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Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\pi: X\to T\) be a proper flat morphism of complex algebraic varieties and \(L\) a line bundle on \(X\) with \(X\) and \(T\) separated, \(T\) irreducible and \(X\) pure dimensional. Suppose that \(k=\dim(T)\) and \(d\) is the relative dimension of \(\pi\). Consider the divisor class \({\mathcal E}(L,F)=\pi_*((rc_ 1(L)-\pi^*c_ 1(F))^{d+1}\cap [X]\in A_{k+1}(T)\) for a coherent subsheaf \(F\) of \(\pi_*(L)\). Suppose that the following conditions hold:
(i) If \(t\) is a general point of \(T\), then \(F_ t\otimes C\subset H^ 0(\pi^{-1}(t)\), \(L_{(\pi^{-1}(t))})\) is base point free, very ample and yields a semi-stable embedding of \(\pi^{-1}(t)\).
(ii) \(L\) is relatively ample.
Then, the main result shows
(1) that \({\mathcal E}(L,F)\) lies in the closure of the cone in the \(A_{k- 1}(T)\otimes \mathbb Q\) generated by the effective Weil divisors.
(2) if \(F\) is locally free, \({\mathcal E}(L,F)\) lies in the closure of the cone generated by the effective Cartier divisors.
A counter example due to Ian Morrison shows that the semi-stability condition is essential.
As a significant application of the above result the authors show: Let \(\overline M_ g\) be the moduli space of stable genus \(g\) curves with \(g\geq 2\) and \(\lambda,\delta \in \text{Pic}(\overline M_ g)\otimes\mathbb Q\) be the Hodge and boundary class. Then the class \(a\lambda-b\delta\) has non-negative degree on every curve in \(\overline M_ g\) not contained in \(\Delta =\overline M_ g- M_ g\) if and only if \(a\geq (8+4/g)b\). Furthermore, \(a\lambda-b\delta\) is ample if and only if \(a>11.b>0\). These results are known in part. Hodge class; divisor class; effective Weil divisors; effective Cartier divisors; moduli space of stable genus g curves; boundary class Cornalba, M; Harris, J, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. École Norm. Sup. (4), 21, 455-475, (1988) Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves Divisor classes associated to families of stable varieties with applications to the moduli space of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies the Chow motive of moduli spaces of stable vector bundles over a smooth projective curve. In particular, he proves that this motive lies in the category generated by the curve, and he computes its class in the Grothendieck ring of the category of motives. These results rely on geometric constructions due to \textit{E. Bifet, F. Ghione} and \textit{M. Letizia} [Math. Ann.~299, 641--672 (1994; Zbl 0840.14003)], and on the author's computation of the Chow motive of a smooth projective variety acted on by the multiplicative group \({\mathbb{G}}_m\) which in turn relies on the work of \textit{A. Białynicki-Birula [Ann. Math.~(2)~98, 480--497 (1973; Zbl 0275.14007)], and which may be considered as a generalization of the reviewer's earlier result \textit{B. Köck} [Manuscr. Math.~70, 363--372 (1991; Zbl 0735.14001)].
As applications the author computes the Poincaré-Hodge polynomial and the number of points over a finite field of these moduli spaces. In particular, he recovers a result of \textit{M. S. Narasimhan} and \textit{S. Ramanan} on the \(\chi_y\)-genus [J. Indian Math. Soc., New Ser.~39, 1--19 (1975; Zbl 0422.14018)].
Finally the author studies some well-known conjectures on algebraic cycles on these moduli spaces, namely the Hodge, Tate and standard conjectures. For instance, he proves that the standard conjecture of Lefschetz type B holds for these moduli spaces, a result which, over the field of complex numbers, has previously been proved by \textit{I. Biswas} and \textit{M. S. Narasimhan} [J. Algebr. Geom.~6, 697--715 (1997; Zbl 0891.14002)].} Chow motive; moduli space; stable vector bundles; Poincaré-Hodge polynomial; symmetric power of a motive; \(\lambda\)-structure on a tensor category; MacDonald theorem; varieties of matrix divisors; standard conjecture of Lefschetz type; semisimplicity of Galois actions; Hodge conjecture; Tate conjecture S. del Baño, \textit{On the Chow motive of some moduli spaces}, J. Reine Angew. Math. \textbf{532} (2001), 105-132. Motivic cohomology; motivic homotopy theory, Algebraic moduli problems, moduli of vector bundles, (Equivariant) Chow groups and rings; motives, Vector bundles on curves and their moduli On the Chow motive of some moduli 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 It is shown that the homology of the spin moduli spaces \({\mathcal M}_ g[\epsilon]\) of Riemann surfaces of genus g with spin structure of Arf invariant \(\epsilon\in {\mathbb{Z}}/2{\mathbb{Z}}\) (resp. of the corresponding spin mapping class groups) is stable, i.e. independent of g and \(\epsilon\) for sufficiently large g. As the author notes, the interest in these moduli spaces comes from fermionic string theory. For a second paper the computation of the first (integer coefficients) and second (rational coefficients) homology, and thus of the Picard group, of the spin moduli spaces is announced. All of this generalizes results and methods (constructing simplicial complexes from configuration of simple closed curves on a surface on which the mapping class groups act, then applying spectral sequence arguments) of two of the author's previous papers in which he obtained analogous results for the ordinary mapping class groups resp. moduli spaces. homology of the spin moduli spaces of Riemann surfaces with spin structure; Arf invariant; spin mapping class groups; fermionic string theory; Picard group; configuration of simple closed curves on a surface Harer J.L. (1990) Stability of the homology of the moduli spaces of Riemann surfaces with spin structure. Math. Ann. 287(2): 323--334 Topology of Euclidean 2-space, 2-manifolds, General low-dimensional topology, Teichmüller theory for Riemann surfaces, Homology of classifying spaces and characteristic classes in algebraic topology, Differential topological aspects of diffeomorphisms, Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) Stability of the homology of the moduli spaces of Riemann surfaces with spin structure | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is known that the ring of modular forms of the Picard modular group \(\Gamma_ K\subset U_ K\) of an imaginary quadratic field K which operates on the 1-ball \(B\subset {\mathbb{C}}^ 2\) is generated by three elements and that the values of certain quotients of these generators, i.e. modular functions, on so-called K-singular moduli \(\tau\in B\), i.e. fixed points of \(U_ K\), are algebraic.
As in the classical cases of the elliptic and Hilbert modular group these singular moduli correspond to ideals in orders of K (which in the classical cases at least furnish an algebraic equation with rational coefficients for the generators of the field of modular functions). Having this final result in mind the author first characterises all K- singular moduli by arithmetic data of K, secondly shows that a certain canonically given jacobian \(Jac(C_{\Phi}^{-1}{}_{(\tau)})\) is simple as long as \(\tau\in B\) is K-singular and finally expresses the number of K-singular moduli in terms of class-numbers of CM-extension, \(K\subset L\) of degree 3 following essentially the ideas of Hecke. elliptic modular group; Picard modular group; imaginary quadratic field; K-singular moduli; Hilbert modular group; jacobian; class-numbers of CM-extension Feustel, J, Eine klassenzahlformel für singuläre moduln der picardschen modulgruppen, Comp. Math, 76, 87-100, (1990) Picard groups, Algebraic moduli problems, moduli of vector bundles, Modular and Shimura varieties, Quadratic extensions, Picard schemes, higher Jacobians, Class numbers, class groups, discriminants Eine Klassenzahlformel für singuläre Moduln der Picardschen Modulgruppen. (A class number formula for singular moduli of Picard modular 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 It is well known in model theory that it is far from being true that elementary equivalence of two fields implies their isomorphism. But this question becomes interesting when restricting to certain classes of fields. For example, it follows from work by Duret and Pierce that if \(K_1\) and \(K_2\) are function fields of curves over an algebraically closed field, then either both are of genus one (with \(K_1\) and \(K_2\) isomorphic in many but not all cases), or both function fields are of genus \(\neq 1\) and isomorphic. Pop has shown that two elementarily equivalent fields that are finitely generated over an algebraically closed field (resp. their prime field) are of the same transcendence degree, from which one can conclude that elementarily equivalent function fields, one of them being of so-called general type, over an algebraically closed field, a number field or a finite field are always isomorphic (Corollary 4 in the present paper in the case of number fields and finite fields).
Here, the author studies this question for function fields of quadrics and Severi-Brauer varieties. He introduces the notion of two fields \(K_1\) and \(K_2\) being (\(k\)-)isogenous, i.e. there exist field homomorphisms \(K_1\to K_2\) and \(K_2\to K_1\) (over a common base field \(k\)). In Theorem 7, he shows that function fields \(K_1\), \(K_2\) over some base field \(k\) of Severi-Brauer varieties of central simple \(k\)-algebras of the same degree and with cyclic division parts are isomorphic iff they are isogenous iff \(\text{Br}(K_1/k)=\text{Br}(K_2/k)\) holds for the Brauer kernels. For function fields \(K_1\) and \(K_2\) of \(n\)-dimensional quadrics (\(n\leq 2\)) over some field \(k\) of characteristic \(\neq 2\), he obtains the same results except for the condition on the Brauer kernels which has to be replaced by \(\text{Br}(lK_1/k)=\text{Br}(lK_2/k)\) for all quadratic extensions \(l/k\). If \(K_1\) is the function field over \(k\) (\(\text{char}(k)\neq 2\)) of an \(n\)-dimensional Severi-Brauer variety and \(K_2\) that of an \(n\)-dimensional quadric (\(n>1\)), then \(K_1\) and \(K_2\) are isomorphic iff they are isogenous iff they have the same Brauer kernels iff they are both rational function fields. He deduces (Corollary 8) that in all these situations, elementary equivalence implies isomorphism whenever \(k\) is algebraic over its prime field. In Theorem 10, he considers the case \(K=k(C)\) of a genus \(1\) curve \(C\) over a number field \(k\) and shows that under certain assumptions on the Jacobian \(J(C)\), any finitely generated field elementarily equivalent to \(K\) is actually isomorphic to \(K\). elementary equivalence; isomorphism; isogeny; function field; Severi-Brauer variety; quadric; elliptic curve; Jacobian Transcendental field extensions, Quadratic forms over general fields, Elliptic curves over global fields, Arithmetic theory of algebraic function fields, Model theory of fields, Grassmannians, Schubert varieties, flag manifolds, Brauer groups (algebraic aspects) On elementary equivalence, isomorphism and isogeny | 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 0672.00003.]
Among the several approaches to the Riemann-Schottky problem of characterizing Jacobian varieties of curves, there is one specific (geometric) attempt based on the Lie-Wirtinger-Poincaré-Tchebotarev theorem on non-developable double translation hypersurfaces in \(\mathbb{C}^ g\). This theorem, worked out in its full generality during the first half of our century, says that the double translation hypersurfaces with special parametrization characterize, in a certain sense, the theta divisors of non-hyperelliptic Jacobians of dimension \(g\). The author of the present paper has studied this approach to the Schottky problem in two previous articles [cf. Compos. Math. 49, 147-171 (1983; Zbl 0521.14017); J. Differ. Geom. 26, 253-272 (1987; Zbl 0599.14038)]. In the present article, he presents both a survey of this (so-called) approach via translation manifolds and a summary of some recent results that indicate possible relations to some other geometric characterizations of Jacobians. After a historical survey on the Lie-Wirtinger-Poincaré- Tchebotarev theorem, the author explains how these ideas can be used to characterize Jacobians by the existence of special parametrizations for their theta divisors. Then he gives a brief sketch of some of his very recent results. More precisely, he discusses the new observation that more general parametrizations for theta divisors also lead to the conclusion of the Lie-Wirtinger-Poincaré-Tchebotarev theorem. At the end of the paper, the author points out a striking connection between this approach and some other geometric approaches to the Schottky problem. He refers to the recent characterization of Jacobians by \textit{J. M. Muñoz Porras} [ Compos. Math. 61, 369-381 (1987; Zbl 0624.14020)] and explains the resulting indications for the fact that the double- translation-manifold property of the theta divisor could be related to the reducibility properties of \(\Theta\cap\Theta_ a\) and the Andreotti- Mayer condition \(\dim(\text{Sing}(\Theta))\geq g-4\) [cf. \textit{A. Beauville} and \textit{O. Debarre}, Invent. Math. 86, 195-207 (1986; Zbl 0659.14021)]. Riemann-Schottky problem; Jacobian varieties of curves; theta divisors of non-hyperelliptic Jacobians; translation manifolds; Lie-Wirtinger- Poincaré-Tchebotarev theorem Theta functions and curves; Schottky problem, Period matrices, variation of Hodge structure; degenerations, Jacobians, Prym varieties, Analytic theory of abelian varieties; abelian integrals and differentials Translation manifolds and the Schottky problem | 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 based on results of \textit{N. Hitchin} [Duke Math. J. 54, 90- 114 (1987; Zbl 0627.14024) and Proc. Lond. Math. Soc., III. Ser. 55, 59- 126 (1987; Zbl 0634.53045)] extended by Simpson. Hitchin proved that for any simple compact Lie group \(G\) the cotangent bundle of the moduli space of stable principal \(G\)-bundles over a compact Riemann surface is a completely integrable system. He showed that when \(G\) is one of the classical groups then the generic level set of this integrable system can be compactified to a Jacobian or Prym variety. Analogous results are obtained in this paper for the geometry of the level sets when \(G\) is the exceptional Lie group \(G_ 2\). exceptional Lie group; moduli space over compact Riemann surface; compactification of level set of integrable system; Prym variety; Jacobian variety; completely integrable system Katzarkov, L.; Pantev, T., Stable \(G_2\) bundles and algebraically completely integrable systems, Compos. math., 92, 43-60, (1994) Families, moduli of curves (algebraic), Picard schemes, higher Jacobians, Lie algebras of linear algebraic groups, Dynamical systems and ergodic theory, Algebraic moduli problems, moduli of vector bundles, Differentials on Riemann surfaces, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Stable \(G_ 2\) bundles and algebraically completely integrable 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 Let \(GL_ n\) be the group of \(n \times n\) invertible complex matrices, and \(P\) a parabolic subgroup of \(GL_ n\). In this paper we give a geometric description of the cohomology ring of a Schubert subvariety \(Y\) of \(Gl_ n/P\). Our main result (theorem 3.1) states that the coordinate ring \(A(Y \cap Z)\) of the scheme-theoretic intersection of \(Y\) and the zero scheme \(Z\) of the vector field \(V\) associated to a principal regular nilpotent element \({\mathfrak n}\) of \({\mathfrak g}{\mathfrak l}_ n\) is isomorphic to the cohomology algebra \(H^*(Y;\mathbb{C})\) of \(Y\). This theorem was conjectured for any reductive algebraic group \(G\) by \textit{E. Akyildiz}, \textit{J. B. Carell} and \textit{D. I. Liebermann} in Compos. Math. 57, 237- 248 (1986; Zbl 0613.14035), and it was proved for the Grassmannian manifolds by \textit{E. Akyildiz} and \textit{Y. Akyildiz} in J. Differ. Geom. 29, No. 1, 135-142 (1989; Zbl 0692.14031). We were recently informed that \textit{D. H. Peterson} has just proved that \(GL_ n\) is exactly the algebraic group \(G\) where the cohomology ring of any Schubert subvariety \(Y\) of the space \(G/B\) is isomorphic to \(A(Y \cap Z)\). Here \(B\) stands for a Borel subgroup of \(G\). It is also interesting to note that the cohomology ring of the union of two Schubert subvarieties in \(GL_ n/P\) may not admit such a description. This result is due to \textit{J. B. Carrell}. cohomology ring of a Schubert variety E. Akyıldız, A. Lascoux, and P. Pragacz, Cohomology of Schubert subvarieties of \?\?_{\?}/\?, J. Differential Geom. 35 (1992), no. 3, 511 -- 519. Grassmannians, Schubert varieties, flag manifolds, (Co)homology theory in algebraic geometry Cohomology of Schubert subvarieties of \(GL_ n/P\) | 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 part I see \textit{B. H. Gross} and \textit{D. B. Zagier}, Invent. Math. 84, 225--320 (1986; Zbl 0608.14019).]
Let \(X_ 0(N)\) be the modular curve of level \(N\) and \(J_ N\) its Jacobian, and denote by \(J^*_ N\) the Jacobian of \(X_ 0(N)/w_ N\) where \(w_ N\) is the Fricke involution. For a fundamental discriminant \(D<0\) and \(r\in {\mathbb Z}\) with \(D\equiv r^ 2 (4N)\) we let \(Y_{D,r}\) be the corresponding Heegner divisor in \(J_ N\) and \(Y^*_{D,r}\) be its image in \(J^*_ N\). Let \(f\) be a normalized newform in \(S_ 2(\Gamma_ 0(N))\) and \(L(f,s)\) be its \(L\)-series. Suppose that the root number of \(L(f,s)\) is \(-1\).
The main result of the paper then states that the subspace of \(J^*_ N({\mathbb Q})\otimes {\mathbb R}\) generated by the \(f\)-eigencomponents of all Heegner divisors \((y^*_{D,r})_ f\) with \((D,2N)=1\) has dimension \(1\) if \(L'(f,1)\neq 0\).
More precisely, \((y^*_{D,r})_ f=c((r^ 2- D)/4N,r)y_ f\), where \(c(n,r)\) is the coefficient of \(e^{2\pi i(n\tau +rz)}\) in a Jacobi form \(\phi_ f\) of weight 2 and index \(N\) corresponding to \(f\) in the sense of \textit{N.-P. Skoruppa} and \textit{D. Zagier} [Invent. Math. 94, 113--146 (1988; Zbl 0651.10020)] and \(y_ f\in (J^*({\mathbb Q})\otimes {\mathbb R})_ f\) is independent of \(D\) and \(r\) with \(\langle y_ f,y_ f\rangle =L'(f,1)/4\pi \| \phi_ f\|^ 2\) \((\langle\cdot, \cdot\rangle\)=canonical height pairing).
This result is in accordance with the conjectures of Birch and Swinnerton-Dyer which in the above situation (i.e. under the assumption \(\text{ord}_{s=1}L(f,s)=1)\) would predict that \(\dim (J^*_ N({\mathbb Q})\otimes {\mathbb R})_ f=1\). derivatives of \(L\)-series; modular curve; Jacobian; Heegner divisor; conjectures of Birch and Swinnerton-Dyer Gross B., Kohnen W. and Zagier D., Heegner points and derivatives of \textit{L}-series. II, Math. Ann. 278 (1987), no. 1-4, 497-562. Arithmetic ground fields for curves, Jacobi forms, Arithmetic aspects of modular and Shimura varieties, Jacobians, Prym varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Heegner points and derivatives of \(L\)-series. II | 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 the coordinate ring of an irreducible affine curve over a perfect field k, and let S be the normalization. It is shown that the canonical map Pic(R)\(\to Pic(S)\) is an isomorphism if (and only if) its kernel is a finitely generated abelian group. Moreover, these conditions hold if and only if the curve has at most one singularity and its local ring is analytically isomorphic to a restricted power series ring \(F+XK[[ X]]\), for suitable finite algebraic extensions \(K\supseteq F\supseteq k\). coordinate ring of an irreducible affine curve; Pic; Picard group DOI: 10.1007/BF01215648 Picard groups, Singularities of curves, local rings Picard groups of singular affine curves over a perfect field | 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 Generalizing methods of \textit{G. Ellingsrud}, \textit{L. Gruson}, \textit{C. Peskine} and \textit{S. A. Strømme} [cf. Invent. Math. 80, 181-189 (1985; Zbl 0629.14026)], we investigate the splitting behavior of the sequence of normal bundles of a Cartier divisor on a projective variety and its geometric interpretation. In particular there are examples showing that the splitting of the sequence in question is not an invariant of the linear equivalence class of a Cartier divisor. splitting of normal bundles of a Cartier divisor Braun, R.: On the normal bundle of cartier divisors on projective varieties. Arch. math. 59, 403-411 (1992) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves On the normal bundle of Cartier divisors on projective varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0626.00011.]
Let \(f: X\to S\) be a proper surjective morphism of normal algebraic varieties, defined over \({\mathbb{C}}\), and let \(D,P,N\in Div(X)\otimes {\mathbb{R}}\) be \({\mathbb{R}}\)-Cartier divisors.
An expression \(D=P+N\) is called the Zariski decomposition of D relative to f if the following conditions are satisfied:
(1) D is f-big;
(2) P is f-numerically effective;
(3) N is effective;
(4) The natural homomorphisms \(f_*{\mathcal O}_ X([mP])\to f_*{\mathcal O}_ X([mD])\) are bijective for all \(m\in {\mathbb{N}}.\)
Let \(\Delta\) be a \({\mathbb{Q}}\)-divisor such that the pair \((X,\Delta)\) is log-terminal. The author proves (theorem 1) that the Zariski decomposition of the log-canonical divisor \(K_ X+\Delta\) with real coefficients, relative to f, implies the finite generatedness, as \({\mathcal O}_ S\)-algebra, of the relative log-canonical ring
\[
R(X/S,K_ X+\Delta)=\sum_{m\in {\mathbb{N}}\cup \{0\}}f_*{\mathcal O}_ X([m(K_ X+\Delta)]).
\]
An analogous result was previously obtained by Benveniste in case \(\dim (X)=3\). - An example by Cutkosky shows that it is necessary to consider divisors with real coefficients.
Note that the Zariski decomposition above is unique if it exists (proposition 4). Hence this definition coincides with the original one by Zariski in the case \(\dim (X)=2\) and \(S=Spec(k)\). Note also that there is another definition due to Fujita. - The existence of the Zariski decomposition in general is an open question. surjective morphism of normal algebraic varieties; Cartier divisors; Zariski decomposition of the log-canonical divisor; log-canonical ring Yujiro Kawamata, The Zariski decomposition of log-canonical divisors, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 425 -- 433. Divisors, linear systems, invertible sheaves, Families, moduli, classification: algebraic theory The Zariski decomposition of log-canonical 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 [For the entire collection see Zbl 0677.00008.]
In [\textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand}, Usp. Mat. Nauk 27, 185-190 (1972; Zbl 0253.58009)] it is proved that the ring \({\mathcal D}(X)\) of differential operators on the cubic cone \(X^ 3_ 1+X^ 3_ 2+X^ 3_ 3=0\) over \({\mathbb{C}}\) is not noetherian, not finitely generated, and not simple. This is in sharp contrast to the ring of differential operators on a smooth affine variety. This example was further developed in [\textit{J. P. Vigué}, C. R. Acad. Sci., Paris, Sér. A 278, 1047-1050 (1974; Zbl 0299.58013)] where it was shown that the same bad properties were possessed by \({\mathcal D}(X)\) for any cone \(X\subset {\mathbb{C}}^ 3\) which is defined by a form of degree \(\geq 3\), and has an isolated singularity at 0.
The present paper further develops this theme. Let \({\mathcal L}\) be an ample line bundle on a smooth projective curve \(\bar X\) of genus g. Define the graded algebra \(R(\bar X,{\mathcal L}):=\oplus_{n\geq 0}H^ 0(\bar H,{\mathcal L}^{\otimes n})\), and let \(X=Spec R\). In all cases X is a normal surface with an isolated singularity (except when \(\bar X={\mathbb{P}}^ 1\) and \({\mathcal L}={\mathcal O}_{\bar X}(1))\); let \({\mathfrak m}=\oplus_{n>0}R_ n\) be the ideal of the singular point. Furthermore R is Cohen-Macaulay, and is Gorenstein if and only if \({\mathcal L}^{\otimes m}\cong \Omega\) the canonical sheaf, for some \(m\in {\mathbb{Z}}\). Let \({\mathcal D}(X)\) be the ring of differential operators on X.
The present paper proves the following results. If \(g\geq 1\), then \({\mathfrak m}\) is a \({\mathcal D}(X)\) submodule of R. Hence \({\mathcal D}(X)\) is not a simple ring. Furthermore, if R is Gorenstein then \({\mathcal D}(X)\) is not noetherian, and is not finitely generated (the proofs are somewhat simpler when \(g=1\), in which case R is Gorenstein for all choices of \({\mathcal L})\). The methods are based on those of the two papers cited above.
The last section gives examples. Let Y be a 2-dimensional normal affine cone with \(Sing(Y)=\{0\}\). Then Y is isomorphic to a quotient \(X/G=Spec R(\bar X,{\mathcal L})^ G\) where G is a finite abelian group acting on \(\bar X,\) and \({\mathcal L}\) is G-invariant. Significantly, \({\mathcal D}(Y)={\mathcal D}(X/G)\cong {\mathcal D}(X)^ G\), and standard results imply that \({\mathcal D}(Y)\) is simple (resp. noetherian) if and only if \({\mathcal D}(X)\) is. Now the earlier results are applied to show that there exist such Y with rational singularities, having the property that \({\mathcal D}(Y)\) is neither simple, nor noetherian. These examples contrast with previous ones which tended to give the (false) impression that rational singularities have better behaved rings of differential operators. ring of differential operators; smooth affine variety; ample line bundle; smooth projective curve; graded algebra; isolated singularity; rational singularities Rings of differential operators (associative algebraic aspects), Group actions on varieties or schemes (quotients), Geometric invariant theory, Sheaves of differential operators and their modules, \(D\)-modules, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Opérateurs différentiels sur les surfaces munies d'une bonne \({\mathbb{C}}^*\)-action. (Differential operators on surfaces with good \({\mathbb{C}}^*\)-action) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a short survey of the most significant results concerning the number of rational points of algebraic curves over finite fields.
Let \(p\) be a prime number, \(X\) a nonsingular, projective, geometrically irreducible curve of genus \(g\) over a finite field \({\mathbb F}_{q}\) with \(q=p^{ \nu}\) elements and \({\#}X({ \mathbb F}_{q})\) the number of \({ \mathbb F}_{q}\)-\textit{rational points} on \(X\). Firstly the author demonstrates several classical results of P. Fermat, L. Euler, C. F. Gauss, J. Lagrange, A. M. Legendre, C. Jacobi et al. related to the number of rational points on an projective curve of a special form defined over \({\mathbb F}_{p}\). The author next discusses a conjecture of E. Artin (1924) on the complex zeroes of the \textit{zeta function} \(Z(X,q,t)\) of \(X\) and then presents the well-known \textit{Hasse-Weil} bound
\[
|{ \#}X({ \mathbb F}_{q})-(q+1)| \leq \lfloor 2g \sqrt{q} \rfloor
\]
for \({\#}X({\mathbb F}_{q})\). Thereafter the author describes a series of distinct proofs of the above bound (H. Hasse, A. Weil, S. A. Stepanov, E. Bombieri et al.) and then demonstrates several its refinements (J. P. Serre, K. Lauter et al.).
Let \(N_{q}(g)\) denote the maximal number of \({\mathbb F}_{q}\)-rational points that a curve \(X\) of genus \(g\) can have. The author discusses several examples of \textit{optimal curves} \(X\) of genus \(g\) (for which \({\#X}({\mathbb F}_{q})=N_{q}(g)\)) and then describes a particular family of optimal curves, the so-called \textit{maximal curves}, whose number of rational points attains the upper Hasse-Weil bound. A distinguished example here is the \textit{Hermitian curve} which is intrinsically determined by its genus and number of rational points. Also, the author shows that there are two important families of optimal curves, namely the \textit{Suzuki curves} and the \textit{Ree curves} with the property that each curve in these families is intrinsically determined by the data: (1) the genus, (2) the number of rational points and (3) the automorphism group.
For applications to coding theory, the key matter is to find a family of algebraic curves \((X_{g})\) (indexed be its genus and defined over a fixed field \({ \mathbb F}_{q}\)) such that
\[
A(q)= \limsup_{g} \frac{N_{q}(g)}{g}
\]
be a large as possible. It was shown by Y. Ihara (with using \textit{supersingular points} on a family of modular curves \((X_{g})\)) that when \(q\) is a square then \(A(q)= \sqrt{q}-1\). Later the same result was obtained by A. Garcia and H. Stichtenoth via modular curves of a different shape defined by ``explicit equations''. The author briefly describes the Goppa construction of so-called \textit{geometric Goppa codes} and then demonstrates the Tsfasman-Vläduţ-Zink result on existence of a family \((C_{i})\) of asymptotically long linear codes (coming from a family of modular curves of growing genus with many \({\mathbb F}_{q}\)-rational points) whose parameters \((R, \delta)\) satisfy
\[
R+ \delta=1-1/( \sqrt{q}-1).
\]
It should be pointed that the last result improves the well-known in coding theory \textit{Gilbert-Varshamov bound}.
The survey is completed with a discussion of Stör-Voloch theory concerning \textit{Weierstrass points} of an algebraic curve and \textit{Frobenius orders}. finite fields; algebraic curves; Riemann-Roch theorem; number of rational points of an algebraic curves over a finite field; Riemann hypothesis; Hasse-Weil bound; asymptotic problems; zeta-functions and linear systems; a characterization of the Suzuki curve; maximal curves; Hermitian curve; Weierstrass points Torres F.: Algebraic curves with many points over finite fields. In: Martínez-Moro, E., Munuera, C., Ruano, D. (eds) Advances in Algebraic Geometry Codes, pp. 221--256. World Scientific Publishing Company, Singapore (2008) Local ground fields in algebraic geometry, Complex multiplication and moduli of abelian varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Research exposition (monographs, survey articles) pertaining to number theory Algebraic curves with many points over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Trigonal curve is a smooth projective curve with \(3:1\) map onto a curve which is isomorphic to \({\mathbb P}^1\). The isomorphism is not part of the data. The stack \({\mathcal T}_g\) of genus \(g\) smooth trigonal curves over fixed field \(k\) of characteristic not equal 2 or 3 is considered. The objects over \(k\)-scheme \(S\) are families \(C \to P \to S\) where \(P\to S\) is a smooth conic bundle, \(C\to P\) smooth \(3:1\) cover and their composite \(C\to S\) is a family of genus \(g\) curves. The forgetful morphism \((C \to P \to S) \mapsto (C\to S)\) takes \({\mathcal T}_g\) to the stack \({\mathcal M}_g\) of smooth genus \(g\) curves. One of central results of the paper is that this morphism is a locally closed immersion whenever \(g\geq 5\). Another central result is a description of the stack \({\mathcal T}_g\) as a quotient \([X_g/\Gamma_g]\) of an appropriate scheme \(X_g\) by the action of a certain algebraic group \(\Gamma_g\). The third central result is a computation of the Picard group of \({\mathcal T}_g\) for \(g\neq 1\). The authors give a description of the stack of vector bundles over a conic as a quotient stack what is of independent interest besides of being one of main tools of the work together with the result of \textit{R. Miranda} [Am. J. Math. 107, 1123--1158 (1985; Zbl 0611.14011)] ``that describes a flat finite triple cover of a scheme \(S\) as given by a locally free sheaf \(E\) of rank two on \(S\), with a section of \(\text{Sym}^3 E \otimes E^{\vee}\)'' (cited from the abstract). Also the stack \(\hat{\mathcal T}_g\) of triple covers which contains \({\mathcal T}_g\) as an open substack, is examined. In particular, the locus of singular curves in \(\hat{\mathcal T}_g\) is analyzed. trigonal curves; algebraic stack; stack of smooth curves; Picard group of a stack; stack of vector bundles on a conic Bolognesi, M; Vistoli, A, Stacks of trigonal curves, Trans. Am. Math. Soc., 364, 3365-3393, (2012) Families, moduli of curves (algebraic), Generalizations (algebraic spaces, stacks), Picard groups Stacks of trigonal 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 totally degenerate curve of genus \(g\) is a stable curve, all of whose components are rational and which has \(g\) additional ordinary double points. Mumford introduced for such a curve \(C\) a generalized Jacobian \(J(C)\), which is given by a totally degenerate symmetric \(g\times g\)- matrix, i.e. a matrix \((v_{ij})\) with \(v_{ii}=0\) and \(v_{ij} \neq 0\) for \(i \neq j\), the period matrix of \(C\). This gives a map \(\text{per}_ g\) from the space of totally degenerate curves of genus \(g\) to the torus of totally degenerate, symmetric \(g \times g\) matrices. The analogue of the classical Schottky problem consists in describing the image of \(\text{per}_ g\). In the paper an equation is derived which describes the locus of period matrices of totally degenerate curves of genus 4 whose intersection graph is simple.
The result allows to deduce systems of equations for the periods of totally degenerate, irreducible curves of genus \(\geq 5\). Moreover, one can specialize the equation in genus 4 to obtain an equation describing the periods of totally degenerate hyperelliptic curves of genus 3. totally degenerate curve; generalized Jacobian; Schottky problem; period matrices L. Gerritzen, Equations defining the periods of totally degenerate curves, Israel Journal of Mathematics 77 (1992), 187--210. Theta functions and curves; Schottky problem, Period matrices, variation of Hodge structure; degenerations Equations defining the periods of totally degenerate 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 part I see Math. Jap. 40, No. 3, 547-552 (1994; see the preceding abstract).]
The classification of minimal elliptic surfaces over a given curve and with a given number of singular fibers has interested various authors. The paper under review is one of a series of articles by the author on this subject. In this paper are classified minimal elliptic surfaces \(\pi : X \to C\), over a genus 2 curve with a section and exactly one singular fiber of type \(I^*_6\). Such classification has been done under the following assumptions:
(a) the \(J\)-map \(J : C \to \mathbb{P}^1\) associated to the fibration factors either through the canonical map \(\varphi_K : C \to \mathbb{P}^1\) or it factors through a triple cover of \(\mathbb{P}^1\);
(b) \(J^{-1} (1)\) is either one point of multiplicity 6 or three points each of multiplicity 2.
The classification was accomplished by finding at first all possibilities for the ramification of the \(J\)-map. These information were then used to build up the desired surfaces. \(J\)-map; classification of minimal elliptic surfaces over a curve; minimal elliptic surfaces; genus 2 curve; singular fiber Elliptic surfaces, elliptic or Calabi-Yau fibrations, Ramification problems in algebraic geometry, Jacobians, Prym varieties, Elliptic curves Elliptic surfaces over a genus 2 curve. II | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a report of the author's recent work [\textit{J. Noguchi}, J. Reine Angew. Math. 487, 61-83 (1997; Zbl 0880.11049)]. In section 2 he discusses the Nevanlinna-Cartan theory over function fields and explains the main results. In section 3 he applies the same idea to obtain a finiteness theorem for \(S\)-unit points of a diophantine equation over number fields. Section 4 is devoted to examples. algebraic variety; \(abc\)-conjecture; finiteness theorem for \(S\)-unit points of a diophantine equation; Nevanlinna-Cartan theory over function fields Varieties over global fields, Rational points, Diophantine approximation, transcendental number theory, Nevanlinna theory; growth estimates; other inequalities of several complex variables Value distribution theory over function fields and a diophantine equation | 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 smooth projective curve, \(G\) a simple and simply connected complex Lie group, \({\mathcal M}_X(G)\) the moduli stack parametrizing principal \(G\)-bundles over \(X\) and \({\mathcal L}_G\) the ample generator of the Picard group of \({\mathcal M}_X(G)\). It is known (see, for example, \textit{G. Faltings} [J. Algebr. Geom. 18, No. 2, 309--369 (2009; Zbl 1161.14025)] or \textit{C. Sorger} [Ann. Sci. Éc. Norm. Supér. (4) 32, No. 1, 127--133 (1999; Zbl 0969.14016)]) that \(\text{dim}\, \text{H}^0({\mathcal M}_X(E_8),{\mathcal L}_{E_8}) = 1\), hence \({\mathcal M}_X(E_8)\) carries a natural divisor \(\Delta\). A geometric interpretation of this divisor is, however, not available.
In the paper under review, the authors study the pullback of \(\Delta\) under the morphisms \({\tilde \phi} : {\mathcal M}_X(P) \rightarrow {\mathcal M}_X(E_8)\) induced by the group homomorphisms \(\phi : P \rightarrow E_8\), where \(P\) is connected, simply connected and semisimple, and \(d\phi : \text{Lie}(P) \rightarrow \text{Lie}(E_8)\) embeds \(\text{Lie}(P)\) as a subalgebra of maximal rank 8 of \(\mathfrak{e}_8\). Actually, \(P\) must be one of the groups \(\text{Spin}(16)\), \(\text{SL}(9)\), \(\text{SL}(5)\times \text{SL}(5)\), \(\text{SL}(3)\times E_6\), \(\text{SL}(2)\times E_7\) and the kernel \(N\) of \(\phi\) is \({\mathbb Z}/2{\mathbb Z}\), \({\mathbb Z}/3{\mathbb Z}\), \({\mathbb Z}/5{\mathbb Z}\), \({\mathbb Z}/3{\mathbb Z}\) and \({\mathbb Z}/2{\mathbb Z}\), respectively. There exists a \textit{canonical} \({\mathcal M}_X(N)\)-linearization of \({\mathcal L}_P\) induced from the isomorphism \({\tilde \phi}^{\ast} {\mathcal L}_{E_8} \simeq {\mathcal L}_P\).
The main result of the paper asserts that the induced map
\[
{\phi}_P : \text{H}^0({\mathcal M}_X(E_8),{\mathcal L}_{E_8}) \rightarrow \text{H}^0({\mathcal M}_X(P),{\mathcal L}_P)
\]
is non-zero and its image coincides with the \({\mathcal M}_X(N)\)-invariant subspace of the space \(\text{H}^0({\mathcal M}_X(P),{\mathcal L}_P)\). The proof uses a result of \textit{P. Belkale} [J. Differ. Geom. 82, No. 2, 445--465 (2009; Zbl 1193.14013)] which implies that, in the case under consideration, \({\phi}_P\) has constant rank when the curve \(X\) varies in a family of smooth curves, and the identification of \(\text{H}^0({\mathcal M}_X(G), {\mathcal L}_G)\) with the space of \textit{conformal blocks} associated to \(X\) (with one marked point labelled with the zero weight) and to \(\text{Lie}(G)\).
The authors also show that if \((A,B)\) is one of the pairs \((\text{SL}(5), \text{SL}(5))\), \((\text{SL}(3),E_6)\), \((\text{SL}(2),E_7)\) and if one considers \textit{any} \({\mathcal M}_X(N)\)-linearization of \({\mathcal L}_{A\times B}\) then a non-zero element of the 1-dimensional \({\mathcal M}_X(N)\)-invariant subspace of:
\[
\text{H}^0({\mathcal M}_X(A\times B),{\mathcal L}_{A\times B}) = \text{H}^0({\mathcal M}_X(A),{\mathcal L}_A) \otimes \text{H}^0({\mathcal M}_X(B),{\mathcal L}_B)
\]
induces an isomorphism:
\[
\text{H}^0({\mathcal M}_X(A),{\mathcal L}_A)^{\ast}\overset{\sim}\longrightarrow \text{H}^0({\mathcal M}_X(B),{\mathcal L}_B)\, .
\]
A similar isomorphism is obtained for the pair \((\text{Spin}(8),\text{Spin}(8))\). The proof of this result uses the representation theory of Heisenberg groups. principal bundle; projective curve; moduli stack; generalized theta function; Strange Duality; conformal embedding of Lie algebras; space of conformal blocks Boysal, A., Pauly, C.: Strange duality for Verlinde spaces of exceptional groups at level one. Int. Math. Res. Not. (2009). 10.1093/imrn/rnp151 Stacks and moduli problems, Vector bundles on curves and their moduli, Holomorphic bundles and generalizations, Exceptional groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Strange duality for Verlinde spaces of exceptional groups at level one | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(N\) be a positive integer and let \(p>2\) be a prime not dividing \(N\). With a normalized eigenform \(f=\sum a_ nq^ n\) on \(X_ 1(N)\) modulo \(p\) of weight \(k\) (and supposed to have nebentypus \(\varepsilon)\) one can associate a representation \(\rho_ f:G=\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\to GL_ 2(E)\), where \(E\) is a finite field of characteristic \(p\). An eigenform \(g=\sum b_ nq^ n\) modulo \(p\) of weight \(k'=p+1-k\) on \(X_ 1(N)\) such that \(na_ n=n^ kb_ n\) is called a companion form of \(f\). A conjecture of Serre says that the existence of such a companion form is equivalent to tame ramification of \(\rho_ f\) above \(p\). The conjecture was proved by \textit{B. H. Gross} [Duke Math. J. 61, 445-517 (1990; Zbl 0743.11030)] in `most cases'. In the underlying paper the assumption Gross had to make is removed, and the following result is proved: For an ordinary cuspidal eigenform \(f\) on \(X_ 1(N)\) modulo \(p\) of weight \(k\) such that \(2<k\leq p\), the representation \(\rho_ f\) is tamely ramified above \(p\) if and only if \(f\) has a companion form.
The results and ideas of Gross (loc. cit.) pervade the proof of the main theorem above, but to circumvent Gross's need for an extra assumption, the Kodaira-Spencer pairing is applied as a useful tool to compute the logarithmic derivative of the Serre-Tate pairings between the Tate module of the reduction of a family of ordinary \(p\)-divisible groups \(G\) over a complete local \(p\)-ring \(R\) and that of its dual \(^ tG\). Here a \(p\)- divisible group \(G\) is said to be ordinary if the dual of the connected subgroup of its special fiber is étale. An explicit general formula for the Kodaira-Spencer pairing can be derived for a semi-stable curve over a one-parameter infinitesimal deformation of a point. Regarding \(X_ 1(pN)\) as a family over \(\text{Spec}(\mathbb{Z}_ p[\zeta_ p])\) with base \(\text{Spec}(\mathbb{Z}_ p)\), this can be applied to modular forms. It leads to a formula for the leading term of the logarithmic derivative of the Serre-Tate pairing for the ordinary factor \(G\) of the Tate module of the Jacobian of \(X_ 1(pN)\) cut out by the natural action of \((\mathbb{Z}/p\mathbb{Z})^*\). The formula shows that the vanishing of the leading term of \(ds/s\) is equivalent to the vanishing of a certain class \(h\) in the de Rham cohomology of the Igusa curve, and this amounts to the existence of a companion form of the modular form \(f\). The last argument necessary for the proof of the theorem now comes from the observation that the vanishing of the leading term is equivalent to the tameness of the ramification of the restriction of \(\rho_ f\) to a decomposition group at \(p\). \(p\)-divisible group; modular elliptic curves; Galois representation; companion form; tame ramification; cuspidal eigenform; Kodaira-Spencer pairing; Serre-Tate pairing; Tate module of the Jacobian; de Rham cohomology of the Igusa curve Robert, F, Coleman and josé felipe voloch, companion forms and Kodaira-spencer theory, Invent. Math., 110, 263-281, (1992) Holomorphic modular forms of integral weight, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, \(p\)-adic cohomology, crystalline cohomology, Elliptic curves, Formal groups, \(p\)-divisible groups Companion forms and Kodaira-Spencer 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 In part I of this paper [\textit{J. Steenbrink} and the author, ibid. 80, 489-542 (1985)] put a mixed Hodge structure on the cohomology groups of a curve with values in \({\mathbb{V}}\), where \({\mathbb{V}}\) is a graded-polarizable variation of mixed Hodge structures. In the paper under review the author considers the case where \({\mathbb{V}}\) arises from geometry. Let \(f: Z\to \bar S\) be a family of quasi-projective varieties over a smooth complete curve, let \(S=\bar S-\Sigma\) be the set of regular values for f, let \(g: U\to S\) be the restriction of f to S, and let \({\mathbb{V}}=R^ ig_*{\mathbb{C}}\). Then the following are morphisms of mixed Hodge structure: \((i)\quad \pi_ i: H^ i(U,{\mathbb{C}})\to H^ 0(S,{\mathbb{V}})\cong H^ 0(\bar S,j_*{\mathbb{V}})\); \((ii)\quad \ker \pi_{i+1}\twoheadrightarrow H^ 1(S,{\mathbb{V}})\) (isomorphism if \(\Sigma\neq 0)\) \((iii)\quad \ker \{H^{i+1}(Z,{\mathbb{C}})\to H^ 0(\bar S,R^{i+1}f_*{\mathbb{C}}\}\twoheadrightarrow H^ 1(\bar S,j_*{\mathbb{V}})\); \((iv)\quad H^ 2(\bar S,j_*{\mathbb{V}})\cong H^ 2(\bar S,R^ if_*{\mathbb{C}})\to H^{i+2}(Z,{\mathbb{C}})\). variation of mixed Hodge structure; mixed Hodge structure on the cohomology groups of a curve Joseph Steenbrink and Steven Zucker, Variation of mixed Hodge structure. I, Invent. Math. 80 (1985), no. 3, 489 -- 542. , https://doi.org/10.1007/BF01388729 Steven Zucker, Variation of mixed Hodge structure. II, Invent. Math. 80 (1985), no. 3, 543 -- 565. Transcendental methods, Hodge theory (algebro-geometric aspects) Variation of mixed Hodge structure. II | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is well known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces, concerning the Abel-Jacobi map from a graph to its Jacobian. As an application of our results, we characterize the existence or non-existence of a winning strategy for a certain chip-firing game played on the vertices of a graph. Graph; Riemann surface; Riemann-Roch theorem; Jacobian of a finite graph; chip-firing games M. Baker and S. Norine, \textit{Riemann--Roch and Abel--Jacobi theory on a finite graph}, Adv. Math., 215 (2007), pp. 766--788, . Paths and cycles, Riemann surfaces; Weierstrass points; gap sequences, Graphs and abstract algebra (groups, rings, fields, etc.) Riemann-Roch and Abel-Jacobi theory on a finite graph | 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 is devoted to the Jacobian conjecture: A polynomial mapping \(f:\mathbb{C}^2 \to\mathbb{C}^2\) with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a non-zero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity. ramified covering; degree of a mapping; Jacobian conjecture; polynomial mapping A. V. Domrina and S. Yu. Orevkov, ''On Four-Sheeted Polynomial Mappings of \(\mathbb{C}\)2. I: The Case of an Irreducible Ramification Curve,'' Mat. Zametki 64(6), 847--862 (1998) [Math. Notes 64, 732--744 (1998)]. Jacobian problem, Ramification problems in algebraic geometry, Coverings in algebraic geometry On four-sheeted polynomial mappings of \(\mathbb{C}^2\). I: The case of an irreducible ramification 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 A \(\mathbb Q\)-curve is an elliptic curve \(E\) over \(\overline{\mathbb Q}\) which is isogenous to all its Galois conjugates. Serre's conjecture implies that \(\mathbb Q\)-curves are modular. This means that \(E\) is a \(\overline{\mathbb Q}\)-simple factor of \(J_1(N)\) for some level \(N\). In this paper we will introduce \(\mathbb Q\)-motives which are generalizations of \(\mathbb Q\)-curves and present basic properties of \(\mathbb Q\)-motives. Their properties are proved under some standard conjectures for motives. Q-curve; abelian variety of GL(2)-type; Serre's conjecture T. Yamauchi, \(\ACHIQ\)-motives and modular forms , Journal of Number Theory 128 (2008), 1485-1505. Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties \(\mathbb Q\)-motives and 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 The paper under review is a thesis, which essentially consists of two papers by the author [see \textit{M. Fujimori}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV Ser. 24, No. 3, 551-569 (1997; Zbl 0916.11035) and Tôkohu Math. J., II. Ser. 46, No. 4, 523-539 (1994; Zbl 0828.11015)]. Let \(k\) be a number field and \(C\) be a (smooth and proper) curve of genus \(g\) over \(k\). Let \(r\) be the rank of the Néron-Severi group of the Jacobian variety \(J\) of \(C\). Consider the image \(I\) of the set \(C(k)\) of algebraic points of \(C\) in \(\mathbb{R}\otimes_{\mathbb{Z}}J(\overline k)\). In the case when \(C\) admits a nontrivial automorphism, the author proves inequalities (in terms of the Néron-Tate pairing of \(J)\) which are valid on \(I\). This result refines a theorem of Manin. As an application, one gets a new proof of the following fact: The number of fixed points of a non-trivial automorphism on a \(k\)-curve of genus \(>1\) is at most \(2g+2\) (the usual proof of this result uses Hurwitz's formula).
The author also studies the case of Thue curves define by equations \(C_a:T(X,Y)=aZ^n\), where \(T(X,Y)\) is a homogeneous polynomial of degree \(n>3\) with a non-zero discriminant: using Mumford's gap principle, he improves (under certain assumptions) the estimates given by Silverman for the number of integral points on \(C_a\). rational points; Néron-Severi group; Jacobian variety; Néron-Tate pairing; number of fixed points; Thue curves; number of integral points Rational points, Jacobians, Prym varieties, Enumerative problems (combinatorial problems) in algebraic geometry, Higher degree equations; Fermat's equation Integral and rational points on algebraic curves of certain types and their Jacobian varieties over number fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The aim of this paper is to give necessary and sufficient conditions on a plane curve \(\Gamma\) of degree \(n(n-1)\) in order that \(\Gamma\) is the ramification curve of the general projection of a smooth surface \(S\subseteq\mathbb{P}^ 3\) onto \(\mathbb{P}^ 2\). The complete characterization of \(\Gamma\) is given by its singularities: it was known [see \textit{A. Libgober}, Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, Part 2, Proc., Sympos. Pure Math. 46, No. 2, 29-45 (1987; Zbl 0703.14007)], that the singular points of \(\Gamma\) must be \(n(n-1)(n-2)(n- 3)\) nodes and \(n(n-1)(n-2)\) cusps; here it is shown that if \(\Gamma\) has precisely those singularities the following conditions are what was asked for:
-- the smallest degree of a curve \(\mu_ 0\) passing through the singular points of \(\Gamma\) is \(n^ 2-3n+2\);
-- there exists a curve \(\mu_ 1\) of degree \(n^ 2-3n+3\) having no common components with \(\mu_ 0\) through the singular points of \(\Gamma\).
This is proved by using in a nice way a very classical construction: The (Cayley) monoidal representation of a space curve \(C\) via the cone \(R\) which projects \(C\) to \(\mathbb{P}^ 2\) and a monoid \(\Psi\) (i.e. a surface of degree \(n\) with an \((n-1)\)-ple point) through \(C\) such that \(\Psi\cap R\) is the union of \(C\) and lines through the singular points of \(\Gamma\). What is proved is that the conditions on \(\Gamma\) are also equivalent to the fact that \(\Gamma\) is the projection of a smooth space curve \(C\) which is the complete intersection of two surfaces \(S, B\) of degrees \(n, n-1\). The surface \(S\) can be chosen in such a way that the (unique) surface \(B\) of degree \(n-1\) is the polar of \(S\) with respect to the center of projection. The interest for this problem lies in the fact that there are few methods to construct singular plane curves, and that the fundamental group of \(\mathbb{P}^ 2-\Gamma\) is a discrete invariant on the moduli space of the surface of general type. ramification curve of the general projection of a smooth surface; singularities; projection of a smooth space; construct singular plane curves D'Almeida, J.: Courbe de ramification de la projectoin su P2 d'une surface de P3. Duke Math. J. 65(2), 229--233 (1992) Singularities of curves, local rings, Special surfaces Ramification curve of the projection on \(\mathbb{P}^ 2\) of a surface of \(\mathbb{P}^ 3\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The fundamental groups of the plane curve complements are of permanent interest.
Definition. We say that a group \(G\) is big if it contains a non-abelian free subgroup.
Main result: Let \(C\subset\mathbb{P}^2\) be an irreducible immersed curve which is neither a line nor a conic nor a nodal cubic. Let \(C^* \subset\mathbb{P}^{2*}\) be the dual curve. Then the group \(\pi_1(\mathbb{P}^{2*} \setminus C^*)\) is big.
If \(C\) is a nodal cubic then \(C^*\) is a three-cuspidal quartic and \(\pi_1(\mathbb{P}^{2*}\setminus C^*)\) is the metacyclic group of order 12. If the geometric genus \(g\) of \(C\) is at least 2, this is clear because a subgroup of \(G\) can be mapped epimorphically onto the fundamental group of the normalization of \(C\). To handle the cases \(g=0, 1\), we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Plücker curves. Such a curve \(C\) can be regarded as a plane section of the corresponding discriminant hypersurface [cf. \textit{I. Dolgachev} and \textit{A. Libgober} in: Algebraic Geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 1-25 (1981; Zbl 0475.14011)]. Applying Zariski-Lefschetz type arguments, we deduce the result from ``the bigness'' of the braid group \(B_{d,g}\), that is, of the group of \(d\)-string braids of a compact genus \(g\) Riemann surface. Zariski problem; fundamental group of the complement of a plane curve; braid group Dethloff, G., Orevkov, S., Zaidenberg, M.: Plane curves with a big fundamental group of the complement. In: Kuchment, P., Lin, V. (eds.) Voronezh Winter Mathematical Schools: Dedicated to Selim Krein, American Mathematical Society Translations-Series 2, vol. 184, pp. 63-84 (1998). Rational and birational maps, Special algebraic curves and curves of low genus, Families, moduli of curves (algebraic), Braid groups; Artin groups, Fundamental groups and their automorphisms (group-theoretic aspects) Plane curves with a big fundamental group of the complement | 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 Connected components of a real \(n\)-dimensional non-singular algebraic variety \(A\) define homology classes in \(H_ n(A(\mathbb{C}),\mathbb{F}_ 2)\). The aim of this paper is to improve the author's previous upper bounds [see Math. USSR, Izv. 22, 247-275 (1984); translation from Izv. Akad. SSSR, Ser. Mat. 47, No. 2, 268-297 (1983; Zbl 0537.14035)] on the number \(k\) of relations between these classes. Here the author uses analogous computations with Galois-Grothendieck cohomology of \(A(\mathbb{C})\) over \(\mathbb{F}_ 2\) with respect to complex conjugation action. Among numerous upper bounds, generalizing the known ones \((k\leq 1\) for curves, \(k\leq 1+q\) for surfaces, where \(q\) is the irregularity) it should be underlined the estimate \(k\leq 1\) for complete intersections. components of a real algebraic variety; homology classes; Galois- Grothendieck cohomology; algebraic variety V. A. Krasnov, ''On homology classes defined by real points of a real algebraic variety,''Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],55, No. 2, 282--302 (1991). Topology of real algebraic varieties, Étale and other Grothendieck topologies and (co)homologies On homology classes determined by real points of a real algebraic variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Der Grad der Inversion einer gegebenen Curve \(n^{\text{ter}}\) Ordnung ist im Allgemeinen \(2n\), aber man kann Curven so classificiren, dass eine gegebene Curve und ihre Inversion zu derselben Familie gehören, und zwar in folgender Weise: \(u_m\) bezeichne die Glieder der Ordnung \(m\) in der Gleichung der Curve. Wenn \(u_m\) eine Potenz von \(x^2+y^2\) als Factor enthält, so ersetzt man ihn durch eine gleiche Potenz von \(r^2\). Betrachtet man \(r^2\) als einen Factor von nur einer Dimension, so kann man den Grad der Gleichung in \(r^2, x\) und \(y\) bestimmen, und nenne dies den circularen Grad der Gleichung. Es wird nun gezeigt, dass eine Curve und ihre Inversion denselben circularen Grad haben. Die Anwendung des circularen Grades wird dann weiter entwickelt und speciell auf die Kreispunkte im Unendlichen und auf Inversion angewandt. Degree of inversion of a curve General theory of nonlinear incidence geometry, Projective analytic geometry, Projective techniques in algebraic geometry, Curves in algebraic geometry Classification of plane curves with reference to inversion. | 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 compute the algebraic equation of the universal family over the Kenyon-Smillie \((2, 3, 4)\)-Teichmüller curve, and we prove that the equation is correct in two different ways. Firstly, we prove it in a constructive way via linear conditions imposed by three special points of the Teichmüller curve. Secondly, we verify that the equation is correct by computing its associated Picard-Fuchs equation. We also notice that each point of the Teichmüller curve has a hyperflex and we see that the torsion map is a central projection from this point. Teichmüller curve; universal family of curves; Picard-Fuchs equation Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Teichmüller theory for Riemann surfaces, Families, moduli of curves (algebraic) The equation of the Kenyon-Smillie \((2, 3, 4)\)-Teichmüller 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 W be a projection of a Veronesian variety. \textit{W. Gröbner} [Arch. Math. 16, 257-264 (1965; Zbl 0135.211)] showed that W can have imperfect defining prime ideals and posed the problem of classifying such projections. Here it is shown that, in the simplicial case, to check if such a projection is arithmetically Cohen-Macaulay or arithmetically Buchsbaum, one needs only finitely many operations. Then a practical criterion for a class of such projections to be arithmetically Cohen- Macaulay or arithmetically Buchsbaum is given. Finally, an upper bound for the difference between the Buchsbaum invariant and the so-called length of its associated semigroup ideal is obtained. classifying projections of a Veronesian variety; arithmetically Cohen- Macaulay; arithmetically Buchsbaum Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Software, source code, etc. for problems pertaining to algebraic geometry, Projective techniques in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Algorithmetical aspects of the problem of classifying multi-projections of Veronesian 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 \(G\) be a commutative algebraic group defined over a number field \(k\). In this paper, the author defines a notion of canonical height \(h_L\) attached to an admissible line bundle \(L\) on a compactification \(\overline G/k\) of \(G\) and gives lower bounds for the height \(h_L(P)\) of a point \(P\in G(k)\) in terms of \(L\), when \(G\) is isogenous to the product of an Abelian variety by a torus. For a point \(P\in G(k)\), denote by \(G(P)\) the smallest algebraic subgroup of \(G\) which contains \(P\), and put \(g(P)= \dim G(P)\). It is shown that, if \(G\) is isogenous to the product of an Abelian variety by a torus, then there is a constant \(c= c(G, \overline G, k)\) such that for any admissible line bundle \(L\) and any point \(p\in G(k)\), we have
\[
h_L(P)^{g(P)}\geq c(L^{q(P)}\cdot G(P)),\tag{\(*\)}
\]
where the right-hand side is an intersection number. In particular, when \(G(P)= G\), we get \(h_L(P)\geq c(L^g)\), where \(g= \dim G\). For a general group \(G\), it is shown that such a constant does not exist. When \(G\) is a split torus/\(k\) or an isotypic Abelian variety, a stronger inequality is proved in the form
\[
h_L(P)^{g(P)}\geq R(G(P)/k, P)(L^{g(P)}\cdot G(P)),
\]
where \(R(G(P)/k, P)\) denotes an intrinsic regulator. An appendix explains how bounds of the type \((*)\) answer a classical problem on linear dependence relations [ see \textit{D. Masser}, New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 248-262 (1988; Zbl 0656.10031)]. group variety; Lehmer's problem; Lang-Silverman conjecture; lower bounds for the height of a point; canonical height; admissible line bundle; Abelian variety; torus Bertrand D., ''Minimal heights and polarizations on group varieties'' Transcendence (general theory), Group varieties, Varieties over global fields Minimal heights and polarizations on group 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 S be a closed subscheme of dimension zero of \({\mathbb{P}}^ 2\), the projective plane over an algebraically closed field. The author first gives a necessary and sufficient condition for there to exist a nonsingular curve containing S. She then shows that if this condition holds, the least degree of such a curve is n-1, where \(n=\deg (S)\). Furthermore, she shows that under certain ``genericity assumptions'' on S this least degree is smaller. - The present work generalizes results obtained by \textit{R. Maggioni} and \textit{A. Ragusa} in the case that S is reduced [J. Algebra 92, 176-193 (1985; Zbl 0556.14010)]. existence of plane curve containing a given subscheme; general position DOI: 10.1080/00927878808823574 Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry Nonsingular curves passing through a zerodimensional subscheme of \({\mathbb{P}}^ 2\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper describes the development of the concept of multiplicity in algebraic geometry and in commutative algebra from the origins up to our days. It brings various concepts of this notion -- both classical and modern -- their comparison and role in algebraic geometry and in algebra of the 20th century. It also presents notable personalities of the history of mathematics who have contributed to the creation and development of the concept of multiplicity. multiplicity of a root of an algebraic equation; multiplicity of a point of an algebraic variety; intersection multiplicity of algebraic varieties at a point; Weil's multiplicity; Hilbert-Samuel's multiplicity History of algebraic geometry, Elementary questions in algebraic geometry, Relevant commutative algebra, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Multiplicity in algebraic geometry (historical development of the concept) | 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 \(\pi:V_ p\to T_ p\) be the universal Teichmüller curve over the Teichmüller space \(T_ p\) of closed Riemann surfaces of genus \(p\geq 2\). The zero set of a suitable holomorphic section of a line bundle \(L\to V_ p\) defines a positive divisor on each closed Riemann surface \(\pi ^{-1}(t)\), \(t\in T_ p\). An explicit construction of such sections by Poincaré series produces divisors of any degree \(\geq 4p-4\), and linear algebra then produces divisors of any degree \(\geq 2p-2\), including divisors of Prym differentials. \textit{L. Bers} [Bull. Am. Math. Soc. 67, 206-210 (1961; Zbl 0102.067)] constructed canonical divisors by essentially the same methods. Teichmüller curve; positive divisor; Poincaré series; divisors of Prym differentials Compact Riemann surfaces and uniformization, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic) Positive divisors and Poincaré series on variable Riemann surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper the authors study the algebraic relations between solutions of autonomous first order non-linear differential equations \[\, P(u, u')=0,\quad P\in\mathcal{C}[X, Y]\,, \,\] where \(P\) is irreducible and \(\mathcal{C}\) is an algebraically closed field of characteristic zero. The main result states that any set of distinct non-constant solutions of such an equation of general type is algebraically independent over \(\mathcal{C}\). To obtain this result the authors use a geometrical approach based on the geometry of curves and their Jacobian varieties. In addition, applying the obtained result the authors prove that any non-constant solution \(\xi\in\mathcal{C}((z))\) of an autonomous first order non-linear equation of general type is not \(D^n\)-finite over \(\mathcal{C}\) for any \(n\). The authors also discuss formal solutions for such autonomous equations. first order ordinary differential equation; autonomous differential equation; non-linear scalar differential equation; differential field; generalized Jacobian; algebraic curve; algebraic dependence Abstract differential equations, Geometric methods in ordinary differential equations, Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain, Jacobians, Prym varieties Autonomous first order differential equations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The theory developed in this paper arose from two main sources. They are the theory of varieties of group representations developed recently by \textit{M. Culler} and the second author [ibid. 117, 109-146 (1983; Zbl 0529.57005)] and Thurston's construction of a compactification of Teichmüller space. As an application of their ideas, the authors give a new construction of this compactification. As they state, their ''methods are drawn from the mathematical mainstream, and therefore help to explain Thurston's results by putting them in a wider framework.''
The central topic of the paper is a construction of compactifications of real and complex algebraic varieties. While there is an obvious way to compactify curves, which was used by Culler-Shalen, the problem of compactification of higher dimensional varieties is anything but routine. The authors' approach to this problem is motivated by the construction of Thurston's compactification of Teichmüller space. They consider an affine algebraic set V and an indexed family \((f_ j)_{j\in J}={\mathcal F}\) with countable index set J of functions which belong to the coordinate ring of V and generate it as an algebra. A compactification of V is canonically defined by \({\mathcal F}\) as follows. Let \({\mathcal P}\) be the quotient of \([0,\infty)^ J\setminus \{0\}\), where \([0,\infty)^ J\) is the Cartesian power, by the diagonal action of positive reals: \(\alpha (t_ j)_{j\in J}=(\alpha t_ j)_{j\in J}.\) Define a map \(\theta\) : \(V\to {\mathcal P}\) by \(\theta (x)=[\log (| f_ j(x)| +2)]_{j\in J}.\) Then the closure of \(\theta\) (V) in \({\mathcal P}\) is compact. This closure is the compactification in question. This compactification is studied on three different levels of generality.
The first one is that of a general variety. This is the theme of Chapter I. The points added to V are interpreted as valuations of the coordinate ring of V over a countable field of definition of V. These valuations are neither discrete nor of rank 1 in general. An important result says that there is a dense subset of added points consisting of discrete, rank 1 valuations.
At the second level V specializes to be the variety of characters X(\(\Gamma)\) of representations of a discrete group \(\Gamma\) in \(SL_ 2({\mathbb{C}})\). In this case there is a natural choice of \({\mathcal F}\). The corresponding \(f_ j\) are the values of characters on conjugacy classes in \(\Gamma\). Now the added points can be interpreted as actions of \(\Gamma\) on some generalized trees. On the vertices of an ordinary tree there is an integer-valued distance function. On generalized trees a similar distance function takes values in an ordered abelian group. The most important case is that of a subgroup of \({\mathbb{R}}\). The theory of such trees is developed from scratch up to a generalization of the well known Bass-Serre theory of trees associated to \(SL_ 2(F)\), where F is a field. While in the Bass-Serre theory a tree is associated with a discrete valuation of F, here the valuation can be nondiscrete. These trees are used for compactification. All this is the theme of Chapter II.
Finally, in Chapter III this theory is applied to the case \(\Gamma =\pi_ 1(S)\), where S is a surface. The Teichmüller space of S turns out to be a component of X(\(\Gamma)\) and the compactification of X(\(\Gamma)\) constructed in Chapter II leads to the Thurston's compactification of Teichmüller space.
In the second part of this paper, existing now in preprint form, these ideas are applied to the study of 3-manifolds. In particular, another important result of Thurston is re-proved from an entirely new point of view. representations of a dicrete group in \(SL_ 2({\mathbb{C}})\); actions on generalized trees; hyperbolic structures on surfaces; varieties of group representations; compactification of Teichmüller space; compactifications of real and complex algebraic varieties; affine algebraic set; valuations of the coordinate ring J. Morgan, P. Shalen. Valuations, trees, and degenerations of hyperbolic structures. I, \textit{Ann. of Math. } 120 (1984), 401--476. General low-dimensional topology, Abelian varieties and schemes, Valuations and their generalizations for commutative rings, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Compactification of analytic spaces, Group rings of finite groups and their modules (group-theoretic aspects), Classification theory of Riemann surfaces, Topology of Euclidean 2-space, 2-manifolds, Topology of general 3-manifolds Valuations, trees, and degenerations of hyperbolic structures. 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 In a series of papers [see, for instance, \textit{V. V. Batyrev} and \textit{Yu. I. Manin}, Math. Ann. 286, 27-43 (1990; Zbl 0679.14008); \textit{E. Peyre}, Duke Math. J. 79, No. 1, 101-218 (1995; Zbl 0901.14025); \textit{V. V. Batyrev} and \textit{Yu. Tschinkel} in: Nombre et repartition de points de hauteur bornées, Astérisque 251, 299-340 (1998; Zbl 0926.11045)], the authors and their collaborators have advanced a conjecture relating the asymptotic behaviour of the counting function for the number of rational points of bounded height on a Fano variety defined over a number field to the geometric invariants of this variety. In this paper, that conjecture is tested numerically for diagonal cubic surfaces \(V_a\) given by the equation \(\sum^3_{i=0}a_i x_i^3=0\), \(a\in \mathbb{Z}^4\), \(\prod^3_{i=0}a_i\neq 0\). Making use of the efficient algorithm developed by \textit{D. J. Bernstein} [Math. Comput. 70, No. 233, 389-394 (2001; Zbl 0960.11055)], the authors count points of height \(\leq 10^5\) for six values of \(a\), they treat, in particular, the surfaces \(V_{(1,1, 1,2)}\) and \(V_{(1,1,1,3)}\) investigated by \textit{D. R. Heath-Brown} [Math. Comput. 59, No. 200, 613-623 (1992; Zbl 0778.11017)] from a somewhat different point of view. Picard group; Tamagawa number; Brauer-Manin obstruction; Zbl 0991.72285; asymptotic behaviour; counting function; number of rational points of bounded height; Fano variety; geometric invariants; diagonal cubic surfaces; algorithm Peyre, E.; Tschinkel, Y., \textit{Tamagawa numbers of diagonal cubic surfaces, numerical evidence}, Math. Comp., 70, 367-387, (2001) Rational points, Fano varieties, Heights, Cubic and quartic Diophantine equations, Arithmetic varieties and schemes; Arakelov theory; heights Tamagawa numbers of diagonal cubic surfaces, numerical evidence | 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 over \({\mathbb{C}}\). We say that C is f-gonal if there exists a surjective morphism \(\phi:C\to {\mathbb{P}}^ 1\) of degree f. We assume C is of genus \(g\geq 2\) and that a base point \(P_ 0\in C\) is fixed. For a positive integer d a morphism I(d) from \(C^{(d)}\) to the Jacobian variety J(C) is defined by \(D\mapsto the\quad linear\quad equivalence\quad class\quad [D-P_ 0].\) For \(r\geq 0\), we put \(W^ r_ d=\{x\in J(C)| \dim(I(d)^{-1}(x))\geq r\},\) which is a Zariski closed subset of J(C). It is easy to see that C is d-gonal if and only if there exists a point in \(W^ 1_ d\) which is not contained in \(W^ 1_{d-1}+W^ 0_ 1\). Now we assume \(f\geq 3\) and that there exists no covering \(\phi:C\to \tilde C\) such that \(1<\deg(\phi)<f.\) Under this assumption the author gives sufficient conditions, concerning \(\dim(W^ r_ d)\) for some r, d, for C to be f-gonal, and he also proves that C is d-gonal for each \(d\geq g-f+2.\) These are the main results in the paper. In the last part of the paper some refinements of the above are tried in some examples. f-gonal smooth curve; Jacobian variety Coppens, M.R.M.: Some sufficient conditions for the gonality of a smooth curve. J. Pure Appl. Alg.30, 5--21 (1983) Jacobians, Prym varieties Some sufficient conditions for the gonality of a smooth 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 authors compute the dimension of the Deligne-Mumford compactification of the universal difference variety. For a curve of genus \(g\), if \(i=\lfloor \frac{g+1}{2}\rfloor\) then we always have a surjective map \(C_i\times C_i\to J(C)\). In the case that \(g\) is even, the map is finite and if \(i\) is odd, then \(C_{i-1}\times C_{i-1}\to J(C)\) gives a natural divisor on \(J(C)\). These maps can be interpreted, on a non-hyperelliptic curve, as a resolution of singularities for a natural divisor on \(J(C)\), defined by setting \(Q_C\) to be the dual of the kernel of \(H^0(C,K_C)\otimes \mathcal{O}_C\to K_C\), and taking the divisor to be \(\Theta_{\bigwedge^{i-1}Q_C}=\{\xi\in J(C)|h^0(C,\bigwedge^{i-1}Q_C\otimes \xi)\geq 1\}\). This resolution is analogous to the resolution of the classical theta divisor by the Abel-Jacobi map \(C_{g-1}\to \mathrm{Pic}^{g-1}(C)\).
The authors are motivated by the connection to Green's conjecture, which on a curve of odd genus and maximal Clifford index is true if and only if the natural map \(\bigwedge^{i-1}H^0(C,K_C)^\vee\to H^0(C,\bigwedge^{i-1}Q_C)\) is an isomorphism, and the codomain is related to the singularities of \(\Theta_{\bigwedge^{i-1}Q_C}\) by [\textit{Y. Laszlo}, Duke Math. J. 64, No. 2, 333--347 (1991; Zbl 0753.14023)].
Additionally, they construct a divisor on \(\overline{\mathcal{M}}_{g,g-3}\), compute its class, and use it to find a class on \(C_{g-3}\) which then they conjecture to span an extremal ray in the effective cone of \(C_{g-3}\). moduli space of curves; Kodaira dimension; difference variety; Jacobian variety Farkas, G.; Verra, A.: The universal difference variety over m\?g. Rend. circ. Mat. Palermo (2) 62, 97-110 (2013) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Theta functions and curves; Schottky problem The universal difference variety over \(\overline{\mathcal M}_g\) | 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 algebraic curve in projective \(n\)-space is called unramified if the degree of any divisor cut out by a real hyperplane overcomes the number of points in the divisor by at most \(n-1\) (here a pair of imaginary conjugate points is thought of as one point). Over the complex field, the unramified curves are known to be rational normal. Real plane unramified curves are conics [\textit{J. Huisman}, An unramified real plane curve is a conic, Matematiche 55, 459--467 (2000); \url{http:// fraise.univ-brest.fr/~huisman/recherche/publications.html}].
In the present note, the author shows that a real plane unramified nonspecial curve in an even-dimensional space is rational normal, and in an odd-dimensional space he shows that it is an \(M\)-curve consisting of pseudo-lines, provided that the curve under consideration has many real branches and, possibly, few ovals. The proof is based on the study of coverings of the real part of the Picard variety by unions of real branches of a curve. rational normal curve; \(M\)-curve; unramified curve; Picard variety Real algebraic sets, Plane and space curves, Coverings in algebraic geometry Unramified nonspecial real space curves having many real branches and few ovals. | 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 the set of all compatible preorders on the Picard group of a commutative ring forms a lattice under given operations. Also we discuss the Grothendieck group of a semigroup. compatible preorders; Picard group of a commutative ring; lattice; Grothendieck group of a semigroup Ordered groups, Semigroups, Picard groups, Class groups, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Grothendieck groups (category-theoretic aspects) Lattices of compatible preorders on Picard groups of commutative rings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper the author constructs an explicit theory of heights for the Jacobian variety \({\mathcal J}\) of a hyperelliptic curve of genus \(g\) defined over a number field \(K\). To define such a height \(H\) one needs to obtain two constants \(C_1\) and \(C_2\) such that for every \(P,G\in{\mathcal J}\) we have \(H(P+Q) H(P-Q) \leq C_1 H(P)^2H(Q)^2\) and \(H(2P) \geq H(P)^4/ C_2 \).
Let \({\mathbf v}=(v_i)\) and \({\mathbf w}=(w_i)\) be indeterminates, denote by \(K \left[\begin{smallmatrix} \mathbf v^2 & \;& \mathbf w^2 \\ & , \end{smallmatrix} \right]\) the ring of quartic polynomials homogeneous of degree 2 in each \(v_0, \dots, v_n\) and \(w_0, \dots, w_n\). Let \(K[\mathbf v^{4}]\) be the ring of quartic polynomials homogeneous of degree 4 in \(v_0, \dots, v_n\) and for any ring \(R\) denote by \(\mathbb{M}^n(R)\) the set of all \((n+1) \times (n+1)\) matrices modulo scalar multiplication. Then he defines heights on \(\mathbb{M}^n(K)\), \(\mathbb{M}^n (K \left[\begin{smallmatrix} \mathbf v^2 & \;& \mathbf w^2 \\ & , \end{smallmatrix} \right])\) and \(\mathbb{P}^n (K[\mathbf v^{4}])\) similar to the naive height on the projective space. There are four steps in the construction.
a) Fix an embedding \(\kappa: {\mathcal J} (K)\to \mathbb{P}^{2^g-1} (K)\) together with \(M \in \mathbb{M}^n (K \left[\begin{smallmatrix} \mathbf v^2 & \;& \mathbf w^2 \\& ,\end{smallmatrix} \right])\) and \(N\in \mathbb{P}^{2^g-1} (K[\mathbf v^4 ])\) satisfying; \((\kappa_i (P+Q) \kappa_j (P-Q)+ \kappa_j (P+Q) \kappa_i (P-Q)= M (\kappa (P), \kappa(Q))\) and \(\kappa (2P)= N(\kappa (P))\) for any \(P,Q \in {\mathcal J}\). The height \(H_\kappa\) is then defined as \(H_\kappa (P)= H(\kappa(P))\).
b) Take \(C_1\) equal to \(2H(M)\).
c) Factor the duplication law \(N\) via isogeny as \(N=W_1\tau W_2\tau W_3\), where \(W_1, W_2, W_3\in \mathbb{M}^{2^g-1}(L)\), for some finite extension \(L\) of \(K\), \(\tau:(v_i) \mapsto (v_i^2) \).
d) Take \(C_2\) equal to \(H(W_1^{-1}) H(W_2^{-1})^2 H(W_3^{-1})^4\).
In the second section of the paper the author demonstrates this process with an elliptic curve and in the third section he computes explicitly the matrices \(W_1\), \(W_2\) and \(W_3\) in the case of \(g=2\). heights; Jacobian variety; hyperelliptic curve E. V. Flynn, An explicit theory of heights , Trans. Amer. Math. Soc. 347 (1995), no. 8, 3003-3015. JSTOR: Curves of arbitrary genus or genus \(\ne 1\) over global fields, Jacobians, Prym varieties An explicit theory of heights | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper we give an explicit construction of a minimal set of generators for the ideal of a monomial curve in \(\mathbb{A}^ 5\). For this, for any element belonging to the associated semigroup, we consider a graph and we interpret the non-connected graphs in arithmetical terms. curve in affine 5-space; semigroup associated to monomial curve; minimal set of generators for the ideal of a monomial curve Campillo, A. and Pisón, P.: Generators of a monomial curve and graphs for the associated semigroup. Bull. Soc. Math. Belg. Sér. A 45 (1993), no. 1-2, 45-58. Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Polynomial rings and ideals; rings of integer-valued polynomials Generators of a monomial curve and graphs for the associated semigroup | 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 asymptotic estimate for large n is given for the number of sections of a Hermitian vector bundle twisted n times with a Hermitian invertible line bundle, given over an arithmetic base variety, in particular showing the existence of a section of norm less than 1. arithmetic variety; number of sections of a Hermitian vector bundle H. Gillet and C. Soulé, Amplitude arithmétique, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 887--890. Arithmetic problems in algebraic geometry; Diophantine geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves Amplitude arithmétique. (Arithmetic ampleness) | 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 reviews the group law algorithms of nonsingular \(C_{ab}\) curves, and provides implementation results comparing it to an algorithm for computing in the generalized Jacobian of \(C_A\) curves. survey of the group law algorithms; generalized Jacobian; hyperelliptic curves; cryptography; Arita-Miura-Sekiguchi algorithm Applications to coding theory and cryptography of arithmetic geometry, Finite ground fields in algebraic geometry, Jacobians, Prym varieties, Cryptography Group law algorithms for Jacobian varieties of curves over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper deals with Hilbert's 16th problem and its generalizations. The configurations of all closed branches of an algebraic curve of degree \(n\) are discussed. The maximum number of sheets for an algebraic equation of degree \(n\) and the maximum number of limit cycles for a planar algebraic autonomous system are achieved. The author also considers different generalizations and some related problems. configuration of branches of an algebraic curve; Harnack theorem; number of limit cycles for a polynomial planar system; Hilbert's 16th problem Enumerative problems (combinatorial problems) in algebraic geometry, Special algebraic curves and curves of low genus, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations Hilbert's sixteenth problem and its generalization | 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 H be a sequence of positive integers which is the Hilbert function of some reduced irreducible arithmetically Cohen-Macaulay curve C in \({\mathbb{P}}^ 3\); the authors find the possible degrees of the elements of a minimal set of generators for the homogeneous ideal of C. - For any such H and choice of allowable degrees they construct arithmetically Cohen-Macaulay curves, union of suitable lines, having minimal generators of the assigned degrees and Hilbert function H. Furthermore, by liaison techniques and Bertini-type theorems, they can get a smooth irreducible curve with assigned Hilbert function and generators' degrees. The case when H is the Hilbert-function of a complete intersection is studied in detail. Hilbert function of reduced irreducible arithmetically Cohen-Macaulay curve; linkage; minimal generators; liaison; Hilbert-function of a complete intersection Maggioni, R.; Ragusa, A.: Construction of smooth curves of P3 with assigned Hilbert function and generators' degrees. Le matematiche 42 (1987) Plane and space curves, Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Complete intersections, Linkage Construction of smooth curves of \({\mathbb{P}}^ 3\) with assigned Hilbert function and generators' degrees | 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 smooth projective curve of genus \(g>1\) over \(\mathbb{C}\) and \({\mathcal U}(r,d)\) the moduli space of semistable vector bundles of rank \(r\) and degree \(d\) on \(X\). The Picard Sheaf \({\mathcal E}_{r,d}\) on \({\mathcal U}(r,d)\) is the direct image sheaf of a universal Poincaré sheaf on \(X\times {\mathcal U}(r,d)\). Drezet and Narasimhan constructed a generalized theta divisor \(\Theta_{{\mathcal U}(r,d)}\) on \({\mathcal U}(r,d)\). The main theorem of the paper under review states that for relatively prime integers \(r\) and \(d\) with \(d>2gr\) the Picard Sheaf \({\mathcal E}_{r,d}\) is \(\Theta_{{\mathcal U}(r,d)}\)-stable. For the proof the author chooses a spectral curve \(Y\) over \(X\). An open subset \(T\) of the Jacobian \(J(Y)\) dominates \({\mathcal U}(r,d)\). It is shown that the pull back of \(\Theta_{{\mathcal U}(r,d)}\) to \(T\) is a power of the usual theta divisor on \(J^ (Y)\) restricted to \(T\). spectral curves; stability of Picard sheaf; generalized theta divisor Li, Y., Spectral curves, theta divisors and Picard bundles, Int. J. Math., 2, 525-550, (1991) Vector bundles on curves and their moduli, Theta functions and abelian varieties, Picard groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Spectral curves, theta divisors and Picard bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\subset\mathbb{P}^ 3\) be a curve. \(X\) is called subcanonical if \(\omega_ X\cong{\mathcal O}_ X(e)\) for some integer \(e\). A classical theorem of Gherardelli and Petri says that an integral arithmetically Cohen-Macaulay space curve is a complete intersection. Here the author proves new and old refinements of this theorem (mainly related to numerically subcanonical curves, i.e. with \(h^ 1(X,{\mathcal O}_ X(t))=h^ 0(X,{\mathcal O}_ X(e-t))\) for every \(t)\). The main point of the author is that all these results follow from Castelnuovo's method based on the study of the Hilbert function of a general hyperplane section. postulation; arithmetically Cohen-Macaulay space curve; complete intersection; numerically subcanonical curves; Hilbert function of a general hyperplane section Plane and space curves, Projective techniques in algebraic geometry, Complete intersections Complements to a theorem of Gherardelli: The postulational viewpoint | 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 With every smooth, projective algebraic curve \(\widetilde C\) with involution \(\sigma : \widetilde C \to \widetilde C\) without fixed points is associated the Prym data which consists of the Prym variety \(P : = (1 - \sigma) J (\widetilde C)\) with principal polarization \(\Xi\) such that \(2 \Xi\) is algebraically equivalent to the restriction on \(P\) of the canonical polarization \(\Theta\) of the Jacobian \(J (\widetilde C)\). In contrast to the classical Torelli theorem the Prym data does not always determine uniquely the pair \((\widetilde C, \sigma)\) up to isomorphism. In this paper we introduce an extension of the Prym data as follows. We consider all symmetric theta divisors \(\Theta\) of \(J (\widetilde C)\) which have even multiplicity at every point of order 2 of \(P\). It turns out that they form three \(P_ 2\) orbits. The restrictions on \(P\) of the divisors of one of the orbits form the orbit \(\{2 \Xi \}\), where \(\Xi\) are the symmetric theta divisors of \(P\). The other restrictions form two \(P_ 2\)-orbits \(O_ 1,O_ 2 \subset | 2 \Xi |\). The extended Prym data consists of \((P, \Xi)\) together with \(O_ 1,O_ 2\). We prove that it determines uniquely the pair \((\widetilde C, \sigma)\) up to isomorphism provided \(g (\widetilde C) \geq 3\). The proof is similar to Andreotti's proof of Torelli's theorem and uses the Gauss map for the divisors of \(O_ 1,O_ 2\). Separate treatment is necessary in the hyperelliptic case, the bi-elliptic case and the case of \(g (\widetilde C) = 5\).
The result is an analogue in genus \(>1\) of the following classically known fact for elliptic curves: Any pair \((E = \mathbb{C}/ \mathbb{Z} \tau + \mathbb{Z}\), \(\mu = {1 \over 2} \tau + {1 \over 2} \pmod {\mathbb{Z} \tau + \mathbb{Z}})\) is determined uniquely up to isomorphism by \(k(\tau) = \lambda (\tau) + 1/ \lambda (\tau)\) where \(\lambda (\tau) = - \theta_{01} (0, \tau)^ 4/ \theta_{10} (0, \tau)^ 4\). polarization of the Jacobian; Prym variety; Torelli theorem; extended Prym data; Gauss map V. Kanev,Recovering of curves with an involution by extended Prym data, Math. Annalen299 (1994), 391--414. Theta functions and abelian varieties, Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Picard schemes, higher Jacobians Recovering of curves with involution by extended Prym data | 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 establish one direction of a conjecture by \textit{N. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45-52 (1990; Zbl 0714.14033)] which describes combinatorially the singular locus of a Schubert variety. We prove that the conjectured singular locus is contained in the singular locus. singular locus of a Schubert variety Gasharov, Vesselin, Sufficiency of Lakshmibai-Sandhya singularity conditions for Schubert varieties, Compos. Math., 126, 1, 47-56, (2001), MR 1827861 Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorics of partially ordered sets Sufficiency of Lakshmibai-Sandhya singularity conditions for 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 An algorithm is presented for computing the topological type of a nonsingular real-algebraic curve on a projective plane. The topological type is a structure including (1) the parity of the degree of the curve; (2) the number of ovals into which the curve splits; (3) partial ordering of ovals by inclusion.
The algorithm works for curves defined by integral homogeneous polynomials. It is based on cylindrical algebraic decomposition and has polynomial complexity assessed as a nice \(O(n^{27}L(d)^ 3)\) where n is the degree of the defining polynomial and L(d) is the total length of the coefficients. computing the topological type of a nonsingular real-algebraic curve on a projective plane; ovals Arnon D., McCallum S.: A polynomial time algorithm for the topological type of a real algebraic curve. J. Symb. Comput. 5, 213--236 (1988) Special algebraic curves and curves of low genus, Real algebraic and real-analytic geometry, Software, source code, etc. for problems pertaining to algebraic geometry, Projective techniques in algebraic geometry A polynomial-time algorithm for the topological type of real algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0639.00032.]
The author develops an interesting formalism based on integration over algebraic variety relative to the measure given by the Euler characteristic. This allows him to treat several classical questions from a unified point of view such as Plücker formulas for plane curves, their analogs for real algebraic curves (Klein's formulas) and for surfaces in \({\mathbb{P}}^ 3.\)
To give some flavor of this approach let X be a projective variety over \({\mathbb{C}}\) or \({\mathbb{R}}\), \({\mathfrak a}\) a collection of semi-algebraic sets in X. Then for \(B\in {\mathfrak a}\) and \(f=\sum \lambda_ A\cdot 1_ A \) where \(1_ C\) is the characteristic function of C, one defines
\[
\int f(x)d\chi (x) =\sum \lambda_ A\chi (A\cup B).
\]
This integral satisfies Fubini's theorem which turns out to be a generalization of the Riemann-Hurwitz formula. Moreover one can define a Radon type transform taking functions of the above type on a projective space for which integration is considered into functions on the dual projective space (i.e. the space of hyperplanes). Example of a typical integrable function is the multiplicity of a point in \({\mathbb{P}}^ 2\) on an algebraic curve A. Suitable integration of a local relation involving such function becomes a Plücker formula. In fact most applications of integration appear to connect local and global data. Perhaps this is why the author views this as a ``much more elementary and easier to use'' counterpart of sheaf theory.
Other applications include Rokhlin's formula on the complex orientations of real curves and Groemer's generalization of Minkowski mixed volume. integration over algebraic variety; Euler characteristic; multiplicity of a point; Plücker formula O. Ya. Viro, ''Some Integral Calculus Based on Euler Characteristic,'' in Topology and Geometry: Rohlin Seminar (Springer, Berlin, 1988), Lect. Notes Math. 1346, pp. 127--138. Topological properties in algebraic geometry, Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures, Integration on manifolds; measures on manifolds Some integral calculus based on Euler 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 [For the entire collection see Zbl 0561.00011.]
The author considers a moduli problem, namely, the classification of nilpotent endomorphisms of indecomposable semi-stable vector bundles of rank 2 over a compact connected Riemann surface. Defining first a global (universal) family of endomorphisms of vector bundles parametrized by a variety, and using the universal property of the Picard variety and the universal family of extensions given by \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math., II. Ser. 89, 14-51 (1969; Zbl 0186.549)], the author constructs a universal family of the above endomorphisms for P(2,d), where P(n,d) denotes the set of isomorphism classes of the pairs [E, ], E being vector bundles of rank n and d being the slope of E. endomorphisms of indecomposable semi-stable vector bundles; compact connected Riemann surface; Picard variety; universal family L. Brambila, Moduli of endomorphisms of vector bundles over a compact Riemann surface, preprint. Sheaves and cohomology of sections of holomorphic vector bundles, general results, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Compact Riemann surfaces and uniformization Endomorphisms of vector bundles over a compact Riemann surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0711.00011.]
Let \(f:X\to B\) be a semi-stable curve of relative genus \(\geq 2\) over the ring of integers of a number field, and let \(\omega_{X/B}\) be its relative dualizing sheaf, endowed with its Arakelov metric. As an analogue of the Bogomolov-Miyaoka inequality for surfaces of general type, \textit{A. N. Parshin} [cf. Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 299-312 (1988; Zbl 0706.14015)] has put forward a conjectural upper bound \(\text{BM}(X)\) for the Arakelov degree of the self-intersection \(e(X)\) of \(\omega_{X/B}\), and shown that the universal truth of \(\text{BM}(X)\) implies a sharpened version \(\text{SZ}(E)\) of a well-known conjecture of Szpiro, bounding the height of a semi-stable elliptic curve \(E\) over \(B\) by its conductor.
In the present paper, the authors consider the case where \(X\) is a curve \(C\) of genus 2 covering \(E\). They make a close study of the Weierstrass sections of \(C\), and relate \(e(C)\) to the degree of their intersections [a similar relation, which holds for all curves of genus 2, is used by \textit{J.-B. Bost}, \textit{J.-F. Mestre} and \textit{L. Moret-Bailly} in Les princeaux de courbes elliptiques, Sémin., Paris/Fr. 1988, Astérisque 183, 69-105 (1990; Zbl 0731.14017) to show how cautious one must be in formulating \(\text{BM}\)]. The authors finally put forward a new conjecture of BM type, bounding \(e(X)\) in terms of the conductor of the jacobian of \(X\), and show that its truth for \(C\) itself suffices to imply \(\text{SZ}(E)\) as soon as the height of the \(j\)-invariant of \(E\) is given by its denominator. For a deduction of \(\text{SZ}(E)\) from other particular cases of \(\text{BM}(X)\), see also \textit{L. Moret-Bailly}, in the same cited seminar, Astérisque 183, 37-58 (1990; Zbl 0727.14015)]. conductor; upper bound for the Arakelov degree; Szpiro conjecture; Arakelov metric; bounding the height of a semi-stable elliptic curve; Weierstrass sections Frey, Gerhard; Kani, Ernst, Curves of genus \(2\) covering elliptic curves and an arithmetical application.Arithmetic algebraic geometry, Texel, 1989, Progr. Math. 89, 153-176, (1991), Birkhäuser Boston, Boston, MA Arithmetic varieties and schemes; Arakelov theory; heights, Elliptic curves, Local ground fields in algebraic geometry, Elliptic curves over global fields Curves of genus 2 covering elliptic curves and an arithmetical application | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author gives a method of computing topological fundamental groups of complex algebraic varieties by expressing them as fibres of good fibrations and relating their fundamental groups to those of the total space of the fibration. Let \(Y\) and \(B\) be nonsingular algebraic varieties over \(\mathbb{C}\). Let \(p: Y\to B\) be a dominant morphism with general fibre \(Y_0\) connected. Assume that \(p\) has a section \(s\) over a classical open set \(U\subset B\). Under certain conditions on \(s\) and \(U\), the author determines the kernel of the morphism \(\pi_1 (Y_0, s(0))\to \pi_1 (p^{-1} (U), s(0))\). He also gives a set of sufficient conditions for this kernel to be trivial. This is used to prove the main theorem:
Let \(F\) be a nonsingular connected quasi-projective variety of dimension \(\geq 2\) and \(\overline {F}\) its nonsingular projective compactification such that \(\overline {F}- F\) is of codimension \(\geq 2\) in \(\overline {F}\). Let \(W\subset F\times \mathbb{A}^n\) be a reduced (possibly reducible) effective divisor, \(W_a=\) fibre over \(a\in \mathbb{A}^n\). Assume that the locus of \(a\in \mathbb{A}^n\) such that \(W_a\) is not a reduced divisor in \(F\) is of codimension \(\geq 2\) in \(\mathbb{A}^n\). Then the natural homomorphism \(\pi_1 (F-W_a)\to \pi_1 ((F\times \mathbb{A}^n- W)\) is an isomorphism for a general \(a\in \mathbb{A}^n\).
Two applications of the theorem are given. For the first, consider general enough hyperplanes \(V_i\subset \mathbb{P}^n\), degree \(V_i= d_i\), \(i=1, \dots, m\). Let \(U\) be the subset of the Grassmannian of lines in \(\mathbb{P}^n\) consisting of all lines which intersect \(V_1\cup \dots \cup V_m\) in distinct \(d_1+ \cdots+ d_m\) points and \(U_p\) the subset of \(U\) consisting of lines passing through a fixed point \(p\in \mathbb{P}^n\). The author shows that for \(n\geq 3\) and \(\sum d_i\geq 3\), the natural monodromy representation of \(\pi_1 (U)\) in the generalized mapping class group is surjective with kernel an abelian group generated by at most \(m\) elements contained in the centre of \(\pi_1 (U)\). Further, if \(p\) is also general, then this representation is an isomorphism. The second application is the determination of the fundamental group of the complement of a projective plane curve \(f^q+ g^p =0\), \(p\geq 2\) and \(q\geq 2\) being degrees of the two general homogeneous polynomials \(f\) and \(g\). Such results are due to \textit{O. Zariski} [Am. J. Math. 51, 305-328 (1929)], \textit{M. Oka} [Math. Ann. 218, 55-65 (1975; Zbl 0335.14005) and J. Math. Soc. Japan 30, 579-597 (1978; Zbl 0387.14004)]\ and \textit{A. Nemethi} [Math. Proc. Camb. Philos. Soc. 102, 453-457 (1987; Zbl 0679.14004)]. Zariski problem; topological fundamental groups of complex algebraic varieties; good fibrations; mapping class group; fundamental group of the complement of a projective plane curve DOI: 10.1016/0040-9383(94)00045-M Homotopy theory and fundamental groups in algebraic geometry, Fundamental group, presentations, free differential calculus Fundamental groups of open 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 In most definitions and notations the author follows \textit{G. Faltings}' paper, Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005). Author's abstract: ``We give a formula for the Arakelov-Green function G of a compact connected Riemann surface X of genus \(g\geq 1\), which involves the integral of a Néron function on the theta divisor of the jacobian variety of X. When \(g=2\), we deduce from this formula an effective estimate of G and integral formulae for the invariant \(\delta\) (X) of Faltings.'' In section \(7\) of the above cited paper Faltings makes things explicit for curves of genus \(1\) in these problems. genus two; Pic; Arakelov-Green function; Riemann surface; theta divisor of the jacobian variety Bost, J.-B.: Fonctions de Green-Arakelov, fonctions thêta et courbes de genre 2. C. R. Acad. Sci. Paris Sér. I Math. \textbf{305}(14), 643-646 (1987) Arithmetic ground fields for curves, Jacobians, Prym varieties, Theta functions and abelian varieties Fonctions de Green-Arakelov, fonctions thêta et courbes de genre 2. (Green-Arakelov functions, theta functions and curves of genus 2) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0755.00010.]
A curve of genus \(g=7\) with the maximal number of automorphisms \((=84(g- 1)=504)\) was given by \textit{A. M. Macbeath} in Proc. Lond. Math. Soc., III. Ser. 15, 527-542 (1965; Zbl 0146.427)]. Its group of automorphisms is the simple group PSL(2,8). The curve can be realized as eight-sheeted covering of the sphere branched over the 7th roots of unity. In the article under review the authors combinatorically represent the monodromy of the covering and use these data to calculate a canonical homology basis of the Macbeath curve. This is done by a computer implementation of an algorithm introduced by \textit{C. L. Tretkoff} and \textit{M. D. Tretkoff} in Contributions to group theory, Contemp. Math. 33, 467-519 (1984; Zbl 0557.30036). With respect to the canonical homology basis the periods of a basis of the holomorphic differentials (given by Macbeath) are calculated. After normaliztion the entries of the \(7\times 7\) period matrix are given. They are rational expressions in a complex number \(t\), where \(t\) is the ratio of periods of a certain elliptic integral. The authors point out that these entries are algebraic numbers if and only if the associated elliptic curve has complex multiplication. This question was not investigated in the paper. periods of a holomorphic differentials; Macbeath curve; canonical homology basis Berry, Kevin; Tretkoff, Marvin, The period matrix of Macbeath's curve of genus seven. Curves, Jacobians, and abelian varieties, Amherst, MA, 1990, Contemp. Math. 136, 31-40, (1992), Amer. Math. Soc., Providence, RI Jacobians, Prym varieties, Period matrices, variation of Hodge structure; degenerations, Differentials on Riemann surfaces The period matrix of Macbeath's curve of genus seven | 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 \(C/\mathbb{Q}\) be a curve of genus 2, and let \(J\) be the Jacobian of \(C\). There is associated to \(J\) a canonical height function \(\hat{h}:J(\mathbb{Q})\to \mathbb{R}\) with the property that \(\hat{h}(nP)=n^2\hat{h}(P)\), and differing by at most a bounded amount from the naive Weil \(h\) height induced on \(J\) by the map to the Kummer surface \(J\to S\subseteq \mathbb{P}^3\). For various applications, such as finding generators for the Mordell-Weil group \(J(\mathbb{Q})\), one would like to be able to compute \(\hat{h}(P)\), and enumerate points \(P\) satisfying \(\hat{h}(P)\leq B\), for specified bound \(B\). There are known explicit bounds on the difference \(\hat{h}-h\), so in principle this is possible, but previous algorithms have been slow in practice.
This paper introduces a practical polynomial-time algorithm addressing the first of these problems. In particular, the authors show how to compute \(\hat{h}(P)\) in time that is quasilinear in the size of the coefficients of \(C\) and coordinates of \(P\), and quasi-quadratic in the desired digits of precision. The authors also present improved methods for enumerating points of bounded canonical height, in practice making previously some infeasible searches for rational points manageable. Although the main result is stated over \(\mathbb{Q}\), most of the results hold in much greater generality, and the main result is expected to hold over an arbitrary number field.
The main improvement to earlier efficient methods is to replace a factorization step, required to find primes of bad reduction, with a factorization into powers of coprimes due to \textit{D. J. Bernstein} [J. Algorithms 54, No. 1, 1--30 (2005; Zbl 1134.11352)]. canonical height; hyperelliptic curve; curve of genus 2; Jacobian surface; Kummer surface J.S. Müller, M. Stoll, Canonical heights on genus two Jacobians. Algebra & Number Theory 10(10), 2153-2234 (2016) Heights, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Abelian varieties of dimension \(> 1\), Arithmetic varieties and schemes; Arakelov theory; heights, Computational aspects of algebraic curves, Rational points Canonical heights on genus-2 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 The author [\textit{Yu. G. Zarhin}, Math. Res. Lett. 7, No. 1, 123-132 (2000; Zbl 0959.14013)] proved that in characteristic 0 the jacobian \(J(C)=J(C_f)\) of a hyperelliptic curve \(C=C_f: y^2=f(x)\) has only trivial endomorphisms over an algebraic closure \(K_a\) of the ground field \(K\) if the Galois group \(\text{Gal}(f)\) of the irreducible polynomial \(f\in K[x]\) is ``very big''. Namely, if \(n=\deg(f)\geq 5\) and \(\text{Gal}(f)\) is either the symmetric group \({\mathfrak S}_n\) or the altnernating group \({\mathfrak A}_n\), then the ring \(\text{End} (J(C_f))\) of \(K_a\)-endomorphisms of \(J(C_f)\) coincides with \(\mathbb{Z}\). Later the author [in: Moduli of abelian varieties, Proc. 3rd Texel Conf., Texel island 1999 Prog. Math. 195, 473-490 (2001; Zbl 1047.14015)] proved that \(\text{End} (J(C_f))=\mathbb{Z}\) for an infinite series of \(\text{Gal} (f)=\text{PSL}_2(\mathbb{F}_{2^r})\) and \(n=2^r+1\) (with \(\dim(J(C_f)) =2^{r-1})\) or when \(\text{Gal}(f)\) is the Suzuki group \({\mathfrak S}{\mathfrak z} (2^{2r+1})\) and \(n=2^{2(2r+1)} +1\) (with \(\dim(J(C_f))= 2^{4r+1})\).
We write \({\mathfrak R}={\mathfrak R}_f\) for the set of roots of \(f\) and consider \(\text{Gal}(f)\) as the corresponding permutation group of \({\mathfrak R}\). Suppose \(q=2^m>2\) is an integral power of 2 and \(\mathbb{F}_{q^2}\) is a finite field consisting of \(q^2\) elements. Let us consider a non-degenerate Hermitian (with respect to \(x\mapsto x^q)\) sesquilinear form on \(\mathbb{F}^3_{q^2}\). In the present paper we prove that \(\text{End}(J(C_f))= \mathbb{Z}\) when \({\mathfrak R}_f\) can be identified with the corresponding ``Hermitian curve'' of isotropic lines in the projective plane \(\mathbb{P}^2 (\mathbb{F}_{q^2})\) in such a way that \(\text{Gal}(f)\) becomes either the projective unitary group \(\text{PGU}_3 (\mathbb{F}_q)\) or the projective special unitary group \({\mathfrak U}_3(q): =\text{PSU}_3 (\mathbb{F}_q)\). In this case \(n=\deg (f)=q^3+1=2^{3m}+1\) and \(\dim(J (C_f))= q^3/2=2^{3m-1}\).
Our proof is based on an observation that the Steinberg representation is the only absolutely irreducible non-trivial representation (up to an isomorphism) over \(\mathbb{F}_2\) of \({\mathfrak U}_3(2^m)\), whose dimension is a power of 2. finite ground field; hyperelliptic Jacobian; endomorphisms of abelian variety; unitary group; Hermitean group; Galois group; Steinberg representation Zarhin Yu.G. (2003). Hyperelliptic jacobians and simple groups U3(2 m ). Proc. AMS 131: 95--102 Jacobians, Prym varieties, Finite ground fields in algebraic geometry, Algebraic theory of abelian varieties, Automorphisms of surfaces and higher-dimensional varieties, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Hyperelliptic Jacobians and simple groups \(U_3(2^m)\) | 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 point of a curve in the complex \(n\)-space is said to be a flattening point if the first \(n\) derivatives of a local regular parametrization are linearly dependent at the point. When a family of curves degenerates, the number of flattening points decreases, and a formula for the number of vanishing flattening points at a non-degenerate quasi-homogeneous singular point of planar type is given. flattening point of a curve; non-degenerate quasi-homogeneous singular point Singularities of curves, local rings, Singularities in algebraic geometry Vanishing flattening points and generalized Plücker formulas | 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 notion of height of varieties generalizes the corresponding notion of height of points. It is a basic arithmetic invariant of a proper variety over $\mathbb{Q}$. The article under review is devoted to the study of heights in toric varieties, a topic that has been treated in great generality by \textit{J. I. Burgos Gil} et al. [Arithmetic geometry of toric varieties. Metrics, measures and heights. Paris: Société Mathématique de France (SMF) (2014; Zbl 1311.14050)]. \par Let $\Sigma$ be a complete fan and let $X_\Sigma$ be the corresponding proper toric variety of dimension $n$ over $\mathbb{Q}$. Denote by $M$ the lattice of characters of the torus of $X_\Sigma$, and denote by $N$ its dual lattice. $M_{\mathbb{R}}$ and $N_{\mathbb{R}}$ will denote the corresponding real vector spaces. Let $\mathfrak{M}$ stand for the set of places of $\mathbb{Q}$. \par The theory developed in [loc. cit.], provides a good amount of height functions, i.e. those that arise from toric line bundles equipped with toric metrics. Moreover, these height functions are described in terms of the underlying combinatorics of $\Sigma$. The machinery developed there, is useful for computing heights of toric subvarieties but is not enough, for instance, to handle the case of a general cycle of codimension $1$ in $X_\Sigma$. This is exactly the main subject of the very well written work of Gualdi. The main result of this article is the following \par Theorem 1. The height of $Y$ with respect to $\overline{D}$ is given by \[ h_{\overline{D}}(Y)=\sum_{v\in \mathfrak{M}}MI_{M_{\mathbb{R}}}(\vartheta_v,\ldots,\vartheta_v,\rho_{f,v}^\vee). \] Here $Y$ is an irreducible hypersurface on $X_\Sigma$ such that the generic point of $Y$ lies in the dense open orbit of $X_\Sigma$, which implies that $Y$ can be described by an irreducible Laurent polynomial $f$ with rational coefficients. By $\overline{D}$ the author denotes a toric divisor on $X_\Sigma$, equipped with an adelic semipositive metric which is invariant under the action of the torus of $X_\Sigma$. Such a metric is associated with a family $\{\vartheta_v\}_{v\in\mathfrak{M}}$ of continuous concave functions on the polytope associated to $D$, such that, $\vartheta_v=0$ for all but finitely many $v$'s. The description of this association is detailed in Section 3. \par The function $\rho_{f,v}$, defined for a fixed place $v$ of $\mathbb{Q}$, is called a $v$-adic Ronkin function of $f$ and constitutes one of the key constructions of the article. It is related with the fibers of the tropicalization of $f$, see Section 2 for the details. The notation $\rho_{f,v}^\vee$ refers to the Legendre-Fenchel dual of $\rho_{f,v}$, which is a concave function on $M_\mathbb{R}$ supported on the Newton polytope of $f$. \par Finally, $MI_{M_{\mathbb{R}}}$ denotes the mixed integral defined by Burgos Gil et al. [loc. cit.]. This is a multilinear symmetric real-valued function obtained from a suitably normalized Haar measure on $M_\mathbb{R}$. \par Furthermore, all the announced results are presented in the general adelic Arakelov framework, i.e. for an arbitrary base adelic field, see Section 3. \par In developing the required tools for the proof of Theorem 1, the author also presents some extra results that are very welcome. They include some new properties of mixed integrals and a uniform study of $v$-adic Ronkin functions, in both, archimedean and nonarchimedean contexts. The author also describes combinatorially the Weil divisor of the rational function defined by a Laurent polynomial on a toric variety, see Section 4. At the end of the paper, Section 6, some well chosen examples are presented in order to relate the presented constructions with previous ones. toric variety; height of a variety; ronkin function; Legendre-Fenchel duality; mixed integral Toric varieties, Newton polyhedra, Okounkov bodies, Heights, Arithmetic varieties and schemes; Arakelov theory; heights, Mixed volumes and related topics in convex geometry Heights of hypersurfaces 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 In 1993 \textit{G. Faltings} [J. Algebr. Geom. 2, 507--568 (1993; Zbl 0790.14019)] gave a ``GIT-free'' construction of the moduli space of semistable vector bundles on smooth projective curve \(X\) over an algebraically clased field \(k\). This construction was presented by him at German-Spanish Workshop on Moduli spaces of Vector Bundles, Essen, February 2007.
The present notes give a self-contained and clear introduction to the moduli spaces of vector bundles on a curve and the generalized \(\Theta\)-divisor. For the sake of simplicity, all the exposition is devoted to the rank 2 case with determinant isomorphic to \(\omega_X\). Also there is a brief discussion of generalizations the arbitrary rank and degree, supplied with necessary references.
Notes contain following sections. -- Outline of the construction. -- Background and notation. -- A nice over-parameterizing family. -- The generalized \(\Theta\)-divisor. -- Raynaud's vanishing result for rank two bundles. -- Semistable limits. -- Positivity. -- The construction. -- Prospect to higher dimension. moduli space; vector bundles on a curve; generalized Theta divisor Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Faltings' construction of the moduli space of vector bundles on a smooth projective curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0707.00010.]
(From the introduction:) Let \(p\) be a rational prime. If \(A\) is an abelian variety with good reduction over an unramified extension \(K\) of \(\mathbb{Q}_ p\), then there is a natural Frobenius-linear endomorphism of \(H^ 1_{DR}(A,K)\). The Jacobian of the Fermat curve \(F_ m:x^ m+y^ m=1\) has good reduction over \(\mathbb{Q}_ p\), so long as \((m,p)=1\). In a previous work [cf. Galois representation and arithmetic algebraic geometry, Proc. Symp., Kyoto/Jap. 1985 and Tokyo/Jap. 1986, Adv. Stud. Pure Math. 12, 21-52 (1987; Zbl 0644.12008)] the author computed the matrix of the corresponding endomorphism with respect to the basis of \(H^ 1_{DR}(A,K)\) represented by the differentials:
\[
\omega_{m,i,j}=x^ iy^ j(y/x)d(x/y),
\]
where \(0<i<m\), \(0<j<m\), \(i+j\not= m\), in terms of the special values of the \(p\)-adic gamma function at rational arguments.
In the general case, the Jacobian of \(F_ m\) has good reduction over an algebraic closure of \(\mathbb{Q}_ p\), but not over an unramified extension of \(\mathbb{Q}_ p\). As a result, one no longer has a canonical Frobenius- linear endomorphism. Instead one has a canonical action of the crystalline Weil group on \(H^ 1_{DR}(F_ m,K)\). The corresponding ``Frobenius matrices'' with respect to the previous basis for cohomology are here calculated. They are expressed in terms of an extension of the Morita gamma function to all of \(\mathbb{Q}_ p\). The resulting formulas resemble the formula for the complex periods in terms of the classical gamma function. This paper is closely related to the paper of \textit{A. Ogus} published in the same volume [\(p\)-adic analysis, Lect. Notes Math. 1454, 319-341 (1990; see the following review)]. Frobenius-linear endomorphism of De Rham cohomology group; Jacobian of the Fermat curve; crystalline Weil group; Frobenius matrices; Morita gamma function R. Coleman, On the Frobenius matrices of Fermat curves, \textit{p}-adic analysis, Lecture Notes in Math. 1454, Springer, Berlin (1990), 173-193. Local ground fields in algebraic geometry, de Rham cohomology and algebraic geometry, Arithmetic ground fields for curves On the Frobenius matrices of Fermat curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be an algebraically closed field of any characteristic, and let \(X\) be a \(k\)-scheme, i.e. a separated scheme of finite type over \(k\). Denote by \(H^ \bullet(X)\) (and \(H^ \bullet_ c(X))\) the \(\ell\)-adic étale cohomology (with compact support) of \(X\). Thus if \(k\) has characteristic zero and is a subfield of \(\mathbb{C}\) one may also take rational singular cohomology. \(H^ \bullet(X)\) and \(H^ \bullet_ c(X)\) carry a weight filtration \(W_ \bullet\), and one defines the (pure) Euler characteristics of \(X\) by \(\chi_ m(X)=\sum_ i(-1)^ i\dim Gr^ W_ mH^ i(X)\), and similarly \(\chi^ m_ c(X)\) for \(Gr^ W_ mH^ i_ c(X)\). The (pure) Poincaré polynomial of \(X\) is defined as \(P_{\text{pur}}(X,t)=\sum_ m\chi_ m(X)t^ m\), and similarly \(P^ c_{\text{pur}}(X,t)=\sum_ m\chi^ m_ c(X)t^ m\). For a \(k\)-scheme \(X\) one can define a stratification or, a little more generally, a (filterable) decomposition. Such a stratification (or filterable decomposition) \(X=\bigcup X_ \alpha\) is called perfect if \(X\) and also the cells \(X_ \alpha\) are smooth and connected and one has the following relation between the Betti numbers of \(X\) and the \(X_ \alpha:b_ i(X)=\sum_ \alpha b_{i-2d_ \alpha}(X_ \alpha)\), where \(d_ \alpha\) is the codimension of \(X_ \alpha\) in \(X\). As a first result it is shown that for a smooth, connected, complete \(k\)-scheme \(X\) with Białynicki-Birula decomposition \(X=\bigcup X_ \alpha\) and with an action of an algebraic torus \(T\), one has \(b_ i(X)-\sum_ \alpha b_{i-2d_ \alpha}(X^ T_ \alpha)\) for all \(i\), where \(X^ T=\bigcup X^ T_ \alpha\) is the decomposition of the \(T\)-fixed part \(X^ T\) of \(X\) into connected components.
Next it is shown that one can define, for a linear \(k\)-algebraic group acting on a \(k\)-scheme \(X\), pure Euler characteristics for the equivariant cohomology
\[
H^ i_ G(X): \chi^ m_ G(X)=\sum_ i(- 1)^ i\dim Gr^ W_ mH^ i_ G(X).
\]
Similarly for the equivariant cohomology with compact support \(H^ i_{G,c}(X)\) one defines \(\chi^ m_{G,c}(X)\). Also, the pure equivariant Poincaré series of \(X\) is defined: \(P^ G_{\text{pur}}(X,t)=\sum_ m\chi^ m_ G(X)t^ m\). One can define (filterable) \(G\)-decompositions and \(G\)-stratifications of the \(G\)-scheme \(X\) in a straightforward way.
The final result says that for a smooth connected \(G\)-scheme \(X\) with \(G\)-decomposition \(X=\bigcup X_ \alpha\) such that the \(X_ \alpha\) are smooth, equidimensional of codimension \(d_ \alpha\) in \(X\), one has \(\chi^ m_ G(X)=\sum_ \alpha\chi_ G^{m-2d_ \alpha}(X_ \alpha)\) and \(P^ G_{\text{pur}}(X,T)=\sum_ \alpha t^{2d_ \alpha}P^ G_{\text{pur}}(X_ \alpha,t)\). This is applied to obtain inductively the Betti numbers of a symplectic or a geometric quotient of a variety. This leads to formulas obtained before by \textit{F. Kirwan}. A second application concerns the calculation of the Betti numbers of geometric quotients as done by Białynicki-Birula and Sommese.
The last section of the paper gives a motivic version of the preceding results. A motivic Euler characteristic \(\chi_{\text{mot}}(X)\) is defined by means of the system of mixed realizations \(SRM^ \bullet(X)\) defined by \(H^ \bullet(X)\). Similarly for compact support. In the situation of the Białynick-Birula decomposition of \(X\) with torus action as sketched above, one finds a relation for the Hodge numbers of \(X\) and the \(X^ T_ \alpha:h^{p,q}(X)=\sum_ \alpha h^{p-d_ \alpha,q-d_ \alpha}(X^ T_ \alpha)\). Also, for the \(G\)-scheme \(X\) with \(G\)-decomposition \(X=\bigcup X_ \alpha\) one finds for the motivic Poincaré series, \(P^ G_{\text{mot,pur}}(X,t)=\sum_ \alpha t^{2d_ \alpha}P^ G_{\text{mot,pur}}(X_ \alpha,t)(-d_ \alpha)\), where \((-d_ \alpha)\) means Tate twist. One deduces a formula for Hodge numbers. stratification; filterable decomposition; Euler characteristics; equivariant cohomology; Betti numbers; quotient of a variety; motivic Euler characteristic; Hodge numbers Navarro Aznar, V.: Stratifications parfaites et théorie des poids. Publ. Mat. 36 (1992), no. 2B, 807--825 (1993) Group actions on varieties or schemes (quotients), Topological properties in algebraic geometry, Stratifications in topological manifolds Perfect stratifications and weight 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 Let \(X\) a smooth genus \(g>2\) complex curve and \(L\) a line bundle on \(X\). Let \(M_L\) denote the moduli space parametrizing all stable pairs \((E, \phi )\) on \(X\) with \(E\) a rank \(2n\) vector bundle over \(X\) and \(\phi : E\otimes E \to L\) skew-symmetric and non-degenerate (\(L\)-symplectic bundles). If \(\deg (E) \geq 4n(g-1)\) there is a projective Picard bundle on \(X\). Fix \(x\in X\) and an \(n\)-dimensional Lagrangian linear subspace \(S\) of the fiber \(E_x\). The associated elementary (or Hecke) transformation is used to prove the stability of the Picard bundle. symplectic bundle on a curve; Picard bundle; moduli spaces; Hecke transformation DOI: 10.1142/S0129167X06003357 Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Hecke correspondence for symplectic bundles with application to the Picard bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0542.00009.]
Almost all completely integrable Hamiltonian systems are presentable as Lax equations with parameter \(\dot A(\xi)=[B(\xi),A(\xi)]\) where \(A(\xi)\) and \(B(\xi)\) are finite Laurent series in \(\xi\) whose coefficients are matrices depending on t. To such an equation, a certain mapping from \({\mathbb{R}}\) into the Jacobian of the curve given by det \(\| \eta I- A(\xi)\| =0\), called the flow, is associated. A cohomological criterion for this flow to be linear is established and exemplified on Euler equations of a free rigid body in \({\mathbb{R}}^ n\), Toda lattice and Nahm's equations. linearization of Hamiltonian systems; Euler equations; Jacobian variety; Lax equations with parameter; Toda lattice; Nahm's equations Griffiths P.A. (1984). Linearizing flows and a cohomological interpretation of Lax equations. Math. Sci. Res. Inst. Publ. 2: 36--46 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Jacobians, Prym varieties, Transformation and reduction of ordinary differential equations and systems, normal forms, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Linearizing flows and a cohomology interpretation of Lax equations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Here we study the Hilbert function of a Cohen-Macaulay homogeneous domain over an algebraically closed field of positive characteristic. The main tool (and an essential part of the main geometrical results) is the study of the Hilbert function of a general hyperplane section \(X\subset\mathbb{P}^r\) of an integral curve \(C\subset\mathbb{P}^{r+1}\), which is pathological in some sense. In \S 1, we study the case when \(C\) is a strange curve, i.e., all tangent lines to \(C\) at its simple points pass through a fixed point \(v\in\mathbb{P}^{r+1}\). In \S 2, we give more refined results under the assumption that the trisecant lemma fails for \(C\), i.e., any line spanned by two points of \(C\) contains one more point of \(C\). Hilbert function of a Cohen-Macaulay homogeneous domain; positive characteristic; Hilbert function of a general hyperplane section; strange curve; trisecant lemma E. Ballico and K. Yanagawa, On the \?-vector of a Cohen-Macaulay domain in positive characteristic, Comm. Algebra 26 (1998), no. 6, 1745 -- 1756. Plane and space curves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Finite ground fields in algebraic geometry On the \(h\)-vector of a Cohen-Macaulay domain in positive 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 The chiral Potts model of statistical mechanics has been studied by a number of mathematical physicists and it has been formulated and solved in one and two-dimensions. In this paper, the authors use the tools of modern algebraic geometry to study the chiral Potts model with three-states. In this case, there appears the curve \(X\equiv x^3+y^3=k(1+x^3y^3)\) of genus 4 where \(k\notin\{0,1,-1\}\).
The authors prove that \(\text{Aut}(X)=S_3\times S_3\) and the Jacobian \(J(X)\) is isogenous to the product of four elliptic curves which occur as pairs. They compute the degree of this isogeny and they calculate the direct image of theta divisor by this isogeny. This computation allows to expand the theta function of \(X\) as a sum of products of four elliptic theta functions and to show that the formula of \textit{V. B. Matveev} and \textit{A. O. Smirnov} [Lett. Math. Phys. 19, 179-185 (1990; Zbl 0715.14024)] is one of an infinite family of such identities.
The chiral Potts model is not solved in this paper, however the authors present a systematic approach to the underlying mathematical aspects of the problem. chiral Potts model; Jacobian; automorphism group of algebraic curve Relationships between algebraic curves and physics, Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, Jacobians, Prym varieties, Automorphisms of curves Algebraic geometry of the three-state chiral Potts model | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the paper there are three results: 1. A refined Noether normalization theorem over an arbitrary infinite field \( \mathbb{K}\) (with an effective algorithm for finding an appropriate linear coordinates system in \(\mathbb{K}^{n}).\) 2. An estimation for the separable geometric degree \(\text{sg}\deg (f):=[\text{Frac}(\mathbb K[X]):\text{Frac}(f^*(\mathbb K[Y]))]_s\) of a dominating morphism \(f=(f_{1},\ldots ,f_{n}):X\rightarrow Y\) of affine irreducible varieties (\(X\subset \mathbb{A}^{m}(\mathbb{K)},\) \(X\subset \mathbb{A}^{n}(\mathbb{K))}\)
\[
\text{sg}\deg (f)\leq \frac{\deg X}{\deg (Y)}(\deg f)^{\dim (X)},
\]
where \(\deg f=\max_{i}(\deg f_{i}).\) 3. An estimation for the degree of the components of the inverse of an isomorphism. Namely, under the above assumptions on \(X\) and \(Y,\) if the morphism \(f\) is an isomorphism and \(g=(g_{1},\ldots ,g_{n})=f^{-1}\) then
\[
\deg (V(g_{i}))\leq \deg (X)(\deg f)^{\dim (X)-1}.
\]
affine algebraic variety; geometric degree of a morphism Adjamagbo, K.; Winiarski, T.: Refined Noether normalization theorem and sharp degree bounds for dominating morphisms. Comm. algebra 33, No. 7, 2387-2393 (2005) Affine geometry Refined Noether normalization theorem and sharp degree bounds for dominating morphisms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A systematic study of topological properties of real Enriques surfaces was started by \textit{V. V. Nikulin} [in: Topology of real algebraic varieties and related topics. Dedic. Memory D. A. Gudkov, Transl., Ser. 2, Am. Math. Soc. 173, 187-201 (1996; Zbl 0869.14017)]. The classification of the real parts of real Enriques surfaces up to homeomorphism was completed by \textit{A. Degtyarev} and \textit{V. Kharlamov} [Topology 35, No. 3, 711-729 (1996)]. The connected components of the real part \(E_{\mathbb{R}}\) of a real Enriques surface can be naturally decomposed into two groups, \(E_{\mathbb{R}}= E_{\mathbb{R}}^{(1)}\cup E_{\mathbb{R}}^{(2)}\), due to the two different liftings of the real structure to the covering K3-surface.
In the present paper, a classification of the triads \((E_{\mathbb{R}}; E_{\mathbb{R}}^{(1)}, E_{\mathbb{R}}^{(2)})\) up to homeomorphism is completed. As a by-product of the investigation, the authors obtain several prohibitions on the topology of generalized Enriques surfaces. These surfaces are quotients of a non singular compact surface \(X\) with \(H_1(X; \mathbb{Z}/2)=0\) and \(w_2(X)=0\) by a fixed point free holomorphic involution. One of the main tools of the paper is Kalinin's spectral sequence used in combination with more traditional tools of topology of real algebraic varieties. real part of a real Enriques surface; generalized Enriques surfaces; quotients Degtyarev, A.; Kharlamov, V., Distribution of the components of a real Enriques surface, (1995) \(K3\) surfaces and Enriques surfaces, Topology of real algebraic varieties, Groups acting on specific manifolds Halves of a real Enriques surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A 4-semigroup means a numerical semigroup whose minimum positive integer is 4. In [\textit{J. Komeda} and \textit{A. Ohbuchi}, Corrigendum for Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve, Serdica Math. J. 30, No. 1, 43--54 (2004; Zbl 1075.14029); corrigendum ibid. 32, No. 4, 375--378 (2006)], we showed that a 4-semigroup with some conditions is the Weierstrass semigroup of a ramification point on a double covering of a hyperelliptic curve. In this paper we prove that the above statement holds for every 4-semigroup. Weierstrass semigroup of a point; double covering of a hyperelliptic curve; 4-semigroup Komeda, J., Ohbuchi, A.: Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve II. Serdica Math. J. \textbf{34}, 771-782 (2008) Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group, Rational and ruled surfaces Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve. II | 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 proposes an algorithm which, given a basis of a subspace of the space of cusp forms of weight 2 for \(\Gamma_0(N)\) which is invariant for the action of the Hecke operators, tests whether the subspace corresponds to a quotient \(A\) of a Jacobian of the modular curve \(X_0(N)\) such that \(A\) is the Jacobian of a curve \(C\). Moreover, equations for such a curve \(C\) are computed which make the quotient suitable for applications in cryptography. One advantage of using such quotients of modular Jacobians is that fast methods are known for finding their number of points over finite fields [\textit{G. Frey} and \textit{M. Müller}, Arithmetic of modular curves and applications. Matzat, B. Heinrich (ed.) et al., Algorithmic algebra and number theory. Heidelberg, 1997. Berlin: Springer, 11-48 (1999; Zbl 0932.11042)]. Our results extend ideas of \textit{M. Shimura} [Tokyo J. Math. 18, 443-456 (1995; Zbl 0865.11052)] who used only the full modular Jacobian instead of Abelian quotients of it. algorithm; Jacobian; modular curve; cryptography; quotients of modular Jacobians; number of points over finite fields Seigo Arita, Construction of secure \?_{\?\?} curves using modular curves, Algorithmic number theory (Leiden, 2000) Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 113 -- 126. Arithmetic aspects of modular and Shimura varieties, Applications to coding theory and cryptography of arithmetic geometry, Holomorphic modular forms of integral weight, Hecke-Petersson operators, differential operators (one variable) Construction of secure \(C_{ab}\) curves using modular 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 \((A, \theta)\) be a complex principally polarized abelian variety (p.p.a.v. for short) of dimension \(g\). For \(0 \leq d \leq g\), the cohomology class \(\theta_d = \theta^d/d!\) is minimal, i.e. nondivisible, in \(H^{2d} (A, \mathbb{Z})\). When \(A\) is the Jacobian variety \(J(C)\) of a curve \(C\) of genus \(g\), the image \(W_{g - d} \subset J(C)\) of \(C^{(g - d)} \) under the Abel-Jacobi map is a subvariety of class \(\theta_d\).
In the paper under review the main result proved is: Any effective cycle in \(J(C)\) with class \(\theta_d\) is a translate of either \(W_{g - d}\) or \(- W_{g - d}\).
The author also proves that: for \(1 < d < g\) the Jacobian locus \({\mathcal J}_g\) (resp. the locus of intermediate Jacobians of cubic 3-folds \({\mathcal C} {\mathcal J}_5)\) is an irreducible component of the set \({\mathcal C}_{g,d}\) of p.p.a.v. of dimension \(g\) for which \(\theta_d\) (resp. \(\theta_3)\) is the class of an effective algebraic cycle.
At the end the author states the following conjecture: If \(1 < d < g\) and \((g,d) \neq (5,3)\) then \({\mathcal C}_{g,d} = {\mathcal J}_g\); furthermore, \({\mathcal C}_{5,3} = {\mathcal J}_5 \cup {\mathcal C} {\mathcal J}_5\). Notice that this holds for any \(g\) and \(d = g - 1\) by Matsusaka's criterion and for \(g = 4\) and \(d = 2\) by a result of \textit{Z. Ran} [Invent. Math. 62, 459-479 (1980; Zbl 0474.14016)]. principally polarized abelian variety; Jacobian variety; effective cycle; locus of intermediate Jacobians of cubic 3-folds Debarre O. (1995). Minimal cohomology classes and Jacobians. J. Algebraic Geom. 4(2): 321--335 Jacobians, Prym varieties, Algebraic cycles Minimal cohomology classes and Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study the relationship between the cohomology of the function field of a curve over a complete discretely valued field and that of the function ring of curves resulting over its residue field. The results are applied to prove the existence of noncrossed product division algebras and indecomposable division algebras of unequal period and index over the function field of any \(p\)-adic curve, generalizing the results and methods of a previous work of the authors and \textit{K. McKinnie} [Adv. Math. 226, No. 5, 4316--4337 (2011; Zbl 1253.16019)]. cohomology of function field of a curve; complete discretely valued field; function ring of curves; existence of noncrossed product division algebras; function field of \(p\)-adic curve E. Brussel and E. Tengan, \textit{Formal constructions in the Brauer group of the function field of a p-adic curve}, Transactions of the American Mathematical Society, to appear. Brauer groups of schemes, Curves over finite and local fields, Brauer groups (algebraic aspects), Finite-dimensional division rings Formal constructions in the Brauer group of the function field of a \(p\)-adic 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 article is dedicated to the memory of Oleg Nikolaevich Vvedenskii (1937--1981). O. N. Vvedenskii was a student of the academician I. R. Shafarevich. O. N. Vvedenskii's research and the results obtained are related to duality in elliptic curves and with the corresponding Galois cohomology over local fields, with Shafarevich-Tate pairing and with other pairings, with local and quasi-local of class fields theory of elliptic curves, with the theory of Abelian varieties of dimension greater than 1, with the theory of commutative formal groups over local fields.
The paper presents both the results obtained by O. N. Vvedenskii, and new selected results, developing research in the directions of the fundamental groups of schemes, the principal homogeneous spaces (torsors), and duality. The first part of the article presented here is an introduction both to the results obtained by O. N. Vvedenskii in the direction of duality of Abelian varieties and formal groups, and in new selected results, developing research in the directions of the fundamental groups of schemes, the principal homogeneous spaces (torsors), and duality. The Introduction gives preliminary information and presents the content of the article. In the first section we give a brief survey of selected results on the theory of algebraic, quasialgebraic and proalgebraic groups and group schemes. Further, in Section 2 we present selected results on fundamental groups of algebraic varieties, on fundamental groups of schemes, and in Section 3 -- selected results on principal homogeneous spaces (torsors), developing research by O. N. Vvedenskii and other authors. In Section 4 we give information on duality, and in Section 5 the paper presents the results by O. N. Vvedenskii on the arithmetic theory of formal groups and their development. The results of this section, represented over local and quasi-local fields $K$, over their rings of integers, and over their residue fields $k$, are connected (1) with the formal structure of Abelian varieties, (2) with commutative formal groups, (3) with corresponding homomorphisms. In the article, algebraic varieties, Abelian schemes, and commutative formal group schemes are defined, as a rule, over local and quasi-local fields, over their rings of integers, and over their residue fields. But these objects are also briey considered over global fields, since O. N. was interested in the subject of algebraic varieties over global fields and he carried out corresponding studies. It is assumed that the characteristic of the residue fields is more than 3, unless otherwise specified. duality; abelian variety; local field; Picard group; formal group; group scheme; fundamental group; torsor; global field; proalgebraic group; group of universal norms History of algebraic geometry, Biographies, obituaries, personalia, bibliographies, History of mathematics in the 20th century, Group schemes, Algebraic theory of abelian varieties, Formal groups, \(p\)-divisible groups Duality in abelian varieties and formal groups over local 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 In this paper the tangent bundle of a flag variety F is given as a quotient of a certain universal bundle on F together with a description of the kernel of this quotient involving exact sequences of all the universal bundles on F. The proof uses a reduction to the case of complete flags and an induction on the dimension of the flag. higher direct image; quotient of universal bundle; tangent bundle of a flag variety Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Embeddings in algebraic geometry On the tangent bundle of a flag variety | 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 \(A\subset\mathbb{Z}^{n-1}\) be a finite subset, \(\mathbb{C}^ A\) the linear \(\mathbb{C}\)-space of Laurent polynomials \(f=\sum_{\omega\in A}a_ \omega X^ \omega\), \(a_ \omega\in\mathbb{C}\) in some indeterminates \(X\) and \(\nabla_ 0\subset\mathbb{C}^ A\) the set of those \(f\) for which there is \(\kappa\in(\mathbb{C}^*)^{n-1}\) such that \(f(\kappa)=(\partial f/\partial X_ i)(\kappa)=0\) for all \(i\). The closure \(\nabla_ A\) of \(\nabla_ 0\) is an irreducible variety defined in fact on \(\mathbb{Z}\). When \(\nabla_ A\) has codimension 1 then an irreducible polynomial \(\Delta_ A\in\mathbb{Z}[a_ \omega;\omega\in A]\), which is zero on \(\nabla_ A\), is unique up to the sign and it is called the \(A\)-discriminant. If \(\text{codim}(\nabla_ A)>1\) then put \(\Delta_ A=1\). The \(A\)- discriminant is homogeneous and satisfies the following quasi homogeneous \((n-1)\)-conditions: ``\(\sum_{\omega\in A}m(\omega)\cdot\omega\in\mathbb{Z}^{n-1}\) is constant for all monomials \(\prod_{\omega\in A}a_ \omega^{m(\omega)}\) which enter in \(\Delta_ A\)''. This notion extends the classical notions of discriminant and resultant. --- Let \(A=\{\omega_ 1,\ldots,\omega_ N\}\) and \(Y_ A\) be the closure of the set \(\{(\kappa^{\omega_ 1},\ldots,\kappa^{\omega_ N}\mid\kappa\in\mathbb{C}^{*n-1}\}\) in \(\mathbb{P}^{N-1}\). Then \(\nabla_ A\) and \(Y_ A\) are dual projective varieties and the description of \(\Delta_ A\) follows if we can describe the equations of the dual projective variety of a given projective one \(Y\subset\mathbb{P}^{N-1}\) [see the authors' previous paper in Sov. Math., Dokl. 39, No. 2, 385-389 (1989); translation from Dokl. Akad. Nauk SSSR 305, No. 6, 1294-1298 (1989; Zbl 0715.14042)].
Let \(G\) be a free abelian group of rank \(n\), \(G_ \mathbb{C}:=\mathbb{C}\otimes_ \mathbb{Z} G\), \(\lambda:G\to(\mathbb{Q},+)\) a nonzero group morphism, \(S\subset G\) a finitely generated semigroup such that \(o\in S\) and \(\lambda(s)\geq 1\) for all \(s\in S\), \(S_ e=\{t\in S\mid\lambda(t)=e\}\) for \(e\in\mathbb{Q}\) and \(A\subset S_ 1\) a finite subset generating in \(G_ \mathbb{R}=\mathbb{R}\otimes_ \mathbb{Z} G\) the same convex cone as \(S\). For \(k\in\mathbb{Z}_ +\), \(e\in\mathbb{Q}\), \(\omega\in A\) let \(\bigwedge^ k(e)\) be the space of all maps \(S_{k+e}\to\bigwedge^ kG_ \mathbb{C}\), \(\partial_ \omega:\bigwedge^ k(e)\to\bigwedge^{k+1}(e)\) the map given by \(\partial_ \omega(\gamma)(u)=\omega\wedge\gamma(u-\omega)\), if \(u-\omega\in S_{k+e}\), otherwise \(\partial_ \omega(\gamma)(u)=0\) and \(\partial_ f=\sum_{\omega\in A}a_ \omega\partial_ \omega\) if \(f=\sum a_ \omega X^ \omega\in\mathbb{C}^ A\). The complex \((\overset{.}\bigwedge (e),\partial_ f)\) is called the Cayley-Koszul complex. --- Choose a basis \(u\) in terms of \(\overset{.}\bigwedge(e)\) and let \(E_ e(f)\) be the determinant of the complex \((\overset{.}\bigwedge (e),\partial_ f)\) with respect to \(u\) [see \textit{F. Fischer}, Math. Z. 26, 497-550 (1927) or \textit{J.-K. Bismut} and \textit{D. S. Freed}, Commun. Math. Phys. 106, 159-176 (1986; Zbl 0657.58037)]. For \(e\) sufficiently high \(E_ A(f):=E_ e(f)\) is a polynomial of \((a_ \omega)\), \(f=\sum a_ \omega X^ \omega\) which depends on \(e\) only by a constant multiple. --- Let \(Q_ A\) be the convex closure of \(A\) in \(G_ \mathbb{R}\). If \(f=\sum a_ \omega X^ \omega\) we can express \(E_ A(f)=\sum_ \varphi c_ \varphi\prod_{\omega\in A}a_ \omega^{\varphi(\omega)}\), where \(\varphi\) runs in the set \(\mathbb{Z}^ A_ +\) of the maps \(A\to\mathbb{Z}_ +\). Let \(M(E_ A)\subset\mathbb{R}^ A\) be the convex closure of those \(\varphi\in\mathbb{Z}^ A_ +\) for which \(c_ \varphi\neq 0\). Then there exists a nice correspondence between the vertices of \(M(E_ A)\) and some special triangulations of \(Q_ A\).
The theory is applied to the following examples: the discriminant of a polynomial in two indeterminates, the resultant of two quadratric polynomials, the elliptic curve in Tate normal form\dots Laurent polynomials; Cayley-Koszul complex; determinant; discriminant of a polynomial in two indeterminates; elliptic curve Gel'fand, I.; Zelevinskiǐand, A.; Kapranov, M., Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Algebra i Analiz, 2, 1, (1990) Toric varieties, Newton polyhedra, Okounkov bodies, Polynomial rings and ideals; rings of integer-valued polynomials, Complexes, Determinantal varieties Discriminants of polynomials in several variables and triangulations of Newton polyhedra | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study the class of admissible linear embeddings of flag varieties in the context of algebraic geometry. They prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra, showing that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags. That is, generalized flags in a countable-dimensional vector space are in a natural one-to-one correspondence with splitting parabolic subgroups \(\mathsf{P}\) of the ind-group \(\mathsf{GL}(\infty)\), and hence the points of homogeneous ind-spaces of the form \(\mathsf{GL}(\infty)/\mathsf{P}\) can be thought of as generalized flags. Ind-varieties are discussed as the ind-group \(\mathsf{SL}(\infty)\), respectively, \(\mathsf{O}(\infty)\) or \(\mathsf{Sp}(\infty)\) for isotropic generalized flags, where the authors construct them in purely algebro-geometric terms. flag variety; homogeneous ind-variety; generalized flag; linear embedding of flag varieties Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) An algebraic-geometric construction of ind-varieties of generalized flags | 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 essential dimension of an algebraic object can be informally viewed as the minimal number of algebraically independent parameters needed to define the object. It was first introduced by Buhler and Reichstein for the class of finite Galois field extensions with a given Galois group, and later extended to the class of \(G\)-torsors for an arbitrary algebraic group \(G\) (see [\textit{J. Buhler} and \textit{Z. Reichstein}, Compos. Math. 106, No. 2, 159--179 (1997; Zbl 0905.12003), and Transform. Groups 5, No. 3, 265--304 (2000; Zbl 0981.20033)], respectively). Many classical objects such as simple algebras, quadratic and hermitian forms, algebras with involutions etc. can be viewed as torsors under algebraic groups. The only property of a class of algebraic objects needed to define the essential dimension is that for every field extension \(K/F\) we must have a set \(\mathcal{F}(K)\) of isomorphism classes of objects, and for every field homomorphism \(K \to L\) over \(F\), a change of field map \(\mathcal{F}(K) \to \mathcal{F}(L)\). In other words, \(\mathcal{F}\) must be a functor from the category \(Fields _ F\) of field extensions of \(F\) to the category of sets. The essential dimension for an arbitrary functor \(Fields _ F \to Sets\) was defined by \textit{G. Berhuy} and \textit{G. Favi} [Doc. Math., J. DMV 8, 279--330 (2003; Zbl 1101.14324)].
The paper under review surveys research on the essential dimension. The studies in this area are motivated by various problems concerning the classification, structure and other properties of algebraic objects. As pointed out by the author, one of the leading contributors to the area, the highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups (the corresponding results are presented with self-contained proofs). In addition, the paper states theorems describing the structure of finite groups of essential dimension \(j\), for \(j = 1, 2\), and provides references to their classification. Lower and upper bounds on the essential dimensions of central simple algebras and exceptional algebraic groups are also presented as well as exact formulae for the canonical \(p\)-dimension of split semisimple algebraic groups (\(p\) is a prime number), and for the essential dimensions of quadratic forms, hypersurfaces and other algebro-geometric objects. The survey gives an idea of various aspects of the relations of the essential dimension and a number of other areas such as birational algebraic geometry, incompressible varieties, Chow motives, equivariant \(K\)-theory, Galois cohomology, representation theory of algebraic groups, fibered categories, valuation theory. The survey is comprehensive with a bibliography of 103 items. essential dimension; essential \(p\)-dimension; functor; canonical \(p\)-dimension of a variety; algebraic group (\(G\)); \(G\)-scheme; \(G\)-torsor; strongly \(p\)-incompressible variety; category fibered in groupoids; group of multiplicative type; central simple algebra; étale algebra; quadratic and hermitian forms A.\ S. Merkurjev, Essential dimension: A survey, Transform. Groups 18 (2013), 415-481. Group actions on varieties or schemes (quotients), Research exposition (monographs, survey articles) pertaining to group theory, Quadratic forms over general fields, Bilinear and Hermitian forms, Galois cohomology of linear algebraic groups, Separable extensions, Galois theory, Transcendental field extensions, Galois cohomology, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Rational and birational maps, Group schemes, Geometric invariant theory, Clifford algebras, spinors, Finite-dimensional division rings, Brauer groups (algebraic aspects), Linear algebraic groups over arbitrary fields, Linear algebraic groups over finite fields Essential dimension: a survey | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians det \(p\) be an odd prime and \(F\) be the \(p\)-th Fermat curve given by \(x^ p+y^ p+z^ p=0\). Let \(\zeta\) be a \(p\)-th root of unity and \(C[p]\) the \(p\)-torsion in the ideal class group \(C\) of \(\mathbb{Q}(\zeta)\). The author proves that, if \(\text{rank}_{\mathbb{Z}/p\mathbb{Z}}C[p]<(p+5)/8\), then the number of \(\mathbb{Q}\)-rational points of \(F\) is at most \(2p-3\). ``Fermat's last theorem'' asserts that the number of \(\mathbb{Q}\)-rational points of \(F\) is at most three. Classically, there are many theorems varifying this statement under various restrictions on \((C[p]\), starting by Kummer's famous theorem that the assertion holds, if \(C[p]=0\) [\textit{E. Kummer}, ``Beweis des Fermatschen Satzes der Unmöglichkeit von \(x^ \lambda+y^ \lambda=z^ \lambda\) für eine unendliche Anzahl von Primzahlen \(\lambda\)'', Collected Papers, Vol. I, pp. 274-281 (1847)]. A result due to Eichler, which can be compared with the author's results, states that, if \(\text{rank}_{\mathbb{Z}/p\mathbb{Z}}C[p]<\sqrt p-2\), then there are no solutions to Fermat's equation, if \(p\nmid xyz\) [\textit{M. Eichler}, Acta Arith. 11, 129-131, 261 (1965; Zbl 0135.094)]. The author remarks that it seems likely that the hypotheses of Eichler's result as well as his result are satisfied for all primes \(p\). For \(p\leq 125.000\), the largest value of the rank is 5. Assuming Vandiver's conjecture, then the rank is equal to the index of irregularity, which is the number of Bernoulli numbers \(B_{_ 2k}\), with \(2\leq 2k\leq p-3\), which are divisible by \(p\). A probabilistic argument shows that this number is \(O(\log p/\log\log p)\) [ see \textit{L. C. Washington} ``Introduction to cyclotomic fields'' (1982; Zbl 0484.12001); example 6.6]. However the best known bound on the rank, as far as the author is concerned, is \(\text{rank}_{\mathbb{Z}/p\mathbb{Z}}C[p]<p/2\).
Usually, arguments on ``Fermat's last theorem'' involved the method of infinite descent, which is done for the Fermat curve itself. In this paper, the author proves his result by using a descent argument on the Jacobian of \(F\) and recovering the information about the curve by applying Coleman's effective version of Chabauty's method [cf. \textit{R. F. Coleman}, Ann. Math., II. Ser. 121, 111-168 (1985; Zbl 0578.14038) and Duke Math. J. 52, 765-770 (1985; Zbl 0588.14015)]. In fact, the author works with quotients of Fermat curves. Let \(a\), \(b\) and \(c\) be integers such that \(\text{gcd}(abc,p)=1\) and \(a+b+c=0\). Let \(F_{a,b,c}\) be the complete nonsingular curve defined over \(\mathbb{Q}\) with affine equation \(y^ p=(-1)^ cx^ a(1-x)^ b\).
The author proves that if \(((a^ ab^ bc^ c)^ p-a^ ab^ bc^ c)/p)\not\equiv 0\pmod p\) and \(\text{rank}_{\mathbb{Z}/p\mathbb{Z}}C[p]<(p+5)/8\), then the number of \(\mathbb{Q}\)-rational points of \(F_{a,b,c}\) is at most \(2p-3\). There is a map \(\varphi:F\to F_{a,b,c}\) defined by \((x,y)\mapsto(-x^ pz^{-p},x^ ay^ bz^ c)\), which is injective on \(F(\mathbb{Q})\). Hence the first result is obtained from the latter and the fact that for \(p\geq 5\) there exists a triple \((a,b,c)\) satisfying the above hypothesis (lemma 6). descent on the Jacobian of Fermat curve; Fermat's last theorem; number of rational points; quotients of Fermat curves McCallum, W.G.: The Arithmetic of Fermat Curves. Math. Ann.294, 503--511 (1992) Rational points, Arithmetic ground fields for curves, Higher degree equations; Fermat's equation, Elliptic curves over global fields The arithmetic of Fermat curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials \(f, g \in \mathbb{F}_{q_0} [x, y]\) and any \(\mathbb{F}_q / \mathbb{F}_{q_0}\), the image of the map \(\mathbb{F}_q^3 \to \mathbb{F}_q^3\) given by \((s, x, y) \mapsto(s, s x + f(x, y), s y + g(x, y))\) has size at least \(\frac{q^3}{4} - O(q^{5 / 2})\) and prove the special case when \(f = f(x), g = g(y)\). We also prove it in the case \(f = f(y), g = g(x)\) under the additional assumption \(f^\prime(0) g^\prime(0) \neq 0\) when \(f, g\) are both affine polynomials. Our approach is based on a combination of Cauchy-Schwarz and Lang-Weil. The algebraic geometry inputs in the proof are various results concerning irreducibility of certain classes of multivariate polynomials. Kakeya problem; image set on \(F_q\)-points; Lang-Weil bound; reducibility of polynomials in several variables; number of irreducible components of a variety; indecomposable polynomials; affine polynomials; permutation polynomials K. Slavov, An algebraic geometry version of the Kakeya problem, in preparation. Configurations and arrangements of linear subspaces, Finite ground fields in algebraic geometry, Finite geometry and special incidence structures An algebraic geometry version of the Kakeya problem | 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 the paper cited in the title by \textit{J. V. Leahy} and the author [Nagoya Math. J. 82, 27-56 (1981; Zbl 0528.14001)] the notion of a c- regular function on an algebraic variety over a closed field k of characteristic zero was introduced. The intention was to describe those k-valued functions on a variety that become regular functions when lifted to the normalization but without any reference to the normalization in the definition. Equivalently, the authors aspired to identify the c- regular functions on a given variety with the regular functions on the weak normalization of that variety. - Originally, a c-regular function was defined to be a continuous k-valued function that is regular at each nonsingular point. This does not provide the desired characterization as is evidenced by the following observation. By deleting finitely many points from the normalization of a weakly normal singular curve X, one obtains a curve X' that is homeomorphic to X via a birational morphism, but is not isomorphic to X as varieties. A regular function on X' that is not the restriction of a regular function on the normalization of X provides an example of a continuous function on X that is regular off the singular locus but does not lift to a regular function on the normalization. This is due to the very special nature of the Zariski topology in dimension one.
For varieties without one-dimensional components, the author offers a new characterization of those functions on a variety that lift to regular functions on the normalization. This characterization makes no reference to the normalization and hinges on the fact that a homeomorphic morphism onto a weakly normal variety without one-dimensional components is an isomorphism (the assumption that the ground field have characteristic zero is necessary here). In order to provide a unified treatment, the author now defines a c-regular function on a variety in terms of its normalization and subsequently offers alternative characterizations. weakly normal variety; homeomorphic morphism of varieties; c-regular function; functions on a variety that lift to regular functions on the normalization Vitulli, M. A.: Corrections to ''seminormal rings and weakly normal varieties''. Nagoya math. J. 107, 147-157 (1987) Varieties and morphisms, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), General commutative ring theory, , Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Corrections to ''Seminormal rings and weakly normal 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 projective curve of genus 2 over a discrete valuation field \(K\) of \(\text{char} (K) \neq 2\), then \(C\) is defined by a hyperelliptic equation \(y^ 2 = a_ 0 x^ 6 + a_ 1x^ 5 + \cdots + a_ 0\), \(a_ i \in K\). Let \({\mathcal X}\) be the minimal regular model of \(C\) over the ring of integers of \(K\). Our purpose is to determine explicitly the special fiber \({\mathcal X}_ s\) in terms of the coefficients \(a_ i\). This is done completely when the residual characteristic of \(K\) is \(\neq 2\). minimal regular model of smooth curve over a discrete valuation field Liu, Qing, Modèles minimaux des courbes de genre deux, J. Reine Angew. Math., 453, 137-164, (1994) Arithmetic ground fields for curves, Local ground fields in algebraic geometry, Coverings in algebraic geometry Minimal models of genus 2 curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review deals with the topology of the hyperplane sections of a smooth affine connected algebraic variety \(V\subset {\mathbb{C}}^ N\). It is well known that the smooth hyperplane sections \(H\cap V\) of V need not form a locally trivial fibration. The main results are devoted to understand what sections should be thrown out in order to get local triviality. It is also shown how to relate the global fibrations with the local Milnor fibrations and the groups \(H_ q(V,V\cap H)\) are computed in terms of Milnor numbers. Finally the theory above is applied to generalize some results due to \textit{S. A. Broughton} [in Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 167- 178 (1983; Zbl 0526.14010)]. topology of the hyperplane sections of a smooth affine connected algebraic variety; locally trivial fibration; Milnor numbers Némethi, A.: Théorie de Lefschetz pour LES variétés algébriques affines. CR acad. Sc. Paris (1986) Structure of families (Picard-Lefschetz, monodromy, etc.), Topological properties in algebraic geometry Théorie de Lefschetz pour les variétés algébriques affines. (Lefschetz theory for affine algebraic varieties) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors consider the problem of determination of the topological type of a plane curve singularity by its Dynkin diagram. As it is known for irreducible singularities, Dynkin diagrams do determine their topological type. The authors construct a Dynkin-type diagram containing additional information, called \(A\Gamma\)-diagram. It is a coloured graph, related to A'Campo's and Gusein-Zade's variants of Dynkin diagrams [\textit{N. A'Campo}, Math. Ann. 213, 1-32 (1975; Zbl 0316.14011)] and \textit{S. M. Gusein-Zade} [Russ. Math. Surv. 32 , No. 2, 23-69 (1977); translation from Usp. Mat. Nauk 32, No. 2(194), 23-65 (1977; Zbl 0363.32010)]. The authors show how to extract the information about intersection multiplicities and \(A\Gamma\)-diagrams of the branches from the \(A\Gamma\)-diagram of the reducible curve, by combinatorial methods. Therefore, the main result of the paper is: The \(A\Gamma\)-diagram of the plane curve singularity determines its topological type. topological type of a plane curve singularity; Dynkin diagrams; \(A\Gamma\)-diagram L. Balke and R. Kaenders. On a certain type of Coxeter-Dynkin diagrams of plane curve singularities. Topology 35 (1996), no. 1, 39--54. Singularities of curves, local rings, Topological properties in algebraic geometry, Singularities in algebraic geometry On a certain type of Coxeter-Dynkin diagrams of plane curve 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 Let $S$ be a Riemann surface of genus $g \geq 2$. Then, the order of the automorphism group of $S$ is at most $84(g-1)$. If additional conditions on the surface or on the group are included, this upper bound is very much shortened. In that sense, deep results have been obtained on families of surfaces of genus $g$ having $4g+4$, or $4g$, automorphisms. In particular, \textit{E. Bujalance} et al. have obtained in [Topology Appl. 218, 1--18 (2017; Zbl 1360.30033)] all the Riemann surfaces of genus $g$ with exactly $4g$ automorphisms. Except the values of $g = 3, 6, 12, 15, 30$, for a given $g$ these surfaces form an equisymmetric one-dimensional family $\mathcal{F}_g$. The automorphism group $G$ of each surface $S$ is isomorphic to the dihedral group $D_{2g}$, and $S/G$ has genus $0$. \par The present article is devoted to study properties of the surfaces $S$ in $\mathcal{F}_g$, and the author restricts himself to $g$ being a prime $q \geq 5$. In Section 3, using that $S \in \mathcal{F}_q$ is hyperelliptic, Theorem 1 gives an algebraic description of the surface, and Theorem 2 provides two explicit generators of its automorphism group. Section 4 is dedicated to study the Jacobian variety $JS$ of $S$, and its group algebra decomposition with respect to $G$. Section 5 considers the fields of definition of $S$. Necessary and sufficient conditions are obtained in Theorems 5 and 6 that characterize when $S$ and $JS$ can be defined by polynomials with real coefficients, resp. algebraic coefficients. Finally, in Section 6 the author studies a special variety in the moduli space of principally polarized abelian varieties of dimension $q$ associated to each $S \in \mathcal{F}_q$, with a special focus on the case $q = 5$. Riemann surfaces; equisymmetric family; Jacobian variety; field of definition Compact Riemann surfaces and uniformization, Automorphisms of curves, Jacobians, Prym varieties On the one-dimensional family of Riemann surfaces of genus \(q\) with 4\(q\) automorphisms | 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 the function field of a smooth projective curve \(C\) over an algebraically closed field \(k\) of characteristic \(p > 0\). Let \(A\) be an abelian variety over \(K\) and let \(f \in K(A)\) be a rational function on \(A\). Fix an integer \(r > 0\). Then there exists \(B(r) \in \mathbb{Z}\) such that for any \(P \in A(K)\) either
\[
P = Q + p^rR,
\]
where \(Q \in A(K) \cap (f)\), \(R \in A(K^{\text{sep}}),\) or, for any \(x \in C(k)\)
\[
\nu_x(f(P)) < B(r).
\]
This result is an analog for characteristic \(p > 0\) of Buium's \(abc\) theorem [\textit{A. Buium}, Int. Math. Res. Not. 1994, No. 5, 219-233 (1994; Zbl 0836.14025)]. function field of a smooth projective curve; characteristic \(p\); \(abc\) theorem [Sc] T. Scanlon: ''The abc theorem for commutative algebraic groups in characteristic p'', Int. Math. Res. Notices, No. 18, (1997), pp. 881--898. Arithmetic ground fields for abelian varieties, Finite ground fields in algebraic geometry, Algebraic functions and function fields in algebraic geometry The \(abc\) theorem for commutative algebraic groups in characteristic \(p\) | 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 0628.00007.]
Let \(X_ 0\) be a proper smooth geometrically connected curve over a finite field. One defines L-functions for certain n-dimensional \(\ell\)- adic representations of the fundamental group \(\pi_ 1(X_ 0)\). Langlands asked whether these functions are automorphic. Drinfeld solved this problem for \(n=2\). He used constructible sheaves and the Shalika transform of the Whittaker functions associated to the representations. The author develops a similar approach for \(n>2\). L-functions for l-adic representations of the fundamental group; curve over a finite field Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Representation-theoretic methods; automorphic representations over local and global fields, Langlands-Weil conjectures, nonabelian class field theory, Finite ground fields in algebraic geometry Constructible sheaves associated to Whittaker functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0553.00001.]
The author describes the recent advances and conjectures in the study of the Mordell-Weil groups of elliptic curves with complex multiplication, and more generally, of elliptic curves that admit Weil parametrizations. The exposition begins with a brief introduction to the theorem of Gross and Zagier giving a formula for the value at \(s=1\) of the derivative of the L-function attached to a Weil curve. The author then gives a description of Greenberg's theory of variation of the Mordell-Weil rank over anti-cyclotomic extensions, and also describes Rohrlich's theory of variation of the rank over the anti-cyclotomic tower, and over cyclotomic towers. Included in this exposition is an introduction to the p-Selmer group, and the Tate-Shafarevich group of an elliptic curve. Of course, the Iwasawa theory of elliptic curves is a central tool in the study of many of the topics mentioned above, and so the author provides a description of some of the important results in this field. This leads, naturally, to the p-adic theory, including the p-adic height pairing of Schneider, the p-adic height pairings of Mazur and Tate, the p-adic L- functions, and p-adic Heegner measures.
The author concludes this paper by asking if there is a p-adic version of the Gross-Zagier theorem. The question has recently been given an affirmative answer by \textit{B. Perrin-Riou} (see her forthcoming papers). Using both the Gross-Zagier theorem, and Perrin-Riou's p-adic version, Karl Rubin has recently proved that if E/\({\mathbb{Q}}\) is an elliptic curve with complex multiplication, and if the Mordell-Weil rank of E is at least 2 then the L-function of E has order of vanishing at least 2.
We should mention that since the Warsaw congress there has been one other striking result in the theory of c. m. curves. Karl Rubin has recently proved the finiteness of the Tate-Shafarevich group for c. m. elliptic curves whose L-functions do not vanish at \(s=1\). Mordell-Weil groups of elliptic curves with complex; multiplication; Weil parametrizations; L-function attached to a Weil curve; anti-cyclotomic tower; Iwasawa theory of elliptic curves; p-adic height pairing; p-adic L-functions; p-adic Heegner measures; finiteness of the Tate-Shafarevich group; Mordell-Weil groups of elliptic curves with complex multiplication B. Mazur, Modular curves and arithmetic, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 185 -- 211. Arithmetic ground fields for curves, Complex multiplication and abelian varieties, Theta series; Weil representation; theta correspondences, Elliptic curves, Global ground fields in algebraic geometry, Local ground fields in algebraic geometry, Special algebraic curves and curves of low genus, Complex multiplication and moduli of abelian varieties Modular curves and arithmetic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper the author studies the automorphism group of a rational curve with only nodes as singularites.
A rational nodal curve of arithmetic genus g (i.e. with g nodes) is isomorphic to the quotient of \({\mathbb{P}}\) (i.e. complex projective line) obtained by identifying g pairs of distinct points. - By \(X=(a_ 1,b_ 1,...,a_ g,b_ g)\) is denoted the rational nodal curve of arithmetic genus g obtained by identifying the points \(a_ i\) and \(b_ i\) of \({\mathbb{P}}\) for \(i=1...g\), and by Aut(X) the automorphism group of X. The author obtains that \(| Aut(X)| \leq 4g(g-1)\) and determines which groups occur as automorphism groups of 2-nodal and 3-nodal rational curves. Thus he obtains that if X is 2-nodal then Aut(X) is the dihedral group \(D_ 2\) or \(D_ 4\) and if it is 3-nodal, Aut(X) is the cyclic group \({\mathbb{Z}}_ 2, D_ 2, D_ 3, D_ 6\) or one of the octahedral groups. automorphism group of a rational curve; nodes; genus Singularities of curves, local rings, Group actions on varieties or schemes (quotients) Automorphism groups of 3-nodal rational 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 0619.00007.]
Let \(M(r,c_ 1,c_ 2)=:M\) be the variety of semi-stable modules over the projective plane \({\mathbb{P}}_ 2({\mathbb{C}})\) of rank r (r\(\geq 2))\) with Chern classes \(c_ 1\), \(c_ 2\). It is a projective, integral and normal variety. The main result of this paper is to prove the theorem: The variety \(M(r,c_ 1,c_ 2)\) is locally factorial. This statement is equivalent to the statement that Pic(M)\(\to Cl(M)\) is bijective.
The author goes on to study Pic(M). Let \(\mu =c_ 1/r\) and \(\Delta:=c_ 2\cdot r^{-1}-(r-1)\cdot (2r^ 2)^{-1}c^ 2_ 1\). There is a unique function \(\delta: {\mathbb{Q}}\to {\mathbb{Q}}\) such that \(\dim (M(r,c_ 1,c_ 2))>0\) if and only if \(\Delta\geq \delta (\mu)\). Then the author's calculations show that: If \(\Delta =\delta (\mu)\), then \(Pic(M)={\mathbb{Z}}\) and if \(\Delta >\delta (\mu)\), then \(Pic(M)={\mathbb{Z}}\oplus {\mathbb{Z}}\). He then describes the generator(s) of Pic(M). generator of Picard group; divisor class group; local factoriality of moduli variety of vector bundle Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Groupe de Picard des variétés de modules de faisceaux semi-stables sur \({\mathbb{P}}_ 2\). (Picard group of moduli varieties of semi-stable sheaves on \({\mathbb{P}}_ 2)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The classical Torelli theorem says that the polarized Jacobian of a curve completely determines the curve. By extension, one talks about Torelli problems as trying to identify a variety through some associated variety. A ``Torelli theorem'' says that the association map is injective while a ``generic Torelli theorem'' says that it is a birational morphism with its image. In this paper, we consider the association of a curve with the set of special divisors of fixed degree and dimension or \(W^ r_ d\). If the dimension of \(W^ r_ d\) is at least two, we show that generic Torelli holds. When the dimension of \(W^ r_ d\) is one, we can only show the result for \(r=1\). We expect it to be true for all values of r. linear series of algebraic curve; divisors; generic Torelli G.P. Pirola, M. Teixidor i Bigas, Generic Torelli for \(\(W^r_d\)\). Math. Z. 209(1), 53-54 (1992) Torelli problem, Jacobians, Prym varieties, Divisors, linear systems, invertible sheaves Generic Torelli for \(W_ d^ r\) | 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 describes a technique, due to him and to Gaveau, which allows to compute the abelian sum of a bundle on a singular curve given by a Cartier divisor, based on a limit procedure. abelian sum of a bundle on a singular curve Algebraic functions and function fields in algebraic geometry On the Abel-Jacobi theorem for singular curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The papers of this third volume of Zariski's work were originally published between 1925 and 1966, but the bulk of them are from the ten year period 1928--1937. Thus they were written during the last year of his stay in Rome and his early years at John Hopkins University. The introduction by M. Artin and B. Mazur contains an illuminating discussion of the impact of these papers on the later work of other mathematicians.
The papers themselves may be broadly divided into three sections: (1) solvability by radicals of the equations of plane-algebraic curves, (2) the fundamental group of the residual space of a plane algebraic curve and (3) the topology of the singularities of plane algebraic curves.
In addition three survey articles are reproduced. The first is Zariski's lecture to the International Congress of Mathematicians at Harvard in 1950 surveying his overall view of algebraic geometry [Zbl 0049.22701]. The second is an introduction to the application of valuation theory to algebraic geometry from lectures given in Rome in 1953. The third on Serre's coherent sheaves is a report of a seminar at an AMS summer institute in 1954 and is still a good place to find an introduction to algebraic sheaf theory. collected papers; solvability by radicals; valuation theory; algebraic sheaf theory; uniformization of algebraic functions; purity of branch locus; fundamental group of a curve O. Zariski,Collected Papers, Vol. III, MIT Press, 1978, pp. 43--49. History of algebraic geometry, Collected or selected works; reprintings or translations of classics, Coverings of curves, fundamental group, Coverings in algebraic geometry, General valuation theory for fields, Singularities of curves, local rings Collected papers. Vol. III: Topology of curves and surfaces, and special topics in the theory of algebraic varieties. Edited and with an introduction by M. Artin and B. Mazur | 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 Segre class of a singular projective variety X is that of the normal cone of the diagonal in the product \(X\times X\). This class was introduced by K. W. Johnson (and W. Fulton) [see \textit{K. W. Johnson}, Acta Math. 140, 49-74 (1978; Zbl 0373.14005)] to study immersions and embeddings. In the author's earlier work [Trans. Am. Math. Soc. 298, 169- 191 (1986; Zbl 0632.14019)], motivated by the remarkable relation between MacPherson's Chern class and Chern-Mather class (i.e., the Dubson formula) he related the Johnson's Segre class and the Segre-Mather class for hypersurfaces with codimension one singularities and \(X^ n\subset {\mathbb{P}}^{2n}\) with isolated singularities.
In the present paper the author generalizes his earlier results to the case of \(X^ n\subset {\mathbb{P}}^ N\) with singularities of codimension N- n (N\(\leq 2n)\). Segre class of a singular projective variety; normal cone of the diagonal; Chern-Mather class Singularities in algebraic geometry, Characteristic classes and numbers in differential topology A formula for Segre classes of singular projective 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 prove Artin's conjecture on the finiteness of the Brauer group for an arithmetic model of a hyperkähler variety \(V\) over a number field \(k\hookrightarrow\mathbb C\) provided that \(b_2(V\otimes_k\mathbb C)> 3\). We show that the Brauer group of an arithmetic model of a simply connected Calabi-Yau variety over a number field is finite. We also prove that if the \(l\)-adic Tate conjecture on divisors holds for a certain smooth projective variety \(V\) over a field \(k\) of arbitrary characteristic \(\mathrm{char}(k)\neq l\), then the group \(\mathrm{Br}^\prime(V\otimes_k k^{\mathrm{s}})^{\mathrm{Gal}(k^{\mathrm{s}}/k)}(l)\) is finite independently of the semisimplicity of the continuous \(l\)-adic representation of the Galois group \(\mathrm{Gal}(k^{\mathrm{s}}/k)\) on the space \(H^2_{\text{ét}}(V\otimes_kk^{\mathrm{s}},\mathbb Q_l(1))\). hyperkähler variety; Calabi-Yau variety; arithmetic model; Brauer group; Artin's conjecture; \(K3\)-surface; abelian surface; Hilbert scheme of points; generalized Kummer variety; Hilbert modular surface Brauer groups of schemes, Algebraic theory of abelian varieties On the Brauer group of an arithmetic model of a hyperkähler variety over a number field | 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 On this joyous occasion of honoring Dick Palais' 60th birthday, it is fitting to discuss a question he raised in 1975: Is the group of polynomial automorphisms of a real compact affine variety \(V\subset\mathbb{R}^n\) necessarily a Lie group? The answer is: ``Sometimes but not always; it is frequently trivial'':
\(\bullet\) always if \(n=2\) (Theorem 7.1).
\(\bullet\) If \(V\) is a hypersurface defined by a proper polynomial map \(f:\mathbb{R}^n\to\mathbb{R}\), then the group \(H=\Aut(\mathbb{R}^n,V)\) of ambient automorphisms is isomorphic to an algebraic subgroup of an orthogonal group \(V\subset O(q)\), and the action of \(H\) on \(V\) is equivalent to that of \(G\) on a variety \(W\subset\mathbb{R}^q\) (Theorem 3.1).
\(\bullet\) Consider the variety \(Y\subset\mathbb{R}^4\times\mathbb{R}^3\) defined by \(\| x\|^4+\| y\|^2=1\) (where \(\| a\|\) denotes the Euclidean norm of a vector \(a)\), diffeomorphic to the 7-sphere. \(\Aut(Y)\) does not act algebraically, and its identity component is not locally compact (section 4).
\(\bullet\) For almost every polynomial \(f:\mathbb{R}^n\to\mathbb{R}\) of degree \(d\) such that \(f^{-1}(0)=V_f\) is hypersurface, \(\Aut(\mathbb{R}^n,V_f)\) is trivial, provided \(d\geq 3\), \(n\geq 2\) (section 6).
\(\bullet\) Let \(X=f^{-1}(0)\subset \mathbb{R}^2\) be a compact connected hypersurface (a curve) obtained from a polynomial \(f:\mathbb{R}^2 \to \mathbb{R}\) of degree \(d\geq 2\). If \(\Aut(X)\) is infinite then \(X\) is isomorphic to the variety \(S^1\subset \mathbb{R}^2\) defined by \(x^2+ y^2=1\) (section 7). group of polynomial automorphisms of a real compact affine variety M. W. Hirsch, Automorphisms of compact affine varieties, in Global Analysis in Modern Mathematics (Orono, ME, 1991; Walthom, MA, 1992), Publish or Perish, Houston, TX, 1993, pp. 227--245. Topological properties of groups of homeomorphisms or diffeomorphisms, Real-analytic and semi-analytic sets, Groups acting on specific manifolds Automorphisms of compact affine 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 deals with the problem of characterizing the pairs \((A,L)\) where A is an abelian surface and L is a very ample line bundle on A. The main result can be expressed by saying that if A is a cyclic covering of order \(n\geq 5\) of the Jacobian variety J of a curve of genus 2 and L is the pull-back of the principal polarization of J then L is very ample, with some cases of exception which can be explicitly described (and which do not occur if, for instance, A does not contain elliptic curves). The main tool in the proof is a close analysis of special divisors on cyclic coverings of order \(\geq 5\) of a curve of genus 2. For \(n=5\) the aforementioned result yields a direct proof of the existence of smooth abelian surfaces of degree 10 in \({\mathbb{P}}^ 4\) which are zero loci of the Horrocks-Mumford bundle. cyclic coverings of a curve; abelian surface; very ample line bundle; Horrocks-Mumford bundle Ramanan, S.: Ample divisors on abelian surfaces. Proc. of London Math. Soc.51, 231--245 (1985) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves, Abelian varieties and schemes Ample divisors on abelian surfaces | 0 |
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