies an order \(d \geq 2\) Laplace equation if its order \(d\) osculating space at its general point has not the expected dimension. Here the author finds more examples of surfaces in \({\mathbb P}^5\) satisfying a Lapace equation. Among them, rational surfaces with genus \(1\) curves as hypersurfaces. Then she studies monomial Togliatti systems of cubics for \(3\)-folds. She conjectures a wide generalization of the \(3\)-fold part. Laplace equation; osculating variety of a projective variety Ilardi, G, Togliatti systems, Osaka J. Math., 43, 1-12, (2006) Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Projective techniques in algebraic geometry Togliatti systems | 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 Gegenstand der Abhandlung ist die Untersuchung der charakteristischen Singularitäten der Modulargleichung
\[
\text{(1)} \quad F(p,q,1)=0,
\]
(wo \(q\) das Modularquadrat einer gegebenen elliptischen Function, \(p\) das Quadrat des transformirten Moduls für eine primäre Transformation ungrader Ordnung \(N\),) un die Untersuchung der in linearen Coordinaten ausgedrückten Gleichung
\[
F(\alpha,\beta,\gamma)=0
\]
der Modularcurve \(C\), welche durch die Substitution \(p=\frac\alpha\gamma, q=\frac\beta\gamma\) mit der Gleichung 1) zusammenhängt. \(P,Q,R\) sind die Spitzen des Dreiecks \(\alpha\beta\gamma\); \(S\) ist der Punkt \(\alpha=\beta=\gamma;p\) und \(q\) sind die Parameter zweier Linienbüschel \(\alpha-p\gamma,\beta-q\gamma\), zwischen deren Strahlen die durch Gleichung (1) ausgedrückte Beziehung besteht: die Modularcurve \(C\) ist der Ort der Durchschnittspunkte entsprechender Straheln beider Büschel. Die Methode der Untersuchung ist schon in früheren Arbeiten von dem Herrn Verfasser befolgt, z. B. in einem der Pariser Akademie überreichten Mémoire: ``Sur les équations modulaires'', das in den Atti a. Acc. d. Lincei (3) I. 1877 gedruckt wurde. Zuerst werden hier die Singularitäten für den Fall untersucht, wo \(N\) keinen quadratischen Theiler hat; dann folgt der entgegengesetzte Fall. Für die Charakteristiken und Singularitäten der Modularcurve \(C\) ergeben sich folgende Resultate. Es sind \(g\) und \(g'\) conjugirte Theiler von \(N\); \(h^2\) ist das grösste in \(N\) enthaltene Quadrat; \(\eta\) der grösste gemeinsame Theiler von \(g\) und \(g'\); \(f(\eta)\) die Anzahl der Zahlen, die \(\leqq\eta\) und relativ prim zu \(\eta; f'(g)\) und \(f'(g')\) sind definirt durch die Gleichung
\[
\frac{f'(g)}g=\frac{f(\eta)}\eta=\frac{f'(g')}{g'};
\]
und es ist
\[
2\nu=\sum f(\eta),\quad A+B=\sum f'(g),\quad A_2+B_2=(A+B)^2,
\]
wo die Summe \(\sum\) sich auf alle Theiler \(g\) von \(N\) erstreckt, wo ferner \(A\) alle \(f'(g)\) umfasst, für welche \(g>\sqrt N\), und zugleich, wenn \(N=\vartheta^2\), den Werth \(\frac12\vartheta'=\frac12 f(\vartheta)\), und wo \(A_2\) alle Terme von \(\sum f'(g_1)\cdot\sum f'(g_2)\), in denen \(g_1g_2>N\), und die Hälfte aller Terme, wofür \(g_1g_2=N\) ist, enthält. Die Definitionen von \(B\) und \(B_1\) ergeben sich analog aus denen von \(A\) und \(A_1\). Ferner bezeichnen \(m,n,K,J,D, T, H,\) resp. die Ordnung, die Klasse, die Spitzen- und Flexions-Singularität, die Discriminanten-Ordnung und Klasse und das Geschlecht der Curve. Die Charakteristiken der Curve genügen dann folgenden Gleichungen:
\[
\begin{matrix} \r &\l \\ m &=2A,\\ n &=3A-B-\vartheta',\\ H &=\frac12(A+B)-3\nu+1,\\ K &=2(A+B)-6\nu+\vartheta',\\ J &=5A-B-6\nu-2\vartheta',\\ J-K &=3A-3B-3\vartheta',\\ D &=4A^2-5A+b+\vartheta',\\ T &=(3A-B-\vartheta')^2-5A+B+\vartheta',\\ T-D &=(3A-B-\vartheta')^2-4A^2. \end{matrix}
\]
Bezeichnen ferner die Symbole \((XXY)\) oder \((YXX)\) einen Zweig, oder ein Aggregat von Zweigen, die die Linie \(XY\) im Punkte \(X\) berühren; die Symbole \(O(XXY), C(XXY), K(XXY), J(XXY), D(XXY), T(XXY)\) die entsprechenden oben angegebenen Charakteristiken der Zweige \((XXY)\), so hat man:
\[
\begin{matrix} \l\\ O(PPQ)=A-B;\\ C(PPQ)=B-\frac12\vartheta',\\ K(PPQ)=A-B-\nu+\frac12\vartheta';\\ J(PPQ)=B-\nu;\\ D(PPQ)=2A^2-A_2-A+\frac12\vartheta';\\ T(PPQ)=B_2-B^2_a\vartheta'B+\frac12\vartheta'+\frac14\vartheta'';\end{matrix}
\]
etc. Auf gleiche Weise werden die Charakteristiker von \((PRR), (QRR), (PSS), (QSS)\) etc. und die Symbole \(O(X), K(X), D(X), C(XY), J(XY), T(XY)\) etc. behandelt, welche eine ähnliche Bedeutung für die Zweige haben, die durch einen gegebenen Punkt \(X\) gehen, oder eine gegebene Linie \(XY\) berühren. Im Folgenden werden die 6 Modularcurven discutirt, welche den 6 Transformationen
\[
x,\quad 1-x,\quad \frac1x, \quad \frac1{1-x}, \quad \frac x{x-1}, \quad \frac{x-1}x
\]
entsprechen, für welche die Modulargleichung ungeändert bleibt. Singularities; elliptic functions; modular curves; characteristic of a curve; modular equations Curves in algebraic geometry, Singularities in algebraic geometry, Elliptic functions and integrals On the singularities of the modular equations and 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 Abelian conformal field theory, that is the theory of free fermions over a compact Riemann surface, has been studied, since the late 1980s, from several points of view. The first rigorous mathematical foundation of abelian conformal field theory was provided by the work of \textit{N. Kawamoto}, \textit{Y. Namikawa}, \textit{A. Tsuchiya} and \textit{Y. Yamada} [Commun. Math. Phys. 116, No. 2, 247-308 (1988; Zbl 0648.35080)], Y. Shimizu and K. Ueno. Their approach uses the framework of Sato's universal Grassmannian manifold, the Krichever correspondence between data on Riemann surfaces and soliton equations, complex cobordism theory, formal groups, and other well-developed theories. In his recent paper ``On conformal field theory'' [in: Vector bundles in algebraic geometry, Proc. 1993 Durham Symp., Lond. Math. Soc. Lect. Note Ser. 208, 283-345 (1995; Zbl 0846.17027)] \textit{K. Ueno} proposed another approach to abelian conformal field theory. Namely, taking the vertex operator algebra constructed from the Heisenberg algebra as a gauge group, and applying the basic constructions in the nonabelian conformal theory to this situation, the relationship between conformal blocks and theta functions of higher level gives another mathematically rigorous description of the physicists' operator formalism in the fermionic theory.
The aim of the present paper is to give a geometric interpretation of one of Ueno's results stated in this context. More precisely, Ueno had described the spaces of vacua (conformal blocks) by imposing an extra gauge condition expressed by vertex operators of even level \(M\) among the Fock spaces. His main result, in this regard, consisted then in establishing an isomorphism between the space of conformal blocks and the space of \(M\)th-order theta functions on the Jacobian of the underlying pointed stable curve \(C\).
In the paper under review, the author points out that Ueno's result can be extended to the case of level \(M = 1\), and he gives another description of the space of conformal blocks in this particular odd-level case.
The basic framework used here is, on the one hand, the Beilinson-Bernstein theory of localizations for representations of certain infinite-dimensional Lie algebras associated with special ``dressed'' moduli spaces and, on the other hand, a version of Skornyakov's theory of the \(\pi\)-Picard group of locally free sheaves on supercurves (for \(N=1\) or \(N=2)\) with symmetry group \(\pi\). The respective general theories of Beilinson-Bernstein [cf. \textit{A. A. Beilinson} and \textit{V. V. Schechtman}, Commun. Math. Phys. 118, No. 4, 651-701 (1988; Zbl 0665.17010)] and of Skornyakov-Manin [cf. \textit{Yu. I. Manin}, Topics in non-commutative geometry (1991; Zbl 0724.17007)] are tailored to the particular context of abelian conformal field theory, and this program forms the main body of the paper. The central results are then the following theorems:
(1) The \(\pi\)-Picard functor for a proper smooth supercurve of dimension \(N = 1\) or \(N = 2\) is representable by a smooth superscheme, the so-called \(\pi\)-Picard variety.
(2) The space of (Ueno's) conformal blocks equals a fiber of the Beilinson-Bernstein localization of the Fock representation on the \(\pi\)-Picard scheme of the respective supercurve.
(3) In the case of level \(M = 1\), the space of conformal blocks is canonically isomorphic to the space of theta functions on the Jacobian of the underlying pointed stable curve \(C\).
As for related and further results, the author refers to two forthcoming papers entitled ``Moduli of \(N = 1\) stable superconformal curves and abelian conformal field theory'' and ``Moduli of \(N = 2\) superconformal curves''. super Riemann surfaces; dressed moduli spaces; Picard group; Picard functor; Picard variety; abelian conformal field theory; vertex operator algebra; Heisenberg algebra; conformal blocks; theta functions of higher level; infinite-dimensional Lie algebras; supercurves; Beilinson-Bernstein localization; Fock representation Virasoro and related algebras, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Riemann surfaces, Algebraic functions and function fields in algebraic geometry, Theta functions and abelian varieties, Families, moduli of curves (algebraic) Abelian conformal field theory and \(N=2\) supercurves | 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 knowledge of the singular locus of an algebraic curve is essential for a variety of algorithms.
Given a defining polynomial $p(x,y)$ for the curve, we simply have to determine the common zeros of $p$ and its partial derivatives.
The situation is more complicated if the curve is given by a rational parametrization; i.e., a pair of rational functions s.t. the curve is the Zariski closure of the image.
In this paper a so-called T-function is determined by means of a resultant computation on the parametrization.
By factorizing this T-function one can obtain the singularities (ordinary or non-ordinary) together with their multiplicities. rational parametrization; singularities of an algebraic curve; multiplicity of a point; ordinary and non-ordinary singularities; T-function; fiber function Computational aspects of algebraic curves, Numerical aspects of computer graphics, image analysis, and computational geometry, Geometric aspects of numerical algebraic geometry An in depth analysis, via resultants, of the singularities of a parametric 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 subfield of \(\mathbb{C}\) which contains all 2-power roots of unity, and let \(K = k(\alpha_1, \alpha_2, \ldots, \alpha_{2 g + 1})\), where the \(\alpha_i\)'s are independent and transcendental over \(k\), and \(g\) is a positive integer. We investigate the image of the 2-adic Galois action associated to the Jacobian \(J\) of the hyperelliptic curve over \(K\) given by \(y^2 = \prod_{i = 1}^{2 g + 1}(x - \alpha_i)\). Our main result states that the image of Galois in \(\operatorname{Sp}(T_2(J))\) coincides with the principal congruence subgroup \(\Gamma(2) \triangleleft \operatorname{Sp}(T_2(J))\). As an application, we find generators for the algebraic extension \(K(J [4]) / K\) generated by coordinates of the 4-torsion points of \(J\). Galois group; elliptic curve; hyperelliptic curve; Jacobian variety; Tate module Yelton, J, Images of \(2\)-adic representations associated to hyperelliptic Jacobians, J. Number Theory, 151, 7-17, (2015) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Jacobians, Prym varieties Images of 2-adic representations associated to 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 In this paper we study several algebraic invariants of a real curve \(X\): the Picard group \(\text{Pic} (X)\), the Brauer group \(\text{Br} (X)\), the \(K\)-theory matrix invariant \(SK_ 1(X)\), the étale cohomology groups \(H^ n (X,\mathbb{Z}/2)\), and the sheaves \({\mathcal H}^ n\) associated to these cohomology groups. We relate these to two topological invariants of the space \(X(\mathbb{R})\) of real points of \(X\): the number \(c\) of components and the number \(\lambda\) of loops. For singular \(X\) we compute \(\text{Pic} X/2 \text{Pic} X\) and construct a Mayer-Vietoris sequence relating \(\text{Br} (X)\) to \(SK_ 1 (X)/2SK_ 1 (X)\) and use it to show that \(\text{Br} (X)=(\mathbb{Z}/2)^ c\) and \(SK_ 1 (X)/2SK_ 1 (X)=(\mathbb{Z}/2)^ \lambda\). space of real points; number of components; number of loops; real curve; Picard group; Brauer group; étale cohomology groups C. Pedrini and C. Weibel, Invariants of real curves , préprint, 1991. Special algebraic curves and curves of low genus, Topology of real algebraic varieties, Picard groups, Brauer groups of schemes, Étale and other Grothendieck topologies and (co)homologies Invariants of real 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 positive characteristic. If \(X\) is an affine curve over \(k\), the algebraic fundamental group \(\pi_1(X)\) is very big in general, as the Artin-Schreier étale covers show. In this lecture of the proceedings the author discusses the cases of the affine line \(\mathbb A^1(k)\) and of the multiplicative group \(\mathbb G_m(k)\). Let \(G\) be a finite group and let \(p(G)\) denote the normal subgroup of \(G\) generated by its \(p\)-Sylow subgroups. Then a conjecture of Abhyankar says in these two cases that:
(1) \(G\) is a quotient of \(\pi_1(\mathbb A^1(k))\) if and only if \(G=p(G)\), and
(2) \(G\) is a quotient of \(\pi_1(\mathbb G_m(k))\) if and only if \(G/p(G)\) is a cyclic group of order prime to \(p\).
The necessity of this conditions was already observed by \textit{A. Grothendieck} in SGA 1 [Séminaire de géométrie algébrique, Bois-Marie 1960-1961, Lect. Notes Math. 224 (1971; Zbl 0234.14002)].
The aim of the lecture is to prove a result of Katz and Gabber which describes the covers of the projective line totally ramified at infinity and tamely ramified at the origin [see \textit{N. M. Katz}, Ann. Inst. Fourier 36, 69-106 (1986; Zbl 0564.14013)], as well as to discuss a method due to Nori for producing étale covers of \(\mathbb A^1(k)\) [see \textit{M. V. Nori}, Algebraic geometry and its applications. Coll. Pap. Abhyankar's 60th birthday, Conf. Purdue Univ. 1990, , 209-212 (1994; Zbl 0811.14031)]. For all these examples the Abhyankar conjecture is verified. fundamental group of a curve; Abhyankar conjecture; affine line; positive characteristic; Artin-Schreier covers Gille P. , Le groupe fondamental sauvage d'une courbe affine , in: Bost J.-B. , Loeser F. , Raynaud M. (Eds.), Courbe semi-stable et groupe fondamental en géométrie algébrique , Progress in Math. , 187 , Birkhäuser , 2000 , pp. 217 - 230 . MR 1768103 | Zbl 0978.14034 Coverings of curves, fundamental group, Local ground fields in algebraic geometry, Arithmetic ground fields for curves The wild fundamental group of an affine group 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 \textit{G. Harder} and \textit{M. S. Narasimhan} [Math. Ann. 212, 215-248 (1975; Zbl 0324.14006)] (and independently \textit{D. Quillen}) have constructed a canonical flag of subbundles on any vector bundle on a complete smooth algebraic curve over a field. This flag measures how far away from semistability the vector bundle is. Influenced by this D. Grayson and U. Stuhler have studied the corresponding situation over number fields. Here one is looking at lattices in some euclidean or hermitean vector space. It turns out that one has again a canonical filtration by a flag of sublattices. If one compares both situations more carefully one sees that the second is in a certain sense more general. This leads one to consider what we call generalized vector bundles in this paper. The idea is to replace locally or better over the completion of the local rings of the curve the lattices given by the stalks of the locally free sheaf associated to the vector bundle by the real valued norms given by these lattices on the associated vector spaces.
It turns out that the Harder-Narasimhan-filtration exists also in this more general context. This is done in section two of this paper and we could follow for this more or less completely the exposition by \textit{D. R. Grayson} [in: Algebraic \(k\)-theory, Proc. Conf., Oberwolfach 1980, Part I, Lect. Notes Math. 966, 69-90 (1982; Zbl 0502.14004)]. We have included in this section a study of the H-N-filtration where the vector bundle is deformed in a family. The main motivation for us to consider this kind of generalisation comes from reduction theory of the general linear group. vector bundle on a complete smooth algebraic curve; flag; generalized vector bundles; Harder-Narasimhan-filtration Hoffmann, N.; Jahnel, J.; Stuhler, U., Generalized vector bundles on curves, Journal für die Reine und Angewandte Mathematik, 495, 35-60, (1998) Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Generalized vector 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 Die Menge \(G^{n,n+1}_{2(n+1)}\) der reellen Kollineationen im \(2(n+1)\)-dimensionalen reellen projektiven Raum \(P_{2(n+1)}\), bei denen zwei reelle unbewegliche windschiefe lineare Unterräume \({\mathcal I}_ n\) und \({\mathcal K}_{n+1}\) (dim \({\mathcal I}_ n=n\), dim \({\mathcal K}_{n+1}=n+1)\) erhalten bleiben, stellt eine \(2(n^ 2+3n+2)\)- Parametergruppe dar. Die Kleinsche Geometrie mit diesen Transformationen \(G^{n,n+1}_{2(n+1)}\) als Fundamentalgruppe heißt Geometrie des verallgemeinerten biaxialen Raumes \(P^{n,n+1}_{2(n+1)}\). In der vorliegenden Arbeit sind einige algebraisch-geometrische Angaben für die Geometrie von \(P^{n,n+1}_{2(n+1)}\) gegeben. Es werden weiterhin einige Probleme betrachtet, die mit einem Einparametersystem \(H=\{{\mathfrak g}_ n\}\) von n-dimensionalen linearen mit \({\mathcal I}_ n\), \({\mathcal K}_{n+1}\) windschiefen Unterräumen \({\mathfrak g}_ n\) verbunden sind. fundamental points for a one-parameter system of n-dimensional linear subspaces; generalized biaxial space Projective analytic geometry, Questions of classical algebraic geometry, Projective techniques in algebraic geometry, Projective differential geometry Fundamental points for a one-parameter system of \(n\)-dimensional linear subspaces in the generalized biaxial space \(P^{n,n+1}_{2(n+1)}\) of even dimension | 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 \(F_ 5\) be the Fermat curve of exponent five: \(X^ 5+Y^ 5+Z^ 5=0\), and let \(J_ 5\) be its Jacobian. Fix a primitive fifth root of unity \(\zeta\). The group of automorphisms of \(F_ 5\) is generated by the elements \(\sigma:(X,Y,Z) \mapsto (\zeta X,Y,Z)\), \(\tau:(X,Y,Z) \mapsto (X,\zeta Y,Z)\), \(\iota:(X,Y,Z) \mapsto (Y,X,Z)\) and \(\rho: (X,Y,Z) \mapsto (Z,X,Y)\). With positive integers \(a,b,c\) such that \(a+b+c=5\), denote the quotient curve \(F_ 5/ \langle \sigma^ b \tau^{-a} \rangle\) by \(F_{a,b,c}\), and its Jacobian by \(J_{a,b,c}\). One has the canonical projection \(\varphi_{a,b,c}:F_ 5 \to F_{a,b,c}\), and an isogeny
\[
\varphi=(1 \times \rho \times \rho^ 2) \circ ((\varphi_{1,3,1})_ * \times (\varphi_{3,1, 1})_ * \times (\varphi_{2,2,1})_ *):J_ 5 \to J^ 3_{1,3,1}.
\]
\(\text{End} (J_{1,3,1})\) may be identified with \(\mathbb{Z}[\zeta]\), thus \(\varphi\) induces a canonical isomorphism
\[
F_ \varphi: \text{End} (J_ 5) \otimes \mathbb{Q} \to M_ 3 (\mathbb{Q} [\zeta]),
\]
the ring of \((3 \times 3)\)- matrices with entries in \(\mathbb{Q} [\zeta]\). -- By very explicit computations the following theorem is proved:
\(F_ \varphi\) maps \(\text{End} (J_ 5)\) onto the subring of \(M_ 3 (\mathbb{Q} [\zeta])\) characterized by the following properties: Let \(\pi=\zeta-1\) and let \(v\) be the row vector (1,1,1). Let \(M \in M_ 3 (\mathbb{Q} [\zeta])\). Then \(M \in \text{End} (J_ 5)\) if and only if there is an integer a such that \(vM \equiv av \equiv vM^ t \pmod \pi\) and \(vMv^ t \equiv (3a) \pmod {\pi^ 2}\).
The theorem can be applied to obtain the following result on the decomposition of \(J_ 5\): There exist abelian varieties \(A\) and \(B\) defined over \(\mathbb{Q}\) such that (i) \(J_ 5 \simeq A^ 2 \times B\) over \(\mathbb{Q}\), (ii) \(A \simeq J_{1,3,1}\) over \(\mathbb{Q} [\zeta]\), (iii) \(B \simeq J_{1,3,1}/ \langle(1+3 \pi)Q \rangle\) over \(\mathbb{Q} [\zeta]\), where \(Q\) is a nontrivial \(\pi^ 2\)-division point on \(J_{1,3,1}\) with complex conjugate equal to \(-Q\). Fermat curve of exponent five; Jacobian; group of automorphisms; canonical isomorphism Lim, C. H.: The geometry of the Jacobian of the Fermat curve of exponent five. J. number theory 41, 102-115 (1992) Global ground fields in algebraic geometry, Jacobians, Prym varieties, Elliptic curves over global fields The geometry of the Jacobian of the Fermat curve of exponent five | 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 deals, using elementary topological methods, with some properties of the branches of a complex algebraic variety along a subvariety. Precisely, let X be an algebraic variety over \({\mathbb{C}}\) and Y the closed subvariety of X described by the non normal points of X. If \(p: \tilde X\to X\) is the normalization morphism, \(\tilde Y=p^{-1}(Y)\) is connected, \(\tilde Y\to Y\) is an unramified Galois covering, and for any point \(z\in Y\) there are n analytic branches of X in z, then n is proved to be equal to the maximum number of connected components of \(\tilde Y\times_ YE\), as \(E\to Y\) varies in the set of all connected coverings of Y. Moreover a covering \(\sigma: \bar X\to X\) of order n is constructed, containing \(\tilde Y\) with n global branches along \(\tilde Y,\) such that \(\sigma_{| \tilde Y}=p_{| \tilde Y}:\) for any \(z\in Y\) and any \(w\in \tilde Y=\sigma^{-1}(Y)\) a (non canonical) bijection can be given between the local branches of X at z and the global branches of \(\bar X\) along \(\tilde Y\) in a neighborhood of w. For recent related algebraic results see \textit{C. Cumino} and \textit{G. Paxia} [Ann. Mat. Pura Appl., IV. Ser. 143, 321-338 (1986; see the following review)]. branches of a complex algebraic variety along a subvariety; local branches; global branches Ramification problems in algebraic geometry, Analytic subsets and submanifolds Parametrization of local branches by global branches | 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 present an algorithm that computes an unmixed-dimensional decomposition of an arbitrary algebraic variety \(V\). Each \(V_ i\) in the decomposition \(V=V_ 1\cup\cdots\cup V_ m\) is given by a finite set of polynomials which represents the generic points of the irreducible components of \(V_ i\). The basic operation in our algorithm is the computation of greatest common divisors of univariate polynomials over extension fields. No factorization is needed. -- Some of the main problems in polynomial ideal theory can be solved by means of our algorithm: we show how the dimension of an ideal can be computed, systems of algebraic equations can be solved, and radical membership can be decided.
Our algorithm has been implemented in the computer algebra system MAPLE. Timings on well-known examples from computer algebra literature are given. decomposition of an algebraic variety; greatest common divisors of univariate polynomials; polynomial ideal theory M. Kalkbrener, \textit{A generalized Euclidean algorithm for computing triangular representations of algebraic varieties}, J. Symbolic Comput., 15 (1993), pp. 143--167. Computational aspects in algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials, Polynomials, factorization in commutative rings, Varieties and morphisms A generalized Euclidean algorithm for computing triangular representations of 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 Let C be a smooth curve of genus g over a field k (algebraically closed and of arbitrary characteristic). One has the natural Abel morphism of degree \(g,\) \(\phi_ g: S^ gC\to J_ g(C),\) from the g-th symmetric product to the Jacobian. We study the structure of the birational morphism \(\phi_ g\). Precisely, we prove the following result:
The birational morphism \(\phi_ g: S^ gC\to J_ g(C)\) is the blowing- up of \(J_ g(C)\) with respect to the prime ideal \({\mathfrak p}\) defining the subvariety \(W^ *_{g-2}\) of special divisors of degree g. - This result allows us to classify all the birational morphisms \(\phi: S^ gC\to J_{g-1}\) such that \(S^ gC\) has a structure of Hilbert scheme of degree \(g\) over C, \(J_{g-1}\) has a structure of Picard scheme of degree \(g-1\) (with \(W_{g-1}\) as natural polarization) and \(\phi\) is the corresponding Abel morphism. This question is closely related with the approaches to the Torelli theorem and Schottky problem performed by the author. Abel morphism; Jacobian; birational morphisms; Torelli theorem; Schottky problem; symmetric product of curve Porras, J. M. Muñoz: On the structure of the birational Abel morphism. Math. ann. 281, 1-6 (1988) Jacobians, Prym varieties, Picard schemes, higher Jacobians, Rational and birational maps, Parametrization (Chow and Hilbert schemes) On the structure of the birational Abel morphism | 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 Given a lattice \(M\), a lattice polytope is a polytope on the real extension \(M_{\mathbb R}\) of \(M\) whose vertices lie in \(M\). A lattice polytope \(\Delta\subset M_{\mathbb R}\) is called reflexive if its dual \(\Delta^* =\{ y\in N_{\mathbb R} : \langle y,x \rangle\geq -1\;\forall x\in M_{\mathbb R}\}\) is a lattice polytope with respect to the lattice \(N\) dual to \(M\).
In the article under review, the authors present an algorithm for the classification of reflexive polytopes in arbitrary dimensions. They also present the results of an application of this algorithm to the case of three dimensional reflexive polytopes.
The study is motivated by the Calabi-Yau compactifications in string theory since the duality of reflexive polytopes corresponds to the mirror symmetry of the resulting Calabi-Yau manifolds. Previously the above mentioned algorithm was used by the authors to reobtain the known 16 two-dimensional reflexive polytopes [Commun. Math. Phys. 185, No. 2, 495-508 (1997; Zbl 0894.14026)]. reflexive polytope; lattice polytope; multiplicities of Picard numbers for reflexive polytopes; multiplicities of point numbers for reflexive polytopes; Newton polytope; mirror symmetry; toric variety; Calabi-Yau manifold; Calabi-Yau compactification Kreuzer, M., Skarke, H.: Classification of Reflexive Polyhedra in Three Dimensions, Advances in Theoretical and Mathematical Physics, 847--864 (1998) Symmetry properties of polytopes, Polyhedral manifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Calabi-Yau manifolds (algebro-geometric aspects), \(3\)-folds Classification of reflexive polyhedra in three dimensions | 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 show that finite fields over which there is a curve of a given genus \(g \geq 1\) with its Jacobian having a small exponent, are very rare. This extends a recent result of Duke in the case of \(g = 1\). We also show that when \(g = 1\) or \(g = 2\), our lower bounds on the exponent, valid for almost all finite fields \(\mathbb F_q\) and all curves over \(\mathbb F_q\), are best possible. Jacobian; group structure; distribution of divisors Curves over finite and local fields, Jacobians, Prym varieties On curves over finite fields with Jacobians of small exponent | 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. Reventós Tarrida is the author resp. co-author of books on (1) plane geometry and linear algebra, (2) axiomatic geometry, (3) projective geometry and (4) affine maps, Euclidean motions and quadrics, all written in Catalonian language. With the present excellently formulated book he turns to the greater English reading mathematical community. The book arose from courses held by the author and his predecessors at the Autonomous University of Barcelona, hence it is, at the one hand, for the advanced undergraduate and beginning graduate students of mathematics, physics, and engineering and, on the other hand, for lecturers of linear algebra. Affine and Euclidean geometry are dealt with as independent applications of linear algebra; apart from two pages in Appendix B, the projective point of view is avoided completely. It is assumed that the reader already knows the basic concepts and results of algebra and linear algebra, in any case the author refers either to the book's appendices or to \textit{F. Cedó} and \textit{A. Reventós} [Geometria plana i àlgebra lineal. Collection Manuals of Autonomous University of Barcelona, Vol. 39 (2004)] written in Catalonian. The clearly arranged book offers modern concepts and notation, a high level of generality, many examples solved in detail, and a great number of exercises.
The text is divided into the preface, 9 chapters, and 4 appendices; at the end of the following short surveys of the chapters and appendices (where the content of the chapters is given -- besides sentences -- by keywords (editorial remark)), \(N_1\) denotes the number of detailed executed examples and \(N_2\) the number of exercises.
Preface. The ideas of Euclid and D. Hilbert are exhibited in a very enlightening way, unfortunately the very first sentence of the book contains a slip because it says: ``About three hundred years before our era, the great Greek geometer Euclid \dots''
Chapter 1. Affine spaces. An affine space is defined as a simply transitive action of the additive group of a \(k\)-vector space (\(k\) is a field) on a set \(\mathbb A\). Thus, the author makes a clear distinction between points and vectors. Dimension. Affine spaces with the same point set, the same action, but different dimensions are given. Linear varieties, affine subspaces, Grassmann formulas, parallel linear varieties. Affine frame and their change, affine coordinates, equations of a linear variety. Barycenter, simple ratio, characterization of a linear segment for \(k=\mathbb R\). The real plane as a complex straight line. Theorem of Thales (= theorem of intersecting lines), its proof is given in the famous Spanish, more exactly, Castillian song `El teorema del Tales' of Le Luthiers. Menelaus' theorem and Ceva's theorem. \(N_1=10\), \(N_2=46\) (among the exercises are the affine versions of Desargues' theorem and of Pappus' theorem).
Chapter 2. Affinities. An affinity is a map between two affine spaces such that the induced map on the corresponding \(k\)-vector spaces is linear. Affine group, isomorphic affine spaces. Affinities and linear varieties. Injective affinities preserve the simple ratio. Equations and compositions of affinities, change of coordinates. Fixed points, invariant varieties. Examples of affinities and their equations, translations, homotheties and their similitude ratio, (axial and mirror) symmetries, projections. Affinities of the line are characterized by the preservation of the simple ratio. If a bijective map between affine spaces of the same dimension \(n\), \(n\geq\,2\), takes collinear points to collinear points, then the map is a semi-affinity except in the case \(k={\mathbb Z}/2{\mathbb Z}\). Example of a bijective map of an affine space over a \(({\mathbb Z}/2{\mathbb Z})\)-vector space taking straight lines to straight lines, but being no semi-affinity. Automorphisms of \(\mathbb Q\), \(\mathbb R\), \(\mathbb C\), and \({\mathbb Z}/p{\mathbb Z}\). \(N_1=2\), \(N_2=26\).
Chapter 3. Classification of affinities. Similar endomorphisms and matrices. Similar affinities and affinely similar matrices. Conditions for homotheties, projections, or symmetries to be similar. The invariance level is the minimum of the dimensions of the invariant linear varieties. Classification of the affinities of the line. Classification of the affinities of the real plane, they are listed, and geometrically interpreted: homologies, elliptic, parabolic, and hyperbolic affinities. Decomposition of the affinities of the real plane. For an example of an affinity of \({\mathbb R}^2\) the canonical matrix is determined in two ways: (1) via eigenvectors (2) via invariant straight lines. \(N_1=1\), \(N_2=21\).
Chapter 4. Classification of affinities in arbitrary dimension. Using the classification of endomorphisms of the vector space the author gives a very general classification of affinities. Jordan matrices, similar endomorphisms and canonical decomposition. Examples of affinities for many different situations that can occur. Classification of affinities of an affine space of dimension \(n\), \(n<\infty\), via Jordan affine frames. Affinities without fixed points. Characterization of similar affinities via their induced endomorphisms and their levels of invariance. \(N_1=N_2=6\).
Chapter 5. Euclidean affine spaces. These are affine spaces whose associated vector spaces are Euclidean. Distance between two points, triangle inequality, Pythagoras' theorem, orthogonal linear varieties, distance between two linear varieties, common perpendicular. \(N_1=2\), \(N_2=19\).
Chapter 6. Euclidean motions. These are the distance preserving affinities; for their study results on isometries of the associated vector spaces are used. Examples of Euclidean motions: translations, symmetries. Similar Euclidean motions. Calculations in coordinates. Glide vector, glide modulus and their computation. Classification of Euclidean motions in arbitrary finite dimension using the canonical expressions of isometries. Invariance of the glide modulus. Two Euclidean motions are similar, iff they have the same glide modulus and their associated endomorphisms have the same characteristic polynomial. \(N_1=0\), \(N_2=13\).
Chapter 7. Euclidean motions of the line, the plane and of space. These are classified without using the results from chapter 6. For the line: translation and symmetry. For the plane: rotation, glide reflection, and translation. For the \(3\)-space: helicoidal Euclidean motion, anti-rotation, and glide reflection. The results are listed in tables and geometrically interpreted. First and second decomposition of a rotation in dimension \(3\) as a product of symmetries. \(N_1=2\), \(N_2=25\) (among these exercises occurs the octahedral group).
Chapter 8. Affine classification of real quadrics. The study of quadrics is reduced to the study of the associated symmetric bilinear maps. Quadratic polynomials and matricial notation. A quadric is the set of points of an affine space of dimension \(n\) whose coordinates, in a given affine frame, are the zeros of a quadratic polynomial in \(n\) variables. Change of affine frames. Image of a quadric under affinity. Equivalent quadratic polynomials, index of a real symmetric matrix, canonical representatives of quadratic polynomials without and with linear part, the classification theorem for quadratic polynomials, the number of equivalence classes of quadratic polynomials, regular zeros of a quadratic polynomial. Affine classification of quadrics, regular point of a quadric, construction of the adapted affine frame. The classification results for the \(2\)- and \(3\)-dimensional cases are listed in tables. Quadrics without regular points, quadrics with center, tangent cone, polar hyperplane. \(N_1=7\), \(N_2=17\).
Chapter 9. Orthogonal classification of quadrics. In essential the spectral theorem from algebra is used. Orthogonally equivalent quadratic polynomials. Criterion for ordering. Canonical representatives of quadratic polynomials with and without linear part. Classification theorem and invariants for quadratic polynomials. The orthogonal classification theorem says: ``Every non-empty quadric of a Euclidean affine space of dimension \(n\), with at least one regular point, is orthogonally equivalent to one and only one of the quadrics given, in a certain orthonormal affine frame, by the following equations
(I) \(\quad x_1^2+d_2x_2^2+\cdots+d_{\rho}d_{\rho}^2=0,\quad\,1\geq\,d_2\geq\cdots\geq\,d_{\rho}\) \quad well ordered
(II/III) \(\quad d_1x_1^2+d_2x_2^2+\cdots+d_{\rho}d_{\rho}^2+1=0,\quad\,d_1\geq\,d_2\geq\cdots\geq\,d_{\rho} \)
(IV) \(\quad d_1x_1^2+d_2x_2^2+\cdots+d_{\rho}d_{\rho}^2+x_{\rho+1}=0,\quad\,d_1\geq\,d_2\geq\cdots\geq\,d_{\rho}\) \quad\, well ordered,
where \(0<\rho\leq\,n\) in the cases (I) and (II/III), and \(0<\rho<n\) in the case (IV).'' The \(2\)- and \(3\)-dimensional cases are listed in tables. Symmetries of a quadric. \(N_1=6\), \(N_2=11\) (among these occur ruled quadrics).
Appendix A. Vector spaces with scalar product. Standard scalar product on \({\mathbb R}^n\), bilinear maps and their matrices, change of basis, symmetric bilinear maps, the radical. Euclidean vector space, the orthogonal group, positive definite matrices, orthogonal vector subspaces. Isometries, polynomials of isometries, the orthogonal groups \(O(2)\) and \(O(3)\), classification of isometries. \(N_1=0\), \(N_2=10\).
Appendix B. Diagonalization of bilinear symmetric maps. The diagonalization theorem, canonical expressions and their uniqueness in the cases \(k={\mathbb C}\), \(k={\mathbb R}\) (Sylvester's theorem), and \(k=\mathrm{GF}(p^N)\), matricial version. The method of completing the squares. Classification of symmetric bilinear maps. Projective classification. \(N_1=2\), \(N_2=8\).
Appendix C. Orthogonal diagonalization. Associated endomorphism, self-adjoint endomorphism, orthogonal diagonalization of symmetric matrices, the spectral theorem, quick calculation of the positivity dimension. \(N_1=2\), \(N_2=6\).
Appendix D. Polynomials with the same zeros. The Nullstellensatz, its quadratic and real quadratic versions. \(N_1=N_2=0\). affine space; linear variety; barycenter; simple ratio; theorem of Thales; theorem of Menelaus; theorem of Ceva; affinity; semi-affinity; affine group; similar affinities; invariance level; Jordan matrix; similar Euclidean motions; glide vector; glide modulus; index of a real symmetric matrix; helicoidal Euclidean motion; anti-rotation; glide reflection Reventós-Tarrida, A.: Affine maps, Euclidean motions and quadrics, Springer undergrad. Math. ser. (2011) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry, Euclidean geometries (general) and generalizations, Affine analytic geometry, Euclidean analytic geometry, Canonical forms, reductions, classification, Quadratic and bilinear forms, inner products, Affine geometry, Forms over real fields Affine maps, Euclidean motions and quadrics | 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 \(I\subseteq k[X_ 1,...,X_ 4]\) be the ideal of the affine cone of a curve in \({\mathbb{P}}_ 3(k)\), where k is an infinite field. Using a generalized form of the Weierstraß preparation theorem [cf. e.g. \textit{H. Grauert}, Invent. Math. 15, 171-198; (1972; Zbl 0237.32011)] the author computes a free resolution of I, which in turn is obviously determined by a finite sequence of matrices over \(k[X_ 1,...,X_ 4]\). The author completely characterizes the matrices arising in this way. With a discussion of some special cases the paper is finished. ideal of the affine cone of a curve; free resolution Amasaki M., Publ. RIMS 19 pp 493-- (1983) Special algebraic curves and curves of low genus, Relevant commutative algebra, Polynomial rings and ideals; rings of integer-valued polynomials, Projective techniques in algebraic geometry Preparatory structure theorem for ideals defining 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 In this article, a noncommutative (regular projective) curve is a category \(\mathcal H\) having the same formal properties as the category \(\mathrm {coh}(X)\) of coherent sheaves on a (regular projective) curve \(X/k\). The formal properties is defined locally by Lenzing-Reiten, and globally by Stafford - van den Bergh: NC1: \(\mathcal H\) is a small, connected, abelian, and each object in \(\mathcal H\) is noetherian. NC2: \(\mathcal H\) is a \(k\)-category with finite dimensional \(\mathrm {Hom}\) and \(\mathrm {Ext}\)-spaces. NC3: There is an autoequivalence \(\tau\) on \(\mathcal H\), the \textit{Auslander-Reiten translation}, such that Serre duality \(\mathrm {Ext}^1_{\mathcal H}(X,Y)=\mathrm {DHom}_{\mathcal H}(Y,\tau X)\) holds, where \(D=\mathrm {Hom}_k(-,k)\). NC4: \(\mathcal H\) contains an object of infinite length.
By Serre duality it follows that \(\mathrm {Ext}^n_{\mathcal H}\) vanishes for all \(n\geq 2\), proving that \(\mathcal H\) is hereditary. \(\mathcal H_0\) denotes the Serre subcategory of \(\mathcal H\) consisting of objects of finite length, \(\mathcal H_+\) objects not containing a simple object. Then every indecomposable object belongs either to \(\mathcal H_+\) or \(\mathcal H_0\), and \(\mathcal H_0=\coprod_{x\in\mathbb X}\mathcal U_x\) for an index set \(\mathbb X\), where \(\mathcal U_x\) are connected, uniserial categories, called \textit{tubes}. By this, it is reasonable to write \(\mathcal H=\mathrm {coh}(\mathbb X)\), and the author add the additional assumption that \(\mathbb X\) consists of infinitely many points. Then \(\mathbb X\) is called a \textit{weighted noncommutative regular projective curve} over \(k\). Because \(\mathbb X\) is assumed to contain infinitely many points, it follows that for each \(x\in\mathbb X\), the number \(p(x)\) of isomorphism classes of simple modules in \(\mathcal U_x\) is finite, and for all but a finite set of \(x\in\mathbb X,\;p(x)=1\). The numbers \(p(x)>1\) are called the weights of \(\mathcal H\), and the corresponding points are called exceptional. A simple simple object \(S\) with \(\mathrm {Ext}^1(S,S)=0\) is called an \textit{exceptional simple sheaf}. An indecomposable object \(L\in\mathcal H\) is called a \textit{line bundle} if it becomes a simple object modulo \(\mathcal H_0\). If, in addition there is upto isomorphism precisely one simple sheaf \(S_x\) concentrated in \(x\) with \(\mathrm {Ext}^1(S_x,L)\neq 0,\) it is called \textit{special}. \(\mathcal H\) is called \textit{non-weighted} (homogeneous) if \(p(x)=1\) for all \(x\), which is equivalent to \(\mathrm {Ext}^1(S,S)\neq 0\) for each simple object \(S\).
The author proves that one can reduce to the non-weighted case: Let \(\mathcal H\) be a weighted noncommuative regular projective curve with the exceptional points given by \(x_1,\dots,x_t,\) with \(p(x_i)>1\). Choose for every \(i=1,\dots,t\) a simple sheaf \(S_i\) concentrated in \(x_i\), and let \(\mathscr{S}\) be the system \(\{\tau^i S_i|i=1,\dots,t;j=1,\dots,p_i-1\}.\) Then the right perpendicular category \(\mathcal H_{nw}=\mathscr{S}^\perp\subseteq\mathcal H\) is a full exact subcategory of \(\mathcal H\) which is a non-weighted noncommutative regular projective curve, and there is a special line bundle \(L\) in\(\mathcal H\).
Each weighted noncommutative regular projective curve \(\mathcal H\) over \(k\) is obtained from a non-weighted noncommutative regular projective curve \(\mathcal H_{nw}\) over \(k\) by insertion of weights into a finite number of points of \(\mathcal H_{nw}\). The authors always consider a pair \((\mathcal H,L)\), \(L\) a special line bundle considered as the structure sheaf. The quotient category \(\tilde H=\mathcal H/\mathcal H_0\) is semisimple with one simple object given by the class \(\tilde L\) pf \(L\) so that \(\tilde H=\mathrm {mod}(k(\mathcal{H}))\) for the skew field \(k(\mathcal H)=\mathrm {End}_{\tilde{\mathcal H}}(\tilde L)\), the \textit{function field}. Also, \(\mathcal H/\mathcal H_0\simeq\mathcal H_{nw}/(\mathcal H_{nw})_0\) implying that \(k(\mathcal H)\simeq k({\mathcal H}_{nw})\). The author proves that if \(\mathcal H\) is non-weighted, then it is uniquely determined by its function field.
The global skewness of \(\mathcal H\) is the number \(s(\mathcal H)=[k(\mathcal H):Z(k(\mathcal H))]^{1/2}\), and \(Z(k(\mathcal H))\simeq k(X)\) for a unique regular projective curve over \(k\), called the centre curve of \(\mathcal H\). In the main part of the text, \(\mathcal H\) is a noncommutative, non-weighted, regular projective curve over a perfect field \(k\), and \(S_x\) denotes the unique simple sheaf concentrated in \(x\).
The aim of the article is to give a detailed introduction, with examples, to noncommutative curves by the approach given with basis in the Auslander-Reiten translation \(\tau\) which is a global datum of the category \(\mathcal H\). The local properties of \(\tau\) is studied by looking into the explicit structure of the tubes \(\mathcal U_x\). The Auslander-Reiten translation \(\tau\) acts on each \(\mathcal U_x\) which is a hereditary category with with Serre duality, and is a basic, non-trivial example of a connected uniserial length category. Such categories where classified by their species by Gabriel. In the case of a homogeneous tube with one simple object \(S\), this species is the \(D-D\)-bimodule \(\mathrm {Ext}^1(S,S)\), \(D=\mathrm {End}(S)\), and these are classified explicitly. This determines the complete local rings as certain twisted power series rings. The core of the main results is stated verbatim as
Theorem. For each point \(x\in\mathbb X\) the full subcategory \(\mathcal U_x\) of skyskraper sheaves concentrated in \(x\) is equivalent to the category of finite length modules over the skew power series ring \(\mathrm {End}(S_x)[[T,\tau^-]]\). Here the twist \(\tau^-\), with \(Tf=\tau^-(f)T\) for all \(f\in\mathrm {End}(S_x)\), is given by the restriction of the inverse Auslander-Reiten translation \(\tau^-;\mathcal H\rightarrow\mathcal H\) to the simple object \(S_x\) concentrated in \(x\).
From this result, the restriction of \(\tau\) to \(\mathcal U_x\) is of order \(e_{\tau}(x)\), the \(\tau\) multiplicity in \(x\). The author study this multiplicity and proves that it has reasonable properties.
From the essential fact that each noncommutative regular projective curve is uniquely determined by its function field, many known results from the theory of orders follows. In particular, the \(\tau\)-multiplicities are just the ramification indices of \(\mathcal A\), the sheaf of \(\mathcal O_X\)-orders.
The author review facts on different and dualizing sheaves which follows after proving that \(\tau\in\mathrm {Pic}(\mathcal H)\). The author shows that \(\mathrm {Pic}(\mathcal H)\) is determined by \(\mathrm {Pic}(X)\), the Picard group over the centre curve \(X\).
The author defines the Euler characteristic and genus of a noncommutative regular projective curve, and proves that it becomes a Morita equivalence. Also, the elliptic case is studied as a particular case. Motivated by the representation theory of finite dimensional algebras, as characterized by admitting tilting objects, a detailed treatment of the genus \(0\) case is given. The main focus is on the ghost group \(\mathcal G(\mathcal H)\), the subgroup of \(\mathrm {Aut}(\mathcal H)\) given by those automorphisms fixing the structure sheaf \(L\) and all simple sheaves \(S_x,\;(x\in\mathbb X)\). The ghost group can be seen as a measure of the failure from the ground field to be algebraically closed, and is then used to study categories of finite dimensional modules.
The above results make possible the study of noncommutative regular projective curves over \(\mathbb R\), which also specializes to the classification of all genus zero and genus one Witt curves.
This is a very extensive article, and it should be mentioned that it also treats tubular curves, the Klein bottle, Fourier-Mukai partners, and that it gives formulas for the normalized orbifold Euler characteristic.
The treatment of noncommutative curves as schemes with coordinate rings and determined by their inclusions in the function field is very algebraic, and the article is a very nice entrance to noncommutative geometry. noncommutative regular projective curve; noncommutative function field; Auslander-Reiten translation; Picard-shift; ghost group; maximal order over a scheme; ramification; Witt curve; noncommutative elliptic curve; Klein bottle; Fourier-Mukai partner; weighted curve; orbifold Euler characteristic; noncommutative orbifold; tubular curve; finite dimensional algebra; Beilinson theorem Kussin, Dirk, Weighted noncommutative regular projective curves, J. Noncommut. Geom., 10, 4, 1465-1540, (2016) Noncommutative algebraic geometry, Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Abelian categories, Grothendieck categories, Elliptic curves, Orders in separable algebras, Klein surfaces Weighted noncommutative regular projective 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 \(Z\) be a real form of \(\mathbb {CP}^n\). Hence either \(Z = \mathbb {RP}^n\) or \(n\) is odd and \(Z\) has no real point. Here the authors classify all algebraic vector bundles on \(Z\) which over \(\mathbb {C}\) are isomorphic to a direct sum of line bundles: their indecomposable factors have rank one or two. real algebraic variety; split vector bundle; real form of a projective space Biswas, I.; Nagaraj, D. S.: Absolutely split real algebraic vector bundles over a real form of projective space, Bull. sci. Math. 131, 686-696 (2007) Real algebraic and real-analytic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Absolutely split real algebraic vector bundles over a real form of 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 In 1973, \textit{V.~M.~Galkin} [Math. USSR, Izv. 7 (1973), 1-17 (1974); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 37, 3-19 (1973; Zbl 0261.12009)] defined a zeta-function for local rings \(\mathfrak O\) of possibly singular complete geometrically irreducible curves over finite fields. His zeta-functions are defined on all of \(\mathbb C\), and if \(\mathfrak O\) is Gorenstein then the associated zeta-function satisfies a certain functional equation. In 1989, \textit{B.~Green} [Manuscr. Math. 64, No.~4, 485-502 (1989; Zbl 0723.11058)] defined zeta-functions for the same type of local rings \(\mathfrak O\). Green's zeta-functions depend on the choice of a dualizing module for~\(\mathfrak O\), but they satisfy a functional equation even when \(\mathfrak O\) is not Gorenstein. In the paper under review, the author defines a new zeta-function for local rings \(\mathfrak O\) as above by the formula
\[
\zeta({\mathfrak O},s) = \sum_{I\supseteq{\mathcal O}} \#(I/{\mathfrak O})^{-s} \text{\quad for \quad } \Re(s) > 0,
\]
where the sum is over all fractional ideals of \(\mathfrak O\) that contain \(\mathfrak O\); this is in contrast to Galkin's zeta-function, where the corresponding sum is over nonzero ideals contained in~\(\mathfrak O\). The author shows that this zeta-function may be extended to the entire complex plane, that it satisfies Green's functional equation, and that it is equal to Galkin's zeta-function precisely when \(\mathfrak O\) is Gorenstein.
The author also defines global zeta-functions for possibly-singular complete geometrically irreducible curves \(X\) over finite fields. His global zeta-functions satisfy a functional equation, and they agree with the usual global zeta-functions when \(X\) is a Gorenstein curve, but they do not necessarily satisfy the Riemann hypothesis. The author's zeta-function and the usual zeta-function share the same residue at~\(s = 0\); this residue determines the number of rational points on the compactified Jacobian of~\(X\). zeta function of Gorenstein ring; singular curve; Gorenstein curve; functional equation; number of rational points on the compactified Jacobian; Riemann hypotheses Karl-Otto Stöhr, Local and global zeta-functions of singular algebraic curves, J. Number Theory 71 (1998), no. 2, 172 -- 202. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Other Dirichlet series and zeta functions Local and global zeta-functions of singular 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 We show that there is a natural decomposition
\[
\text{Pic}(A[t,t^{- 1}]) \cong \text{Pic}(A) \bigoplus N \text{Pic}(A) \bigoplus N \text{Pic}(A) \bigoplus H^1(A)
\]
for any commutative ring \(A\), where \(\text{Pic}(A)\) is the Picard group of invertible \(A\)-modules, and \(H^1(A)\) is the étale cohomology group \(H^ 1(\text{Spec} (A), \mathbb{Z})\). A similar decomposition of \(\text{Pic} (X[t,t^{-1}])\) holds for any scheme \(X\). This makes Pic a ``contracted functor'' in the sense of \textit{H. Bass} [cf. ``Algebraic \(K\)-theory'' (1968; Zbl 0174.30302); p. 670]. \(H^1(A)\) is always a torsionfree group, and is zero if \(A\) is normal. For pseudo-geometric rings, \(H^1(A)\) is an effectively computable, finitely generated free abelian group. We also show that \(H^1(A[t,t^{-1}]) \cong H^ 1(A)\), i.e., \(NH^1=LH^1=0\). This yields the formula for group rings:
\[
\text{Pic} \bigl (A[t_ 1,t_1^{-1}, \dots, t_m, t_m^{-1}] \bigr) \cong \text{Pic}(A) \bigoplus \coprod^ m_{i=1} H^1(A) \bigoplus \coprod^ m_{k=1} \coprod^{2^ k \binom{m}{k}}_{i=1} N^k \text{Pic}(A). \]
[See also the author's paper in C. R. Acad. Sci., Paris, Sér. I 310, No. 2, 57--59 (1990; Zbl 0695.14010)]. Pic is a contracted functor; algebraic \(K\)-theory; Picard group of the Laurent polynomial ring Weibel C.: Pic is a contracted functor. Invent. Math. 103, 351--377 (1991) Grothendieck groups, \(K\)-theory and commutative rings, Picard groups, Computations of higher \(K\)-theory of rings, Formal power series rings Pic is a contracted functor | 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 Authors' abstract: ``We extend a result of [\textit{W. Fulton} et al., Proc. Amer. Math. Soc. 92, No. 3, 320--322 (1984; Zbl 0549.14004)] to secant loci in symmetric products of curves. We compare three secant loci and prove that the dimensions of bigger loci can not be excessively larger than the dimension of smaller loci.'' These results seems to be very useful to extend their paper [\textit{M. Aprodu} and \textit{E. Sernesi}, Algebra Number Theory 9, No. 3, 585--600 (2015; Zbl 1320.14067)] and they were applied to the syzygies of curves in [\textit{M. Kemeny}, ``The extremal secant conjecture for curves of arbitrary gonality'', Preprint \url{arXiv:1512.00212}]. curves; excess linear series; secant bundle; secant space to a curve; symmetric product of a curve \textsc{M. Aprodu and E. Sernesi,} Excess dimension for secant loci in symmetric products of curves, E. Collect. Math. (2016). 10.1007/s13348-016-0166-2. Special divisors on curves (gonality, Brill-Noether theory), Plane and space curves, Determinantal varieties Excess dimension for secant loci in symmetric products 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 A submonoid of non-negative integers is called a numerical semigroup if its complement is a finite set. The cardinality of the complement is called the genus. A numerical semigroup obtained by an algebraic curve and its point is called Weierstrass semigroup. A numerical semigroup is said to be of double covering type if it is obtained by an algebraic curve which is double covering of a curve and its ramification point. Numerical semigroups of double covering type obtained by double covering of curves of genus 0 and 1 were determined completely. In this paper, the authors determined numerical semigroups of double covering type obtained by double covering of curves of genus 2. numerical semigroup; Weierstrass semigroup; double cover of a curve; curve of genus two Harui, T., Komeda, J., Ohbuchi, A.: The Weierstrass semigroups on double covers of genus two curves, preprint Riemann surfaces; Weierstrass points; gap sequences, Special algebraic curves and curves of low genus, Commutative semigroups The Weierstrass semigroups on double covers of genus two 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 \(E\) be a rank \(n\) vector bundle on a smooth projective curve \(X\). It is known that \(E\) may be obtained from a splitted bundle \(\bigoplus_{1\leq i\leq n} L_i\), \(\text{rank} (L_i)=1\), by a finite number of elementary transformations. Here we give upper bounds for their minimal number. If \(n=2\) this is related to the order of stability of \(E\). splitting of vector bundles; vector bundle on a smooth projective curve Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Splitting of vector bundles on algebraic curves and elementary transformations | 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 curve over a global field \(k\). In this paper we study conjectures of Bloch and Vaserstein [\textit{L. N. Vaserstein} [in: Quadratic and Hermitian forms, Conf. Hamilton/Ont. 1983, CMS Conf. Proc. 4, 131--140 (1984; Zbl 0577.20033)] and \textit{S. Bloch} [Ann. Math. (2) 114, 229--265 (1981; Zbl 0512.14009)] on the structure of the group \(K_1(X)\). When \(X=\mathrm{Spec}(A)\) is affine (not necessarily regular) Vaserstein's conjecture says that the group \(SK_1(A)=: \operatorname{Ker}[K_1(A)\to A^*]\) is a torsion group. More concretely, this conjecture says that given a \(3\times 3\) matrix \(M\) of determinant 1 with coefficients in \(A\), some power of \(M\) is a product of elementary matrices. Bloch's conjecture predicts that if \(X\) is smooth and projective then the group \(V(X)=: \operatorname{Ker}[SK_1(X)\to k^*]\) is a torsion group.
In the appendix to this paper (p. 191--193), \textit{C. Weibel} shows that the conjectures of Bloch and Vaserstein are equivalent. This is proved in the paper when \(A\) is regular. This being the case, we focus on the case of a smooth, projective, geometrically connected curve \(X\) over a global field \(k\).
The main result of this paper is that when \(k\) is a number field then \(V(X)\) is an extension of a uniquely divisible group by a torsion group. The uniquely divisible part is the set of Galois invariants \(V(\bar X)^G\), where \(\bar k\) is an algebraic closure of \(k\), \(\bar X=X\times_k\bar k\) and \(G=\mathrm{Gal}(\bar k/k)\). Thus either \(V(\bar X)^G=0\) and Bloch's conjecture is true or the conjecture fails badly. The proof of the main theorem uses Saito's local class field theory of curves [\textit{S. Saito}, J. Number Theory 21, 44--80 (1985; Zbl 0599.14008)] and a theorem of Jannsen on the Galois cohomology of number fields [\textit{U. Jannsen}, in: Galois groups over \(\mathbb{Q}\), Proc. Workshop, Berkeley/CA 1987, Publ., Math. Sci. Res. Inst. 16, 315--359 (1989; Zbl 0703.14010)]. It is surprising that such an innocent statement has a proof which is not very innocent.
For a smooth, projective curve \(X\) over a global field of finite characteristic, one can easily show that \(V(X)\) is either a torsion group or it is quite large. Because we know so much more about \(K_1\) of surfaces over finite fields than arithmetic surfaces, one can prove the conjecture in some cases using results of \textit{C. Soulé} [Math. Ann. 268, 317-345 (1984; Zbl 0573.14001)]. curve over a global field; \(K_1\); local class field theory of curves; Galois cohomology; surfaces over finite fields Wayne Raskind, On \?\(_{1}\) of curves over global fields, Math. Ann. 288 (1990), no. 2, 179 -- 193. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Arithmetic ground fields for curves, Global ground fields in algebraic geometry, \(K\)-theory in geometry On \(K_1\) of curves over global fields. Appendix by C. Weibel | 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,x) be a germ of a normal complex surface singularity. In this note the author studies the set of isomorphism classes of vector bundles on the punctured spectrum \(X\setminus \{x\}\). These isomorphism classes are in a natural bijection with isomorphism classes of reflexive modules over the local ring \({\mathcal O}_{X,x}.\)
The first two chapters are a survey of the main facts about normal surface singularities, reflexive modules over the local ring of such a singularity and the Auslander-Reiten theory for modules.
The next chapters are the main technical kernel of this work. Let \(\pi\) : \(\tilde X\to X\) be a resolution of the singularity with exceptional fiber E and let Z be an effective divisor with support E. For any coherent reflexive sheaf M on X denote by \(R_ Z(M)=(\pi^*M)^{\vee}|_ Z\) (i.e. the restriction to Z of the bidual of \(\pi^*M)\). This is a locally free sheaf on Z. For a proper choice of the cycle Z (the so called reduction cycle), \(R_ Z\) defines an injective map from the set of isomorphism classes of reflexive \({\mathcal O}_{X,x}\)-modules to the set of isomorphism classes of locally free \({\mathcal O}_ Z\)-modules. The image of this map is characterized and explicitly described in the case of the rational singularities and minimal-elliptic singularities. In the case of the simply-elliptic singularites, when \(\pi\) is the minimal resolution, the exceptional curve E is a smooth irreducible elliptic curve and coincides with the reduction cycle. A complete characterization of the isomorphism classes of indecomposable reflexive modules over \({\mathcal O}_{X,x}\) is obtained using Atiyah's classification of the indecomposable vector bundles on an elliptic curve.
In chapter 6 the author proves that each vector bundle over the punctured spectrum of a simply-elliptic singularity is associated to a finite dimensional representation of the local fundamental group, which is a discrete Heisenberg group. A complete description of those representations which correspond to irreducible vector bundles is achieved.
The last chapter uses the above results for the study of the torsion free modules over plane curve singularities of type \(E_ 6\) and \(E_ 7\). Auslander-Reiten quiver; germ of a normal complex surface singularity; isomorphism classes of vector bundles; reflexive modules; reduction cycle; rational singularities; minimal-elliptic singularities; simply- elliptic singularites; torsion free modules over plane curve singularities Kahn, Reflexive Moduln auf einfach-elliptischen Flächensingularitäten, Dissertation (1988) Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Representation theory of associative rings and algebras Reflexive Moduln auf einfach-elliptischen Flächensingularitäten. (Reflexive modules on simply elliptic surface singularities) | 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 0723.14004. smooth irreducible curve on a Del~Pezzo surface; linear system; Clifford index; syzygies of curves; Kodaira divisor; gonality Pareschi G. (1991). Exceptional linear systems on curves on Del Pezzo surfaces. Math. Ann. 291: 17--38 Divisors, linear systems, invertible sheaves, Fano varieties, Vector bundles on curves and their moduli Exceptional linear systems on curves on Del Pezzo 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 Dans la première partie de cet article on va donner un algorithme pour trouver les équations d'une variété torique (affine ou projective) et en la deuxième partie nous allons définir les variétés projectivement toriques et on demontrera que dans certains conditions ces variétés sont toriques. projectively toric varieties; equations of a toric variety Toric varieties, Newton polyhedra, Okounkov bodies, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The ideal of a toric variety and the projectively toric 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 \(Y\) be an irreducible and smooth curve of genus \(g\geq 4\). Denote \(W_ n\) the image of the n-th symmetric product of Y in the Jacobian J(Y) under the Abel-Jacobi map \(\mu_ n.\) \(Y\) is called elliptic-hyperelliptic (e. h.) if there exists a degree two morphism \(Y\to E\) onto an elliptic curve. Looking for a characterization of an e. h. curve from the point of view of \(W_ n(\text{some }n)\) one is led soon to the following: if Y is nonhyperelliptic then it is e. h. if and only if there exists an irreducible curve
\[
Z\quad \subset \quad \{\mu_ 2(P_ 1+P_ 2)\in W_ 2| \quad h^{\bullet}({\mathcal O}_ Y(2P_ 1+2P_ 2))=2\} \subset W_ 2.
\]
The aim of the paper is to give a characterization of an e. h. curve in terms of a differential geometric property of the embedding \(W_ 2\hookrightarrow J(Y)\) along \(Z\). If we call asymptotic direction of \(W_ 2\) at a point \(x\) the germ of a ''line'' \(L\subset J(Y)\) which intersects \(W_ 2\) at \(x\) with (at least) third order, then we prove: If Y is nonhyperelliptic the following are equivalent:
(i) \(W_ 2\) has two asymptotic directions at \(x=\mu_ 2(P+Q);\)
(ii) \(h^{\bullet}({\mathcal O}_ Y(2P+2Q))=2\). elliptic-hyperelliptic curves; n-th symmetric product of curve; Jacobian; asymptotic direction Special algebraic curves and curves of low genus, Jacobians, Prym varieties, Elliptic curves On a characterization of elliptic-hyperelliptic 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 deals with polynomials of the form
\[
p(x,y)=ax^n+by^m+\sum_{im+jn \leq mn}c_{ij}x^iy^j
\]
\(a,\;b,\;c_{ij} \in K\), a field of characteristic 0, and in case \(K=\mathbb{C}\), the field of complex numbers, with their 0-loci in \({\mathbb{C}^2}\). The Newton polygon of these polynomials are triangles or line segments. In theorem 1.1 the authors give conditions on the coefficients so that two of these polynomials are not equivalent under automorphisms of \(K[x,y]\).
In this paper the problem of finding a canonical model of a given polynomial, is arranged by considering certain elementary transformations of \(K[t] \times K[t]\) and by Whitehead's idea of \textit{peak reduction}. An application, on polynomials whose 0-loci admit a one-variable parametrization, is discussed in proposition 1.2 and in theorem 1.3, where in particular, using a result of \textit{V. Ya. Lin} and \textit{M. G. Zaidenberg} [in: Voronezh Winter Math. Sch., Transl., Ser. 2, Am. Math. Soc. 184(37), 111-130 (1998; Zbl 0955.32013)], irreducible simply connected curves are related to certain polynomials of the above form.
Another application concerns the number of inequivalent embeddings of algebraic curves (with a place at infinity) in \({\mathbb{C}^2}\) (here equivalent means: under automorphisms of \({\mathbb{C}^2}\)). By a result of \textit{S. S. Abhyankar} and \textit{B. Singh} [Am. J. Math. 100, 99-175 (1978; Zbl 0383.14007)]\ an irreducible algebraic curve with a place at infinity cannot have infinitely many inequivalent embeddings in \({\mathbb{C}^2}\), while the first example of such a curve with at least two inequivalent embeddings in \({\mathbb{C}^2}\) is contained in a paper by \textit{S. S. Abhyankar} and \textit{A. Sathaye} [Proc. Am. Math. Soc. 124, No. 4, 1061-1069 (1996; Zbl 0880.14012)]. The authors prove that for any \(k \geq 2\) there is an irreducible algebraic curve (with a place at infinity), which has at least \(k\) inequivalent embeddings in \({\mathbb{C}^2}\), i.e. there are arbitrary (but finitely) many isomorphic algebraic curves in \({\mathbb{C}^2}\), belonging to different orbits under the action of \(\Aut({\mathbb{C}^2})\). canonical model of a given polynomial; peack reduction; irreducible algebraic curve; number of embeddings; automorphism Shpilrain V, Yu J-T. Embeddings of curves in the plane. J Algebra, 1999, 217: 668--678 Singularities of curves, local rings, Embeddings in algebraic geometry, Plane and space curves, Polynomials in real and complex fields: location of zeros (algebraic theorems), Special algebraic curves and curves of low genus Embeddings of curves in the plane | 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 \textit{R. Pellikaan} [IEEE Trans. Inform. Theory 35, 369-381 (1989; Zbl 0694.94015)] has given a noneffective maximal decoding algorithm of a geometric code. To this end, our purpose is the determination of the minimal integer \(s\), such that the maps \(\Psi^s_{g-k} (k=1,2,)\), defined in Pellikaan's paper, are surjective. Then, on the one hand, we show that the theta divisor of the Jacobian variety of an algebraic curve provides partial answers. On the other hand, for the Klein quartic defined over \(\mathbb{F}_8\), we determine explicitly divisors of degree 8 which allows us to decode up to 5 errors. decoding algorithm; geometric code; theta divisor; Jacobian variety; algebraic curve; Klein quartic Geometric methods (including applications of algebraic geometry) applied to coding theory, Decoding, Curves in algebraic geometry The theta divisor of a Jacobian variety and the decoding of geometric Goppa codes | 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 deals with singularities of genus \(2\) curves on a general \((d_1; d_2)\)-polarized abelian surface \((S; L)\). In analogy with Chen's results concerning rational curves on \(K3\) surfaces [\textit{X. Chen}, J. Algebr. Geom. 8, No. 2, 245--278 (1999; Zbl 0940.14024); Math. Ann. 324, No. 1, 71--104 (2002; Zbl 1039.14019)], it is natural to ask whether all such curves are nodal. We prove that this holds true if and only if \(d_2\) is not divisible by \(4\). In the cases where \(d_2\) is a multiple of \(4\), we exhibit genus \(2\) curves in \(|L|\) that have a triple, \(4\)-tuple or \(6\)-tuple point. We show that these are the only possible types of unnodal singularities of a genus \(2\) curve in \(|L|\). Furthermore, with no assumption on \(d_1\) and \(d_2\), we prove the existence of at least one nodal genus \(2\) curve in \(|L|\). As a corollary, we obtain nonemptiness of all Severi varieties on general abelian surfaces and hence generalize [\textit{A. L. Knutsen} et al., J. Reine Angew. Math. 749, 161--200 (2019; Zbl 1439.14021), Thm. 1.1] to nonprimitive polarizations. abelian surface; singluarities of curves; curves of low genus; isogeny; Jacobian variety; nonprimitive polarization Subvarieties of abelian varieties, Singularities of curves, local rings, Special algebraic curves and curves of low genus, Isogeny, Jacobians, Prym varieties Genus two curves on abelian 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 Given a proper smooth curve \(X\) over a complete discrete valuation field \(K\) with characteristic zero. Suppose \(X\) has a semistable model over the valuation ring of \(K\). In this paper, the authors discuss these three categories over \(X\) in a unified manner, the Kummer étale category, the de Rham category, and the crystalline category. For the Tannakian categories, they study the corresponding fundamental groups, particularly, the unipotent \(p\) -adic étale fundamental group \(G^{\text{ét}}\) and the unipotent de Rham fundamental group \(G^{\mathrm{dR}}\) of \(X\), respectively.
Here in the paper, \(G^{\text{ét}}\) is said to be a crystalline fundamental group if \(G^{\text{ét}}\) and \(G^{\mathrm{dR}}\) respectively tensor the Fontaine's rings \( B_{\mathrm{crys},K}\) and \(B_{\mathrm{st},K}\) are \(G\left( \bar{K}/K\right) \)-equivariant isomorphic group schemes. Then, the authors obtain the main theorem of the paper, i.e., a criterion for good reduction of curves: The curve \(X\) has good reduction if and only if the fundamental group \(G^{\text{ét}}\) is crystalline. This result is a generalization of many known results on good reduction. good reduction of a curve; Fontaine's rings; crystalline site; unipotent fundamental group F.~Andreatta, A.~Iovita, and M.~Kim, \emph{A \(p\)-adic non-abelian criterion for good reduction of curves}, Duke Math. J. \textbf{164} (2015), no.~13, 2597--2642. DOI 10.1215/00127094-3146817; zbl 1347.11051; MR3405595 Curves over finite and local fields, \(p\)-adic cohomology, crystalline cohomology, Rigid analytic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) A \(p\)-adic nonabelian criterion for good reduction 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 [For the entire collection see Zbl 0632.00004.]
This note is a continuation of the previous work of the author [Invent. Math. 75, 1-8 (1984; Zbl 0616.14007)] and is concerned with the application of the following results of this work: If a finite group G is a quotient of the algebraic fundamental group of a complete irreducible non-singular algebraic curve X of genus g over an algebraically closed field k (i.e. the Galois group of the maximal unramified extension of the function field \(k(X)^{ur}/k(X))\) then the minimum number t(G) of the generators of the augmentation ideal of the group ring k[G] does not exceed g.
Here the author describes several propositions on calculation of t(G) which result in a restriction on the quotients of the algebraic fundamental group of a curve which allows one to give examples of finite groups which cannot appear as such quotients. An alternative direct proof of this proposition is given. So, as the author points out, the strength of the quoted theorem is unclear as he could not find examples of finite groups ruled out by this theorems as quotients of the algebraic fundamental groups of curves and which cannot be ruled out by other means. algebraic fundamental group of a complete irreducible non-singular algebraic curve Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group On the quotients of the fundamental group 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 The paper under review takes up a question formulated by \textit{G. Frey} and \textit{M. Jarden} [Proc. Lond. Math. Soc. (3) 28, 112-128 (1974; Zbl 0275.14021)]: If \(A\) is an abelian variety of positive dimension over some number field \(k\), does \(A\) acquire infinite Mordell-Weil rank over the maximal abelian extension \(k^{ab}\) of \(k\)?
Frey and Jarden showed that the answer is yes for elliptic curves, even with the maximal elementary abelian 2-extension instead of \(k^{ab}\). This was generalized to Jacobians of certain cyclic covers of the projective line having a \(k\)-rational Weierstrass point by \textit{H. Imai} [Kodai Math. J. 3, 56-58 (1980; Zbl 0439.14003)], \textit{J. Top} [Tôhoku Math. J. (2) 40, 613-616 (1988; Zbl 0688.14027)], and \textit{N. Murabayashi} [Acta Arith. 64, 297-302 (1993; Zbl 0785.14011)].
The authors now extend these result by proving the following theorem.
Let \(C \to {\mathbb P}^1\) be a geometrically irreducible cyclic cover of degree \(n\), defined over \(k\), such that \(C\) has positive genus. Then there exists a divisor \(d\) of \(n\), which can be taken either as a power of 2 or as an odd prime such that the Jacobian of \(C\) has infinite rank over the maximal abelian extension of \(k\) of exponent \(d\).
This implies a positive answer to Frey and Jarden's question for absolutely simple principally polarized abelian surfaces.
In a similar vein, the authors obtain a result on general abelian varieties as follows. If \(A\) is a \(d\)-dimensional abelian variety over \(k\) (\(d > 0\)) having a degree-\(n\) projective embedding over \(k\), then \(A\) acquires infinite rank over the compositum of all extensions of \(k\) of degree \(< n(4d+2)\). This result is obtained by studying ramification properties of division points on \(A\). abelian variety; Jacobian variety; Mordell-Weil rank; cyclic cover of the projective line; infinite field extension; idempotent relation DOI: 10.1006/jnth.2001.2692 Abelian varieties of dimension \(> 1\), Jacobians, Prym varieties, Arithmetic ground fields for abelian varieties, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rational points, Global ground fields in algebraic geometry The rank of abelian varieties over infinite Galois extensions | 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, projective non-singular algebraic curves of genus \(3\) over an algebraically closed field of characteristic \(2\) are studied. The techniques of \textit{K.- O. Stöhr} and \textit{J. F. Voloch} [J. Reine Angew. Math. 377, 49--64 (1987; Zbl 0605.14023)] are applied to analyze the order sequence and weight of the Weierstrass points. The locus of the curves whose canonical theta characteristic is totally supported at one point is seen to have dimension \(4\). The locus of those curves whose canonical theta characteristic is represented by a positive divisor supported at one point having two Weierstrass directions towards it, is seen to have dimension \(2\). curve of genus three: moduli space; Weierstrass point; Cartier operator Special algebraic curves and curves of low genus, Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Theta functions and curves; Schottky problem The moduli of certain curves of genus three in characteristic two | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.