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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 \(\Theta\) be an arbitrary variety of algebras and \(H\) be an algebra in \(\Theta\). Along with algebraic geometry in \(\Theta\) over the distinguished algebra \(H\) we consider logical geometry in \(\Theta\) over \(H\). This insight leads to a system of notions and stimulates a number of new problems. We introduce a notion of logically separable in \(\Theta\) algebras and consider it in the frames of logical-geometrical relations between different \(H_1\) and \(H_2\) in \(\Theta\). The paper aims at giving a flavor of a rather new subject in a short and concentrated manner. variety of algebras; free algebra; algebraic geometry in a variety; logical geometry in a variety; geometrically equivalent algebras; logically equivalent algebras Categories of algebras, Free algebras, Foundations of algebraic geometry Locicaly separable algebras in varieties of algebras | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Each smooth elliptic Calabi-Yau 4-fold determines both a three-dimensional physical theory (a compactification of ``\(M\)-theory'') and a four-dimensional physical theory (using the ``\(F\)-theory'' construction). A key issue in both theories is the calculation of the ``superpotential'' of the theory, which by a result of Witten is determined by the divisors \(D\) on the 4-fold satisfying \(\chi({\mathcal O}_D)=1\). We propose a systematic approach to identify these divisors, and derive some criteria to determine whether a given divisor indeed contributes. We then apply our techniques in explicit examples, in particular, when the base \(B\) of the elliptic fibration is a toric variety or a Fano 3-fold.
When \(B\) is Fano, we show how divisors contributing to the superpotential are always ``exceptional'' (in some sense) for the Calabi-Yau 4-fold \(X\). This naturally leads to certain transitions of \(X\), i.e., birational transformations to a singular model (where the image of \(D\) no longer contributes) as well as certain smoothings of the singular model. The singularities which occur are ``canonical'', the same type of singularities of a (singular) Weierstrass model. We work out the transitions. If a smoothing exists, then the Hodge numbers change.
We speculate that divisors contributing to the superpotential are always ``exceptional'' (in some sense) for \(X\), also in \(M\)-theory. In fact we show that this is a consequence of the (log)-minimal model algorithm in dimension 4, which is still conjectural in its generality, but it has been worked out in various cases, among which are toric varieties. \(F\)-theory; superpotential; Calabi-Yau 4-fold; divisors; elliptic fibration; toric variety; Fano 3-fold; transitions; smoothings of the singular model; Hodge numbers A. Grassi, \textit{Divisors on elliptic Calabi-Yau} 4\textit{-folds and the superpotential in F-theory, I}, alg-geom/9704008. Calabi-Yau manifolds (algebro-geometric aspects), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Divisors, linear systems, invertible sheaves, Supermanifolds and graded manifolds Divisors on elliptic Calabi-Yau 4-folds and the superpotential in F-theory. 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 There is the problem of expressing topological invariants of maps or of their germs (say, multiplicity, real index, and so on) in analytic terms. Here we discuss this problem for the zeta-function of a monodromy of a plane algebraic curve (or, in other terms, of a polynomial in two variables) with one branch at infinity. We show that this zeta-function coincides with the Poincaré series of an appropriate ring of functions (or of an appropriate semigroup). topological invariants of maps; zeta-function; monodromy of a plane algebraic curve; Poincaré series Campillo, A.; Delgado, F.; Gusein-Zade, S. M.: On the monodromy at infinity of a plane curve and the Poincaré series of its coordinate ring, Topology 8, No. 2, 1839-1842 (1998) Plane and space curves, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) On the monodromy at infinity of a plane curve and the Poincaré series of its coordinate ring | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this paper is to initiate a study of the cohomology rings of invariant subvarieties of a smooth projective \(variety\quad X\) with a holomorphic vector field V having nontrivial zero \(set\quad Z.\) We will first consider the case in which V is generated by a torus action on X, showing that if V is tangent to the set of smooth points of a closed subvariety Y of X such that \(Y\cap Z\) is finite, then the graded ring \(i^*H^{\bullet}(X;{\mathbb{C}})\), \(i: Y\to X\) being the inclusion, is the image under a \({\mathbb{C}}\)-algebra homomorphism \(\psi\) of the graded algebra associated to a certain filtration of \(H^ 0(Y\cap Z;{\mathbb{C}})\). In certain cases, for example when Z is finite and \(i^*\) surjective, \(\psi\) is an isomorphism.
Applying this to the vector fields on flag varieties \(X=G/B\) gives a surprising description of the cohomology algebra of a Schubert variety which is now explained. Suppose G is a semi-simple complex Lie group, B a Borel subgroup and \(X=G/B\) the associated flag variety. Let \({\mathfrak h}\) be a Cartan subalgebra of Lie(G) and Lie(B), and let W be the associated partially ordered Weyl group of G. For any regular element \(h\in {\mathfrak h}\), consider the regular orbit \(W\cdot h\subset {\mathfrak h}\) as a finite reduced subvariety of \({\mathfrak h}\) with ring of regular functions \(A(W\cdot h)=A({\mathfrak h})/I(W\cdot h)\), the ring of complex polynomials on \({\mathfrak h}\) modulo those vanishing on \(W\cdot h.\)
The ascending filtration on A(\({\mathfrak h})\) coming from the degree of a polynomial gives an ascending filtration F of A(W\(\cdot h)\) whose associated graded ring Gr A(W\(\cdot h)\) is isomorphic with \(H^{\bullet}(X;{\mathbb{C}})\). The upshot of our result on torus action is that if \(X_ w=\cup_{v\leq w}BvB/B\) is the generalized Schubert variety in X determined by \(w\in W\), then \(H^{\bullet}(X_ w;{\mathbb{C}})\cong Gr A([e,w]\cdot h)\), where \([e,w]\cdot h=\{v\cdot h| \quad v\leq w\}\) and the \({\mathbb{C}}\)-algebra on the right is the graded algebra associated to the ring of regular functions on the subvariety [e,w]\(\cdot h\) of \(W\cdot h\) with natural ascending filtration defined as above. In addition, the natural map \(i^*: H^{\bullet}(X;{\mathbb{C}})\to H^{\bullet}(X_ w;{\mathbb{C}})\) is precisely the restriction \(j^*_ h: Gr(A(W\cdot h))\to Gr(A([e,w]\cdot h))\) where \(j_ h: [e,w]\cdot h\to W\cdot h\) is the inclusion. toric action; cohomology rings of invariant subvarieties; holomorphic vector field; cohomology algebra of a Schubert variety Akyıldız E. , Carrell J.B. , Lieberman D.I. , Zeros of holomorphic vector fields on singular spaces and intersection rings of Schubert varieties , Compositio Math. 57 ( 2 ) ( 1986 ) 237 - 248 , MR827353 (87j:32086). Numdam | Zbl 0613.14035 Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homogeneous spaces and generalizations, Geometric invariant theory Zero of holomorphic vector fields on singular spaces and intersection rings of 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 We study an invariant of plane algebraic curves with several components. Such invariant, called here a characteristic variety, is a collection of subtori in the group of characters of the fundamental group of the complement to the curve. This invariant is a generalization of one variable Alexander polynomial. The paper discusses the basic properties of characteristic varieties and their calculation in terms of position of the singularities of the curve in the plane. plane algebraic curves; characteristic variety; fundamental group of the complement to the curve A. Libgober, Characteristic varieties of algebraic curves, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 215 -- 254. Plane and space curves, Homotopy theory and fundamental groups in algebraic geometry, Singularities of curves, local rings Characteristic varieties of algebraic curves. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Threefolds with a pencil of del Pezzo surfaces with \(1\le K^2\le 5\) are studied. The curve which parametrizes the exceptional curves on the fibers together with the incidence correspondence, induced by the intersection of the exceptional curves on the fibers, determines a Prym-Tjurin variety which is an abelian subvariety of the Jacobian of this curve. Under certain restrictions on the singular fibers of the pencil the author proves that the Prym-Tjurin variety, the intermediate Jacobian of the three-fold and the Chow group of algebraic one-cycles algebraically equivalent to zero are isomorphic. Moreover, the first two are isomorphic as principally polarized abelian varieties (see theorem 5.7 and theorem 7.3). In section 6 the author discusses the isomorphism between the Prym variety and the intermediate Jacobian and gives geometrical conditions on the singular fibers, under which this isomorphism holds (see proposition 6.1), and describes appropriate examples of del Pezzo fibrations with \(K^2_{\text{fiber}}=2, 3\). threefolds; pencil of del Pezzo surfaces; exceptional curves; Prym-Tyurin variety; intermediate Jacobian; Chow group Kanev V., Intermediate Jacobians and Chow groups of threefolds with a pencil of del Pezzo surfaces, Ann. Mat. Pura Appl., 1989, 154, 13--48 Picard schemes, higher Jacobians, Parametrization (Chow and Hilbert schemes), \(3\)-folds, Jacobians, Prym varieties Intermediate Jacobians and Chow groups of threefolds with a pencil of del Pezzo surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper considers a topological definition for ''branch of a complex algebraic variety along a subvariety''. Connections with classical algebraic branch theory are discussed, and examples are given. branch of a complex algebraic variety Singularities in algebraic geometry, Topological properties in algebraic geometry Branches of a variety along a subvariety from topological point of view | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 variant of Torelli's theorem due to \textit{A. Andreotti} [Am. J. Math. 80, 801-828 (1958; Zbl 0084.173] states that the birational equivalence class of a curve of genus \(g\) is determined by that of its (g-1)-fold symmetric product. The reviewer [Am. J. Math. 87, 257-261 (1965; Zbl 0137.405)] showed that the same is true for the p-fold symmetric product, \(1\leq p\leq g-1\). Independently the author recently discovered this generalisation of Andreotti's version, and gives a neat proof depending only on Poincaré's formula, and points out some immediate consequences. Torelli's theorem; birational equivalence class of a curve; symmetric product Z. Ran, On a theorem of Martens, Rend. Sem. Mat. Univ. Politec. Torino, \textbf{44} (1986), 287-291. Jacobians, Prym varieties, Rational and birational maps, Compact Riemann surfaces and uniformization, Families, moduli of curves (analytic) On a theorem of Martens | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Prym-Tyurin abelian variety were initially by Tyurin as a natural generalization of Prym varieties.
1)
\textit{A. Beauville} [in: Theta functions, Proc. 35th Summer Res. Inst. Bowdoin Coll., Brunswick/ME 1987, Proc. Symp. Pure Math. 49, Pt. 1, 607--620 (1989; Zbl 0736.14020)] showed that if some endomorphism \(u\) a Jacobian \(J(C)\) for curve \(C\) has connected kernel, the principal polarization on \(J(C)\) induces a multiple of the principal polarization on the image of \(u.\) The author reformulates and completes this theorem proving ``constructively'' the following
Theorem. Let \(Z \subset J(C)\) be an abelian subvariety and \(Y\) its complementary variety. Then \(Z\) is a Prym-Tyurin variety with respect to \(J(C)\) if and only if the following sequence
\[
0 \rightarrow Y \rightarrow J(C) \rightarrow Z \rightarrow 0
\]
is exact.
Theorem 2.1'. Let \(Z\) be an abelian subvariety of \(J(C).\) If \(Ker(N_{Z})\) is connected, then \(Z\) is a Prym-Tyurin variety.
2) \textit{E. Izadi} [in: Applications of algebraic geometry to coding theory, physics and computation. Proceedings of the NATO advanced research workshop, Eilat, Israel, February 25-March 1, 2001. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 36, 207--214 (2001; Zbl 1006.14015)] set the question whether every principal polarization abelian variety is a Prym-Tyurin variety for a symmetric fixed point free correspondence. In this work, the author provides a contribution to a possible negative answer to this question by building a classical Prym-Tyurin variety explicitly, but this variety can never be defined through a fixed point free correspondence, consequently confirming that Kanev's condition is only sufficient. Prym-Tyurin abelian variety; Jacobian \(J(C)\) for curve \(C\); principal polarization on \(J(C)\) Vector bundles on curves and their moduli, Families, moduli of curves (algebraic), Jacobians, Prym varieties, Automorphisms of curves, Subvarieties of abelian varieties Some remarks on Prym--Tyurin 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 extract the Abhyankar-Moh-Suzuki theorem from the Lin-Zaidenberg theorem. simply connected curve; zero locus of a primitive polynomial Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) One more proof of the Abhyankar-Moh-Suzuki theorem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this brief note the author announces a series of results on duality in the category \(D^ b_ c(X,S)\) of bounded complexes of sheafs of \(\mathbb{C}\) vector spaces on a finite CW-complex \(X\) which have constructible on the strata of a stratification \(S\) of \(X\) cohomology sheafs (the stratification compatible with the CW-structure) and the category \(D^ +_ c(X,S)\) of infinite on the right complexes on \(X\) with constructive on strata of \(S\) cohomology. The results are stated in terms of the operation of reconstruction introduced in the previous more detailed work of \textit{A. Bondal} and the author [Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 6, 1183-1205 (1989; Zbl 0703.14011)] which in turn generalized the work of \textit{A. L. Gorodentsev} and \textit{A. N. Rudakov} [Duke Math. J. 54, 115-130 (1987; Zbl 0646.14014)]. The author states two results which guarantee existence of the Serre functor (i.e. a functor \(F:{\mathcal A} \to {\mathcal A}\) such that \(\hbox{Hom}_{\mathcal A}(A,B)=\hbox{Hom}_{\mathcal A}(B,F(A))^*\). In category \(D^ +_ c(X,S)\) this is the case if \(S\) is stratification with simply-connected strata and in \(D^ b_ c(X,S)\) if \(S\) is stratification of complex toric variety by the orbits of the torus. Explicit calculation of the Serre functor in the case \(X=\mathbb{C}^ n\) using geometric Fourier transform of Brylinski is stated. duality; bounded complexes of sheafs; finite CW-complex; strata of a stratification; cohomology sheafs; Serre functor; complex toric variety Kapranov, M, Mutations and Serre functors in constructible sheaves, Funct. Anal. Appl., 24, 155-156, (1990) Sheaf cohomology in algebraic topology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Mutations and Serre functors on constructive 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 This paper contains the proof of the following result: the general Enriques threefold \(V\subset {\mathbb{P}}^ 4({\mathbb{C}})\) is nonrational. This proof was announced by the author in a previous note and is founded on the following preliminary and nice result: the general Enriques threefold V admits a standard model \(\pi_ W:\quad W\to S_ W\) where \(S_ W\) is the blow-up of the product \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) in four general points and the degeneracy curve \(C_ W\) is a smooth curve of genus 5 (proposition (2.1)). The notion of standard model was defined by \textit{V. G. Sarkisov} [Russ. Math. Surv. 34, No.4, 183-184 (1979); translation from Usp. Mat. Nauk. 34, No.4, 207-208 (1979; Zbl 0426.14017)]. The result of the present paper is founded on a result of Shokurov on the intermediate Jacobian \(J_ 3(W)\) and a result of \textit{C. H. Clemens} and \textit{P. A. Griffiths} [Ann. Math., II. Ser. 95, 281-356 (1972; Zbl 0214.483)]. nonrationality of the general Enriques variety; standard model; intermediate Jacobian Эндрюшка, С. Ю., Нерациональность общего многообразия энриквеса, Матем. сб., 123(165), 2, 269-275, (1984) Rational and unirational varieties, \(3\)-folds, Picard schemes, higher Jacobians Nonrationality of the general Enriques 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 In the present paper, the author continues his vast investigations on finite automorphism groups of compact topological surfaces. His general interest includes, in particular, the automorphism groups of Riemann surfaces and Klein surfaces, as well as direct applications to the classification of the real forms of a given complex algebraic curve.
The first part of the paper, i.e., the first four sections, contains a detailed study of the properties of finite homeomorphism groups of compact oriented surfaces. A special emphasis is put on those groups which contain reflections, i.e., orientation reversing involutions with fixed points. The main results here include the description of the connected components of the fixed point sets of reflections, as well as an estimate for the number of those connected components.
The second part of the paper (i.e., the remaining five sections) deals with concrete applications of the fore-going results to the study of real forms of complex algebraic curves. Section 5 clarifies the link between reflections, automorphisms of Riemann surfaces, and real algebraic curves. Section 6 gives constructive methods for obtaining Riemann surfaces with antiholomorphic automorphisms. The problem of determining the number of the connected components of the set of real points in complex curves is discussed in section 7, and section 8 deals with properties of families of Riemann surfaces (of given genus \(g)\) admitting special antiholomorphic reflections. The concluding section \(9\) contains quantitative and qualitative results on particular real algebraic curves and their automorphism groups. More precisely, the author studies here the possible real forms of the so-called complexified real (M-1)-curves, which he had introduced and investigated already in some earlier papers. finite automorphism groups of compact topological surfaces; Riemann surfaces; Klein surfaces; real forms of a given complex algebraic curve; connected components of the fixed point sets of reflections; antiholomorphic automorphisms; complexified real (M-1)-curves Natanzon, S. M.: Finite groups of homeomorphisms of surfaces, and real forms of complex algebraic curves. (Russian). Trudy Moskov. Mat. Obshch. 51 (1988), 3-53, 258. Translation in Trans. Moscow Math. Soc. (1989), 1-51. Rational and birational maps, Compact Riemann surfaces and uniformization, Real algebraic and real-analytic geometry, Families, moduli of curves (analytic) Finite groups of homeomorphisms of surfaces and real forms of complex algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We present an \(0(n\cdot d^{0(1)})\) algorithm to compute the convex hull of a curved object bounded by 0(n) algebraic curve segments of maximum degree d. convex hull of a curved object; algebraic curve C. Bajaj and M.-S. Kim: \(Convex Hulls of Objects Bounded by Algebraic Curves\). Algorithmica 6(1991), pp. 533-553. Computer graphics; computational geometry (digital and algorithmic aspects), Analysis of algorithms and problem complexity, Computational aspects of algebraic curves Convex hulls of objects bounded by algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians An ideal I of a local ring (R,\({\mathfrak m})\) is called a generalized Cohen- Macaulay ideal if \(S=R/I\) is a generalized Cohen-Macaulay ring, i.e. \(\ell (H^ i_{{\mathfrak m}}(S))<\infty\) for \(i=0,...,\dim S-1.\) The class of such ideals is rather large, e.g. it includes the defining prime ideals of isolated singularities. Inspired of an article of \textit{J. Sally} on Cohen-Macaulay ideals [Pac. J. Math. 63, 517-520 (1976; Zbl 0314.13009)], this paper will give bounds for the minimum number of generators v(I) of a generalized Cohen-Macaulay ideal I in terms of the embedding dimensions, the dimensions, the multiplicities of R, S, and \({\mathcal L}(H^ i_{{\mathfrak m}}(S))\). There are some interesting applications. For example, if I is the defining prime ideal of an arithmetically Buchsbaum curve C in \(P^ 3_ k\), i.e. \(H^ 1_ m(S)\) is a vector space over the field k, where S is the local ring of the vertex of the affine cone over C, then \(v(I)\leq 3e/2+1,\) where e denotes the degree of C. multiplicity; local cohomology modules; generalized Cohen-Macaulay ideal; minimum number of generators; embedding dimensions; arithmetically Buchsbaum curve Ngô Viá»\?t Trung, Bounds for the minimum numbers of generators of generalized Cohen-Macaulay ideals, J. Algebra 90 (1984), no. 1, 1 -- 9. Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Multiplicity theory and related topics, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Bounds for the minimum numbers of generators of generalized Cohen- Macaulay ideals | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 reductive connected algebraic group over an algebraically closed field, and let u be a unipotent element of G. Let \(A_ G(u)\) be the group of components of the centralizer \(Z_ G(u)\). \(A_ G(u)\) acts naturally by permutations on the set of irreducible components of the variety of Borel subgroups containing u and Springer has shown that (with some restrictions on the characteristic) the irreducible representations of \(A_ G(u)\) appearing in this permutation representation for various u (up to conjugacy) are in 1-1 correspondence with the irreducible representations of the Weyl group. However, in general, not all irreducible representations of \(A_ G(u)\) appear in this permutation representation. In this paper, the author investigates the missing representations.
Let P be a parabolic subgroup of G with Levi decomposition \(P=LU_ P\), and let v be a unipotent element in L. Let \(Y_{u,v}=\{gZ^ 0_ L(v)U_ p|\quad g\in G,\quad g^{-1}ug\in vU_ P\}.\) Then \(\dim Y_{u,v}\leq d=1/2(\dim Z_ G(u)-\dim Z_ L(v)).\) The group \(Z_ G(u)\) acts naturally on \(Y_{u,v}\) by left translation. This induces an action of the finite group \(A_ G(u)\) on the finite set \(S_{u,v}\) of irreducible components of dimension d of \(Y_{u,v}\). When P is a Borel subgroup and \(v=1\), this is just the action considered by Springer. An irreducible representation of \(A_ G(u)\) is said to be cuspidal if it does not appear in the permutation representation on \(S_{u,v}\) for any \(P\), \(v\) as above with \(P\neq G\). The author shows that very few representations of \(A_ G(u)\) are cuspidal. More precisely, for a fixed character \(\chi\) of the group \(\Gamma\) of components of the center of G, and for a field k of good characteristic, there is at most one pair \((u,\rho)\) with u unipotent in G (up to conjugacy) such that \(\rho\) is an irreducible cuspidal representation of \(A_ G(u)\) on which \(\Gamma\) acts according to \(\chi\). Given a pair \((u,\rho)\), the author defines a triple \((L,v,\rho')\) up to conjugacy, where \(L\) is the Levi subgroup of a parabolic subgroup of \(G\), \(v\) a unipotent element in \(L\), and \(\rho'\) is a cuspidal representation of \(A_ L(v)\), and he shows that the set of pairs \((u,\rho)\) giving rise to a fixed triple \((L,v,\rho')\) as above may be naturally put into 1-1 correspondence with the set of irreducible representations of the group of components of the normalizer of \(L\) which is shown to be a Coxeter group. It reduces to the correspondence described originally by Springer, in the case where \(L\) is a maximal torus, and it is called generalized Springer correspondence. The author determines in a combinatorial way this correspondence in the case of symplectic and special orthogonal groups in odd characteristic. This generalizes the main result of Shoji on the usual Springer correspondence for these groups. Throughout the paper, the intersection cohomology theory of Deligne-Goresky-MacPherson is used extensively. Using the result of this paper, by Spaltenstein, the correspondence for exceptional groups in arbitrary characteristic has also been determined explicitly in almost all cases. reductive connected algebraic group; unipotent element; irreducible components; variety of Borel subgroups; irreducible representations; permutation representation; Levi decomposition; irreducible cuspidal representation; Coxeter group; generalized Springer correspondence; special orthogonal groups; intersection cohomology G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205-272. Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, (Co)homology theory in algebraic geometry Intersection cohomology complexes on a reductive group | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 connected reductive algebraic group and let \(B\) be a Borel subgroup of \(G\). A spherical variety is a normal algebraic variety with an action of \(G\) and a dense orbit of \(B\). The author surveys three aspects of the theory of spherical varieties which are fields of active research: line bundles over spherical varieties, classification of spherical varieties, and orbits of a Borel subgroup in spherical varieties. group action; spherical variety; line bundles; orbits of a Borel subgroup Brion, Michel, Spherical varieties, 2 (Zürich, 1994), pp. 753-760. Birkhäuser, Basel (1995) Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations Spherical 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 As it was proved by \textit{V. V. Nikulin} [in: Algebraic Geometry and its Applications, Aspects Math. E 25, 113-136 (1994; Zbl 0840.14024)] for a non-singular real algebraic variety \(X\) the étale cohomology \(H^n_{\text{et}}(X,\mathbb{Z}_2)\) coincides with the equivariant cohomology \(H^n(X)(\mathbb{C});G,\mathbb{Z}_2)\), where \(G=G(\mathbb{C}/ \mathbb{R})\) is the Galois group. In the present paper, the author uses this relation to obtain several results in problems that were initially formulated in terms of étale cohomology. In particular, he computes the Witt group of a real Enriques surface. equivariant cohomology; real algebraic variety; étale cohomology; Witt group of a real Enriques surface V. A. Krasnov, ''Étale and equivariant cohomology of a real algebraic variety,'',Izv. Ross. Akad. Nauk Ser. Mat., [Russian Acad. Sci. Izv. Math.] (to appear)''. Topology of real algebraic varieties, Étale and other Grothendieck topologies and (co)homologies, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Equivariant homology and cohomology in algebraic topology, \(K3\) surfaces and Enriques surfaces The étale and equivariant cohomology 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 Let \(\widetilde C\) be a smooth projective curve with an involution \(\sigma\), let \(C=\widetilde C/ \sigma\) and let \(\delta\) be the invertible sheaf of \(C\) such that \(\pi_*{\mathcal O}_{\widetilde C}\cong {\mathcal O}_C \oplus \delta\). For \(A=K_C(-D)\) or \(K_C\otimes \delta(-D)\), where \(D\geq 0\) one considers the set \(W^r_A= \{L\in\text{Pic} (\widetilde C)\mid \text{Nm}(L) \cong A\), \(h^0 (\widetilde C)\geq r+1\}\). It is proven the non-emptiness of \(W^r_{K_C(-D)}\) and there are given some estimates from below for \(\dim W^r_A\) which are sharper than those obtained by Brill-Noether theory. For instance if \(n=\deg (\delta)\), \(d=\deg (D)\) then
\[
\dim W^r_{K_C(-D)} \geq g(C)- 1+n- (d+n) (r+1)- r(r+1)/2.
\]
This exends and gives a new proof of a theorem of \textit{A. Bertram} [case \(\delta=0\), \(D=0\), see: Invent. Math. 90, 669-671 (1987; Zbl 0646.14006)]. An application to the Brill-Noether theory of vector bundles of rank 2 with canonical determinant is given. Picard group; Prym variety; curve with an involution; Brill-Noether theory V. Kanev,special line bundles on curves with involution, Math. Z.222 (1996), 213--229. Vector bundles on curves and their moduli, Determinantal varieties, Picard groups, Jacobians, Prym varieties, Picard schemes, higher Jacobians, Divisors, linear systems, invertible sheaves Special line bundles on curves with involution | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this article, we consider the Jacobian variety \(A_ r\) of the algebraic curve \(C_ r\) with equation \(y^ p=x(1-x)^ r\) over the complex number field \(\mathbb C\), where \(r\in \mathbb Z\) and \(1\leq r\leq p-2\). \(C_ r\) is a quotient of the Fermat curve \(F_ p\) of exponent \(p: X^ p+Y^ p+Z^ p=0\), and the Jacobian of \(F_ p\) is isogenous over \(\mathbb Q\) to \(A_ 1\times\ldots\times A_{p-2}\). It is well known that \(A_ r\) admits complex multiplication by the ring of integers \(\mathfrak O\) of the \(p\)-th cyclotomic field \(K\), and that \({\mathfrak O}\) is equal to the endomorphism ring \(\text{End}(A_ r)\) of \(A_ r\) when \(A_ r\) is simple.
The object of the paper is to determine \(\text{End}(A_ r)\) whenever \(A_ r\) is nonsimple. From results due to Koblitz-Rohrlich and Schmidt, such an \(A\) is isogenous to a cube of a simple abelian variety over \(K\). We use our result on \(\text{End}(A_ r)\) to prove that \(A_ r\) is isomorphic over \(K\) to a cube of a simple abelian variety. endomorphism ring of Jacobian; Fermat curve; complex multiplication Hai Lim, C.: The Jacobian of a cyclic quotient of the Fermat curve. Nagoya Math. J. 125, 73--92 (1992) Jacobians, Prym varieties, Arithmetic ground fields for curves, Arithmetic ground fields for abelian varieties The Jacobian of a cyclic quotient of a Fermat 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 It is a classical result due to \textit{G. Castelnuovo} [Rend. Circ. Mat. Palermo 7, 89-110 (1893), see also Memorie scelte (1973)] that the arithmetic genus of an irreducible, non degenerate curve of degree \(d\) in a projective space \({\mathbb{P}}^ r\) (over an algebraically closed field of characteristic zero) has an upper bound which is a function of d and r. The original idea of Castelnuovo, essentially based on the so called ''uniform position lemma'', is briefly reviewed in this paper and some other applications of his method are given in order to prove a few uniqueness theorems for certain linear series on some classes of curves, like curves of high genus with respect to the degree and the dimension of the projection space in which they lie, subcanonical curves, complete intersection curves. In particular a proof is given of a result stated, but never proved, by \textit{M. Noether} [''Zur Grundlegung der Theorie der algebraischen Raumkurven'', Math.-Phys. Abhdl., Kgl. Preuss. Akad. Wiss. Berlin (1883)] which gives a maximum for the dimension of linear series of given degree on a smooth plane curve of degree d\(>3\), and a characterization of the linear series for which the maximum is attained. This theorem has been recently proved in a different way, and in a more general form, by \textit{R. Hartshorne} [J. Math. Kyoto Univ. 26, 375-386 (1986; Zbl 0613.14008)]. Results related to the present paper are also contained in the note by the author and \textit{R. Lazarsfeld} in Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 198-213 (1984; Zbl 0548.14016). maximum for the dimension of linear series of given degree on a smooth plane curve of degree d C. Ciliberto , Alcune applicazioni di un classico procedimento di Castelnuovo , Sem. di variabili Complesse, Univ. di Bologna, 1982-83,17-43. Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry Some applications of a classical method of Castelnuovo | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 cross ratios which are applied in this paper may usefully be thought of as an algebraic analogue for an algebraic moduli problem of the periods of a transcendental moduli problem (such as abelian varieties up to isomorphism). The relation is roughly \(\log (\text{cross ratio}) = \text{period}\). There are also coincidences between the two. A typical such moduli problem illustrates this well: At the generic point, the moduli problem of six points on \(\mathbb{P}^1\) up to isomorphism, which has cross ratios as invariants, coincides with that of genus two curves (given by the double cover of \(\mathbb{P}^1\) branched at the given six points), and by means of the Jacobian translates into the moduli problem of principally polarised abelian surfaces, for which the usual periods with values in the Siegel upper half space of degree two are the invariants. So the GIT quotient of the first problem (with semistable points acting as the boundary) is birational to the Siegel threefold (with Baily-Borel compactification). Letting \(\mathbb{P} (n,k)\) be the moduli space of \(k\) points in \(\mathbb{P}^n\), then for \(\mathbb{P} (2,6)\), \(\mathbb{P} (2,7)\), \(\mathbb{P} (2,8)\) one can describe the corresponding ratios in terms of the root systems of type \(E_6\), \(E_7\) and \(E_8\). This correspondence is originally due to A. Coble, who related the corresponding symmetry groups of birational transformation of \(\mathbb{P}^2\), and has first given a general presentation by Manin deriving from the Picard group of a Del Pezzo surface.
This paper gives a systematic presentation of these and more general moduli spaces related to certain root systems. In particular, for any subroot system of type \(A_3\) or \(D_4\) in a root system \(\Delta\), one can define a meromorphic function on the algebraic torus \(Z=: \mathbb{C}^{\text{rank} (\Delta )}-\)\{reflection hyperplanes for reflections in \(W (\Delta)\}\) \((W (\Delta)\) is the Weyl group) associated with that root system, and taking the Cartesian product over all (finitely many) such subroot systems embeds \(Z\) in \((\mathbb{P}^1)^{\# \text{subroot systems}}\); the algebraic closure of the image is the cross ratio variety.
The author considers the following two types of loci on these cross ratio varieties: (a) boundary (value of some cross ratios are 0 or 1), and (b) subloci which are themselves cross ratio varieties for \(\Delta' \subset \Delta\). These are quite analogous to boundary components (of Baily-Borel type compactifications of arithmetic quotients of period domains), and to modular subvarieties, i.e., subvarieties defined by subgroups of the automorphism group of the period domain, respectively. Let \(C (\Delta, A_3)\) and \(C (\Delta, D_4)\) denote the corresponding cross ratio varieties. Then \(C (\Delta (A_n), A_3)\) turns out to be closely related to Appell-Lauricella hypergeometric differential equations. The interesting cases are where \(\Delta\) is of type \(E_k\), \(k = 6,7,8\). For \(\Delta (E_6)\), the variety \(C ( \Delta (E_6), D_4)\) was considered already in an earlier paper [\textit{I. Naruki}, Proc. Lond. Math. Soc., III. Ser. 45, 1-30 (1982; Zbl 0508.14005)]. Here the author determines several subvarieties of the type \(C (\Delta', D_4)\) for \(\Delta' \subset \Delta (E_6)\). The main emphasis of the paper is the case \(\Delta = \Delta (E_7)\), and the variety \(C (\Delta (E_7)\), \(D_4)\). Here the boundary and subloci are studied and the author offers a list for the boundary components, which he conjectures is complete. periods of a transcendental moduli problem; moduli problem of principally polarized abelian surfaces; periods; root systems; algebraic torus; cross ratio variety Sekiguchi, J.: Cross-ratio varieties for root systems. Kyushu J. Math. 48, 123--168 (1994) Algebraic moduli problems, moduli of vector bundles, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Geometric invariant theory, \(n\)-folds (\(n>4\)) Cross ratio varieties for root 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 This article contains the details of the following two announcements: C. R. Acad. Sci., Soc. R. Can. 3, 273-278 (1981; Zbl 0495.14015) by these authors and Can. J. Math. 34, 169-180 (1982; Zbl 0477.14019) by the second author. The authors construct the coordinate ring A of a curve with one singular point P via Cartesian squares \(A\to^{f}\bar A\to^{\pi}\prod^{s}_{i=1}k[t_ i]/t^ n_ i;\quad A\to D\to^{g}\prod^{s}_{i=1}k[t_ i]| t^ n_ i\) (f and g are inclusions and \(\pi\) is onto), where \(\bar A\) is the normalization of A and k is a field. They compute several invariants of the local ring \(A_ P\), which only depend on g. In particular, there is shown an algorithm for computing the Hilbert function H of P. Thus they decided in several cases when H can have a temporary decrease. In particular, there is a generic homogeneous ordinary singular point such that H can decrease and \(H(1)=4\). Furthermore, the authors discuss the Cohen-Macaulay type of \(A_ P\) and its relationship to the Cohen-Macaulay type of its reduced tangent cone. In the case of generic homogeneous ordinary singularities both are the same, but in other cases, they may differ. This answers some questions posed by \textit{A. V. Geramita} and \textit{F. Orecchia} in J. Algebr 78, 36-57 (1982; Zbl 0502.14001) and by \textit{A. V. Geramita} and \textit{P. Maroscia} in C. R. Math. Acad. Sci., Soc. R. Can. 4, 179-184 (1982; Zbl 0493.14001). Furthermore the authors show that the K-theory of A, \(K_ i(A)\), \(i\leq 1\), depends on the number of irreducible components of Spec A (and on the inclusion). ordinary singularities of curves; coordinate ring of a curve; Hilbert function; Cohen-Macaulay type; K-theory Gupta, S. K.; Roberts, L. G., Cartesian squares and ordinary singularities of curves, \textit{Commun. Algebra}, 11, 2, 127-182, (1983) Singularities of curves, local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Grothendieck groups, \(K\)-theory and commutative rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry Cartesian squares and ordinary singularities 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 Shimura variety associated to an indefinite quaternion algebra over a totally real field is considered. It is assumed that there is a prime number p which stays prime in the extension field and where the quaternion algebra ramifies. The level structure is such that it does not involve the prime p. Using some of the results of \textit{T. Zink} [Compos. Math. 45, 15-107 (1981; Zbl 0483.14006)] on the structure of the bad reduction of these varieties at p the semi-simple local zeta function is expressed in terms of automorphic L-functions. The semi-simple zeta function is not the correct factor for the functional equation of the Hasse-Weil zeta function.
It is shown, however, that assuming Deligne's conjecture on the purity of the monodromy filtration the correct local factor may be deduced from this result. The results in this paper generalize the results pertaining to quaternionic Shimura varieties in the paper by the author and \textit{T. Zink} [Invent. Math. 68, 21-101 (1982; Zbl 0498.14010)]. - In addition, the conjecture of \textit{R. P. Langlands} and the author on the reduction modulo p of a Shimura variety [J. Reine Angew. Math. 378, 113-220 (1987; Zbl 0615.14014)] is verified in this special case. Shimura variety associated to an indefinite quaternion algebra over a totally real field; semi-simple local zeta function; automorphic L- functions; purity of the monodromy filtration; local factor Rapoport, M.: On the local zeta function of quaternionic Shimura varieties with bad reduction. Math. Ann. 279, 673--697 (1988) Local ground fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Quaternion and other division algebras: arithmetic, zeta functions On the local zeta function of quaternionic Shimura varieties with bad reduction | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author considers a smooth projective complex manifold X, which is a double covering of a non singular three-dimensional quadric Q with branch locus a surface \(S\subset Q\) of degree 8. We shall call such a manifold a double quadric. In the classification of Fano's threefolds due to \textit{V. A. Iskovskih} [cf. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 12, 59-157 (1979; Zbl 0415.14024)] the double quadric X is a Fano variety of \(index\quad 1,\) i.e. the anticanonical \(class-K_ X\) generates the Picard group Pic(X). Let F(X) be the Hilbert scheme parametrized by the conics on X. When F(X) is a smooth variety, A(F(X)) is its Albanese variety. J(X) denotes the Jacobian variety of X.
Theorem: For a general double quadric X the scheme F(X) is a smooth, connected, projective surface and A(F(X)) is isomorphic to J(X).
In the work of \textit{M. Letizia} [''The Abel-Jacobi mapping for the quartic threefold'' (preprint, Univ. Utah 1982); see also the same named paper in Invent. Math. 75, 477-492 (1984; Zbl 0571.14022)] it was proved that for a general hypersurface \(Y\subset {\mathbb{P}}^ 4\), of degree 4, the mapping \(H_ 1(F(Y), {\mathbb{Z}})\to H_ 3(Y, {\mathbb{Z}})\) is an isomorphism. Here F(Y) is the surface of the conics on the quartic Y and the proof of the above theorem is a reduction to the case of the quartic Y. double quadric; Fano's threefolds; Picard group; Hilbert scheme; Albanese variety; Jacobian variety \(3\)-folds, Picard groups, Picard schemes, higher Jacobians, Parametrization (Chow and Hilbert schemes) On conics on a double 3-dimensional quadric | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a positive integer \(\alpha\) an \(\alpha\)-dot over a point \(p \in \mathbb{P}^n = \mathbb{P}^n_\mathbb{C}\) is the point \(p\) together with an \((\alpha - 1)\)-dimensional tangent space at \(p\). As a scheme, an \(\alpha\)-dot \(Y\) over \(p\) is isomorphic to \(\text{Spec} (\mathbb{C} [z_1, \ldots, z_{\alpha - 1}]/(z_1, \ldots, z_{\alpha - 1})^2)\), the first infinitesimal neighbourhood of a point in \(\mathbb{A}_\mathbb{C}^{\alpha - 1}\), with \(Y_{\text{red}}\) identified with the point \(p\). The main result is as follows:
Theorem. Let \(\Gamma\) be a collection of \(d\) \(\alpha\)-dots over points which are in linear general position in \(\mathbb{P}^n\), \(\alpha \geq 2\). Then the Hilbert function \(h_\Gamma\) of \(\Gamma\) satisfies
\[
h_\Gamma (r) \geq \min (rn + 1,2d) + (\alpha - 2) \min \bigl( (r - 1) n - 1, d \bigr)
\]
for \(r \geq 3\). Equality occurs for some \(r\) with \(rn + 2 \leq 2d\) if and only if \(\Gamma_{\text{red}}\) is contained in a rational normal curve \(C\), and the tangent directions to this curve at these points are all contained in \(\Gamma\). Equality occurs for some \(r\) with \((r - 1) n \leq d\) if and only if \(\Gamma\) is contained in the first infinitesimal neighbourhood of \(C\) with respect to a subbundle, of rank \(\alpha - 1\) and of maximal degree, of the normal bundle of \(C\) in \(\mathbb{P}^n\).
The theorem implies an upper bound on the degree of a subbundle of rank \(\alpha - 1\) of the normal bundle of a curve of degree \(d\) in \(\mathbb{P}^n \).
[See also the generalization of this paper in Trans. Am. Math. Soc. 347, No. 3, 767-784 (1995; see the following review)]. \(\alpha\)-dot; Hilbert function; normal bundle of a curve Karen A. Chandler, Hilbert functions of dots in linear general position, Zero-dimensional schemes (Ravello, 1992) de Gruyter, Berlin, 1994, pp. 65 -- 79. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Hilbert functions of dots in linear general position | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 surface and \(D\subset X\) a smooth curve of genus \(g\). Set \(r:= \dim (| D| )\) and let \(\nu _D : | D| \dasharrow \mathcal {M}_g\) be the moduli map. Here the authors study pairs \((X,D)\) with large \(r\), greatly improving classical works by Castelnuovo and Enriques. For instance they define an integer \(\tau (k,g)\), \(k \geq 1\), and for all \(k = 2,3,4\), classify the pairs \((X,D)\) such that \(\tau (k-1,g) > r \geq \tau (k,g)\). As a corollary they prove that if \(g\geq 22\), then \(\dim (\mathrm{Im} (\nu _D)) \leq 2g+1\) and classify the pairs for which equality holds (the image is the trigonal locus). For background and older questions on the topic (see [\textit{C. Ciliberto} and \textit{F. Russo}, Adv. Math. 200, No. 1, 1--50 (2006; Zbl 1086.14043)]). rational surface; divisor; curves in a surface; number of moduli; plane curve; Castelnuovo's inequality; trigonal locus Castorena, A.; Ciliberto, C.: On a theorem of Castelnuovo and applications to moduli, Kyoto J. Math. 51, No. 3, 633-645 (2011) Rational and ruled surfaces, Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic) On a theorem of Castelnuovo and applications to moduli | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 investigate the problem to determine the defining equations of the algebraic variety of Hankel two-planes in the projective space. We compute the first and the second partial lifting of the Machado's binomial relations, by applying tecniques of Sagbi bases theory. Sagbi basis; Grassmann variety; Hankel variety; toric deformation of a variety G. Failla, \textit{Combinatorics of Hankel relations}, Ann. Acad. Rom. Sci. Ser. Math. Appl. 9 (2017) 2 (to appear). Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Determinantal varieties, Toric varieties, Newton polyhedra, Okounkov bodies Combinatorics of Hankel relations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 algebraic number field of characteristic zero and \(f \in K(X)\) a monic, separable polynomial of odd degree \(d\). The equation \(Y^2 = f(X)\) defines a hyperelliptic curve \(C\) over \(K\) of genus \({d - 1 \over 2}\). By the Mordell-Weil theorem the group of \(K\)-rational points \(J(K)\) of the Jacobian variety of \(C\) is a finitely generated abelian group. The paper in question represents a new method for computing the rank of the free part of \(J(K)\), the Mordell-Weil rank of \(J\) over \(K\), or to be more precise the rank of \(J(K)/2J (K)\). The method is applied to compute \(J(\mathbb{Q})/2J (\mathbb{Q})\) for the hyperelliptic curve \(Y^2 = X(X - 2) (X - 3) (X - 4) (X - 5) (X - 7) (X - 10)\). rank of group of rational points; Jacobian; Mordell-Weil rank; hyperelliptic curve Schaefer, Edward F., \(2\)-descent on the Jacobians of hyperelliptic curves, J. Number Theory, 51, 2, 219-232, (1995) Rational points, Jacobians, Prym varieties, Elliptic curves 2-descent on the Jacobians of hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians From author's abstract: This is a survey of the topology of real plane curves, concentrating on the constructive aspects of this theory, i.e., the problem of constructing curves of a given degree with a prescribed arrangement of its components. A large part of the paper is concerned with introductory material - the formulation of the basic problems and the history of the early development of the subject - so that the exposition is essentially self-contained. A detailed presentation is given of the technique of perturbing singular curves with controlled variation of the topology. The author plans to publish the final part of the survey in the following issues.
Curves with complicated singularities are perturbed, namely semi- quasihomogeneous singularities, and dissipations of such singularities are classified.
At the end of the paper counterexamples are constructed to Ragdale's conjecture for a non-singular curve of even degree \(m: p\leq (3m^ 2- 6m+8)/8\) and \(n\leq (3m^ 2-6m)/8\) where p denotes the number of even ovals (which envelop an even number of other ovals) and n denotes the number of the other ones. real algebraic plane curve; Hilbert 16th problem; singularities of plane curve; constructing curves of a given degree; prescribed arrangement; perturbing singular curves with controlled variation of the topology; Ragdale's conjecture Brugallé, E.: Tropical curves, notes from introductary lectures given in July 2013 at Max Planck Institute for Mathematics, Bonn. http://erwan.brugalle.perso.math.cnrs.fr/articles/TropicalBonn/TropicalCurves Topology of real algebraic varieties, Singularities of curves, local rings, Enumerative problems (combinatorial problems) in algebraic geometry, Topological properties in algebraic geometry Real algebraic plane curves: Constructions with controlled topology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 under review, the authors study the inverse Jacobian problem for the case of Picard curves over \({\mathbb C.}\) More precisely, the authors provide an algorithm (see Section 3) accepting input and returning output as follows:
``Input: A period matrix \(\Omega\in \mathbf{H}_3\) of the Jacobian of a Picard curve \(C\), and the transposed rational representation \(N\in {\mathbb Z}^{6\times 6}\) of the automorphism of the Jacobian \(\rho_*\) induced by the curve automorphism \(\rho(x,y)=(x,z_3y).\)
Output: The complex values \(\lambda\) and \(\mu\) in a Legendre-Rosenhain equation \(y^3=x(x-1)(x-\lambda)(x-\mu)\) for the Picard curve \(C\),'' where \(\mathbf{H}_3\) is the Siegel upper half-space, and \(z_3\) is a primitive third root of unity.
The results follow in a great part from earlier work of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)], but these have been presented to refine, correct or generalize the results of Koike and Weng.
One of the key reasons that the inverse Jacobian problem can be done for Picard curves is the following proposition in Section 3 (cf. Lemma 1 of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)]).
\textbf{Proposition 1.} Let \(X\) be a simple principally polarized abelian variety of dimension \(3\) defined over an algebraically closed field \(k\). If \(X\) has automorphism \(\phi\) of order \(3\), then we have that \(X\) is the Jacobian of a Picard curve. Furthermore, let \(\rho\) be the curve automorphism \(\rho(x,y)=(x,z_3y),\) and let \(\rho_*\) be the automorphism of the Jacobian that it induces. Then we get \(\langle\phi\rangle=\langle\rho_*\rangle.\)
The authors of the paper under review mentioned that their proof followed the idea of Koike and Weng, but they fixed a gap by their reference to a result of Estrada. Furthermore, they remarked that the result can be seen by applying the classification of plane quartics and genus-3 hyperelliptic curves by their automorphism group [\textit{D. Lombardo} et al., ``Decomposing Jacobians via Galois covers'', Preprint, \url{arXiv:2003.07774}].
The algorithm mentioned above was made possible by establishing the following two theorems which are refined statements or generalizations of the results in [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)].
\textbf{Theorem 3.} Let \(C\) be a Picard curve defined over \({\mathbb C}\) given by \[y^3=x(x-1)(x-\lambda)(x-\mu),\] and consider the branch points \(P_0=(0,0),P_1=(1,0),P_{\lambda}=(\lambda,0),P_{\mu}=(\mu,0),\) and \(P_{\infty}\) at infinity. Let \(J(C)\) be the Jacobian of \(C\) with period matrix \(\Omega\), let \(\alpha\) be the Abel-Jacobi map with base point \(P_{\infty},\) and let \(\Delta\in J(C)\) be the associated Riemann constant. Then, for \(\eta\in {\lambda,\mu},\) we have \[\eta=\varepsilon_{\eta}\left(\frac{\theta[\widetilde{P_1}+2\widetilde{P_{\eta}}-\widetilde{P_0}-\widetilde{\Delta}](\Omega)} {\theta[2\widetilde{P_1}+\widetilde{P_{\eta}}-\widetilde{P_0}-\widetilde{\Delta}](\Omega)}\right)^3,\] where \(\varepsilon_{\eta}=\exp(6\pi i((\widetilde{P_{\eta}}-\widetilde{P_1})_1(\widetilde{P_0})_2+\widetilde{\Delta_1}(3\widetilde{P_1}+3\widetilde{P_{\eta}}-2\widetilde{\Delta})_2)).\)
The proof of the above theorem (see Section 2) uses Riemann's Vanishing Theorem, a result of Siegel, and a transition step (see Corollary 1 for details and for other undefined notations) which converts a product of Riemann theta constants as a product of theta constants with characteristics. For \(P\in C\), \(\widetilde{P}\) is a shorthand notation for the image under Abel-Jacobi map with identification of \(J(C)\) with \({\mathbb R}^{2g}/{\mathbb Z}^{2g}\) and reduction (so \(\widetilde{P}\in [0,1)^{2g},\) see (6) for details). The formula for \(\eta\) (i.e. \(\lambda\) or \(\mu\)) is to be compared with that of Corollary 11 of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)], where the correction factor \(\varepsilon_{\eta}=1,\) because the authors of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)] took a four-element subset \(\{D_1,\cdots,D_4\}\) (of which \(\widetilde{P_{\eta}},\widetilde{P_1}\), etc, are analogues) of a set analogous to \(\Theta_3\) in Theorem 4 below with \(\sum_i D_i=\mathbf{0}.\) The authors of the paper under review clarified the existence of such four-element subset and its relation to the image of branch points under the Abel-Jacobi map. Accordingly Theorem 4 below (see Section 3) gives a refinement, clarification, or generalization of Corollary 11 of [\textit{K. Koike} and \textit{A. Weng}, Math. Comput. 74, No. 249, 499--518 (2005; Zbl 1049.14014)].
\textbf{Theorem 4.} Let \(J(C)\) be the Jacobian of a Picard curve \(C,\) let \(\rho_*\) be the automorphism of \(J(C)\) induced by the curve automorphism \(\rho(x,y)=(x,z_3y).\) Let \(\mathcal{B}\) be the set of affine branch points of \(C,\) let \(\alpha\) be the Abel-Jacobi map with base point \(P_{\infty}=(0:1:0),\) let \(\Delta\) be the Riemann constant with respect to \(\alpha\) and define \[\Theta_3:=\{x\in J(C)[1-\rho_*]:\theta[\underline{x}+\underline{\Delta}](\Omega)=0\}.\] Then \(\alpha(\mathcal{B})\) and \(-\alpha(\mathcal{B})\) are the only subsets \(\mathcal{T}\subset J(C)\) of four elements such that:
\noindent (i) the sum \(\sum_{x\in \mathcal{T}}x\) is zero,
\noindent (ii) \(\mathcal{T}\) is a set of generators of \(J(C)[1-\rho_*],\) and
\noindent (iii) the set \(\mathcal{O}(\mathcal{T}):=\{\sum_{x\in \mathcal{T}}a_xx:a\in {\mathbb Z}^4_{\geq 0},\sum_{x\in \mathcal{T}}a_x\leq 2\}\) satisfies \[\mathcal{O}(\mathcal{T})=\Theta_3.\]
In Section 4, the authors of the paper under review applied their revised algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most \(4\).
In the Appendix, the third named author of the paper under review applied the tools used in Section 2 to correct a sign in the generalization of Takase's formula [\textit{K. Takase}, Proc. Japan Acad., Ser. A 72, No. 7, 162--165 (1996; Zbl 0924.14016)] for the inverse Jacobian problem for hyperelliptic curves, given in [\textit{J. S. Balakrishnan} et al., LMS J. Comput. Math. 19A, 283--300 (2016; Zbl 1404.11085)]. Picard curve; hyperelliptic curves; genus 3; inverse Jacobian; explicit algorithm Arithmetic ground fields for curves, Complex multiplication and moduli of abelian varieties, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Special algebraic curves and curves of low genus, Theta functions and abelian varieties, Computational aspects of algebraic curves An inverse Jacobian algorithm for Picard curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article is a comprehensive review on the study of projective hypersurfaces of low degrees, and in particular, degree \(3\). The author starts by introducing the family of projective linear space, and in particular, lines, contained in a smooth hypersurface. The most typical example given is the \(27\) lines on a cubic surface. The article goes on to discuss rationality and unirationality of cubic hypersurfaces, and includes the Clemens-Griffiths proof of the irrationality of cubic threefolds. For this purpose, the author gives a brief introduction on several fundamental algebro-geometric objects, such as the intermediate Jacobian, the Albanese varieties, principally polarized abelian varieties, and Prym varieties. The article also discusses the variety of lines contained in a smooth cubic fourfold. hypersurfaces; Schebert calculus; cubic hypersurfaces; cubic threefolds; cubic fourfolds; Pfaffian cubics; unirationality; rationality; Picard group; intermediate Jacobian; Albanese variety; Abel-Jacobi map; conic bundles; abelian varieties; Prym varieties; Hilbert square; varieties with vanishing Chern class; Calabi-Yau varieties; holomorphic symplectic varieties; Beauville-Bogomolov decomposition theorem Hypersurfaces and algebraic geometry, Rational and ruled surfaces, \(3\)-folds, \(4\)-folds, Calabi-Yau manifolds (algebro-geometric aspects), Jacobians, Prym varieties, Picard schemes, higher Jacobians, Rationality questions in algebraic geometry, Rational and unirational varieties On the geometry of hypersurfaces of low degrees in the projective space | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0721.00014.]
The author shows that there is a natural action of the Hecke algebra on the singular homology groups of a suitable non-singular projective toroidal compactification of a Siegel modular variety of genus \(g\geq 2\) and on the spaces of harmonic forms on this compactification. This action is studied in detail. In particular, it is shown that a certain family of integrals of a holomorphic Siegel cusp form is contained in a \(\mathbb{Z}\)- module of finite rank; this generalizes a well-known theorem of \textit{Yu. I. Manin} in the case \(g=1\) [Math. USSR, Izv. 6 (1972), 19-64 (1973); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 36, 19-66 (1972; Zbl 0243.14008)]. action of Hecke algebra on the singular homology groups; toroidal compactification of a Siegel modular variety Hatada, K.: Homology groups, differential forms and Hecke rings on Siegel modular varieties. To appear in the volume Topics in Mathematical Analysis [to the memory of A. L. Cauchy] (ed. T. M. Rassias), World Scientific Publishing Co., Singapore (1989). Modular and Shimura varieties, Toric varieties, Newton polyhedra, Okounkov bodies, Classical real and complex (co)homology in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Homology groups, differential forms and Hecke rings on Siegel modular 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 a series of previous papers dating back from the last decade of the XXth century [Geom. Dedicata 64, No. 2, 229--251 (1997; Zbl 0897.14008); Trans. Am. Math. Soc. 348, No. 10, 4185--4197 (1996; Zbl 0878.14005); Geom. Dedicata 74, No. 1, 37--47 (1999; Zbl 1049.14033)], the author of the present paper studied smooth projective surfaces \(X\) endowed with a nef and big line bundle \(L\).
He showed that for Kodaira dimension of \(X\) \(\leq 1\) or Kodaira dimension of \(X=2\) and \(h^0(L)>0\), \(p_a(L)\geq q(X)\) always holds (where \(p_a(L)\) denotes the arithmetic genus of \(L\): \(=1/2(K_XL+L)^2+1\) and \(q(X):=h^1(X,\mathcal O_X\))). Furthermore, he completely classified the case \(p_a(L)=q(X)\), when the Kodaira dimension of \(X\) is \(\leq 1\). For the case of Kodaira dimension of \(X=2\) and \(h^0(L)>0\), the complete classification was not achieved.
Recently, in the article [``A characterization of the symmetric square of a curve'', Int. Math. Res. Not. 2011, \url{arXiv:1008.1790} (2011), \url{doi:10.1093/imrn/rnr025}] by the reviewer, \textit{R. Pardini} and \textit{G. P. Pirola}, surfaces of general type containing a 1-connected effective divisor satisfying \(D^2>0\) and \(p_a(D)=q(X)\) were completely characterized.
In this note the author shows that the techniques and results of the above mentioned paper allow him to complete the classification of surfaces \(S\) of general type with a nef and big divisor \(L\) such that \(h^0(L)\geq 1\) and such that \(p_a(L)=q(S)\). surface of general type; sectional genus; irregularity; quasi-polarized surface; symmetric square of a curve Surfaces of general type, Divisors, linear systems, invertible sheaves A note on quasi-polarized surfaces of general type whose sectional genus is equal to the irregularity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(Y\) be a smooth Fano \(n\)-fold with \(n\geq 3\), \(b_2(Y)=1\) and of index \(n-1\). Let \(A\) be a smooth \(n\)-fold which is a finite double cover of \(Y\). Suppose that \(A\) is embedded in a smooth projective manifold \(X\) as a very ample divisor. This paper gives a precise classification of such \(X\) for each pair \((A, Y)\). Note that \({\text{Pic}}(Y)\cong {\text{Pic}}(A)\cong {\text{Pic}}(X)\cong {\mathbb Z}\) by the Lefschetz theorem. Moreover, if \(h\) and \(H\) are the ample generators of \({\text{Pic}}(Y)\) and \({\text{Pic}}(X)\) respectively, then \(h_A=H_A\) and this is the ample generator of \({\text{Pic}}(A)\). The branch locus of \(A\to Y\) is a member of \(|2bh|\) for some natural number \(b\) and \(A\) is a member of \(|aH|\) on \(X\) for some natural number \(a\). The authors enumerate all the possible values of \((a,b)\) for each \(d=d(Y,h)=h^n\) (note that \(d\leq 5\) by the classification theory of Del Pezzo manifolds), and describe the structure of \(X\) for each possible case. In particular \(X\) does not exist when \(d=1\).
The proof depends on various results in the theory of polarized manifolds, including those on cases of \(\Delta\)-genus two. The assumption that \(A\) is very ample (i.e., not merely ample) is essential and makes the argument very subtle in several cases.
\{ Reviewer's remark: The authors seem to assert lemma 2.2 under a weaker assumption that \(A\) is merely ample. But this is not true. The claim ``\(\phi\) factors through \(\pi\)'' is false. In fact, when \(d=1\) and \(b=1\), \(A\) becomes a double cover of \(\mathbb{P}^n\), which can always be embedded as an ample divisor. However, one can easily show that such \(A\) cannot be a very ample divisor, so the main result can be justified\}. Picard group; double cover of Fano \(n\)-fold; embedded as a very ample divisor; branch locus; Del Pezzo manifolds; polarized manifolds; \(\Delta\)-genus Fano varieties, Coverings in algebraic geometry, Divisors, linear systems, invertible sheaves, \(n\)-folds (\(n>4\)), \(3\)-folds Double covers of some Fano manifolds as hyperplane sections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 proves the following result: If \(k\) is any uncountabe field and \(\mathrm{Spec}(A)\) is an affine open subscheme of a curve \(C/k\) of genus at least 1, then \(A\) is not a Euclidean domain. - The paper contains also an extension of a paper of \textit{P. Samuel} [J. Algebra 19, 282--301 (1971; Zbl 0223.13019)], concerning finitely generated abelian subgroups of a linear group. Euclidean domains; affine open subscheme of a curve Brown, M.L.: A note on Euclidean rings of affine curves. J. Lond. Math. Soc.29, 229--236 (1984) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Euclidean rings and generalizations, Special algebraic curves and curves of low genus, Relevant commutative algebra A note on Euclidean rings of affine 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 \(\pi: \tilde C\to C\) be an unramified double cover of a nonsingular complete curve C of genus \(g\geq 2\) on an algebraically closed field K of characteristic \(\neq 2\). Let \(Pic_ d(C)\) be the variety of isomorphism classes of invertible sheaves of degree d on C, and \(\sigma \in Pic_ 0(C)\) be the 2-division point associated with \(\pi\). We will say that a curve is k-gonal if it has a \(g_ k^ 1\) (i.e. a linear series of degree k and dimension 1), but not a \(g^ 1_{k'}\) with \(k'<k\) (hence a \(g^ 1_ k\) on a k-gonal curve must be complete and without fixed points). Let us denote by \(W_ d\) the subset of \(Pic_ d(C)\) of invertible sheaves associated to effective divisors. We prove the following:
(1) if C is k-gonal then \(\tilde C\) is k-gonal if and only if k is even and \(W_{k/2}\cdot (W_{k/2}+\sigma)\neq \emptyset.\)
(2) if C is a general point in \({\mathcal M}^ 1_{g,k}\) (the moduli space of curves of \(genus\quad g\) with a \(g^ 1_ k)\) and \(g\geq 4k-5\) then \(\tilde C\) is 2k-gonal.
A curve C is said to be elliptic-hyperelliptic if there exists a degree two morphism \(\epsilon: C\to E\) onto an elliptic curve E. Applying (1), together with the description of the Prym-canonical map, we get the following:
(3) if C is elliptic-hyperelliptic then \(\tilde C\) is elliptic- hyperelliptic if and only if \(\sigma = \epsilon^*(\eta)\) where \(\eta\) is a 2-division point in \(Pic_ 0(E).\)
At the end, we show that given an elliptic-hyperelliptic curve \(\epsilon: C\to E,\) we can build on it, in a natural way, a ''tower'' of unramified double covers, having on each ''floor'' an elliptic-hyperelliptic curve. k-gonal curve; double cover of a nonsingular complete curve; linear series Del Centina, A.: \(g{\kappa}1\) on an unramified double cover of a k-gonal curve and applications. Rend. sem. Mat. Torino 41, 53-63 (1983) Coverings of curves, fundamental group, Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus, Coverings in algebraic geometry \(g_ k^ l\)'s on an unramified double cover of a k-gonal curve and applications | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a principally polarized abelian variety \(A\) of dimension 4 with theta divisor \(\Theta\), let \(| 2 \Theta |_{00}\) be the linear system of divisors linearly equivalent to \(2 \Theta\) which have multiplicity at least 4 at the origin (an element of \(| 2 \Theta |_{00}\) is called a \(\Gamma_{00}\)-divisor). We use Clemens' double solids and the Prym map to associate to \((A, \Theta)\) a cubic hypersurface \(T\) in \(| 2 \Theta |_{00}\) whenever \((A, \Theta)\) is neither decomposable (in which case \(| 2 \Theta |_{00} \cong \mathbb{P}^ 4)\) nor the jacobian of a hyperelliptic curve. We determine the singularities of \(T\): for instance, we show that \(T\) is smooth if \((A, \Theta)\) is not a jacobian and does not have a vanishing theta-null. We also characterize \(T\) as the dual variety of a component of the branch locus of the natural rational map (which we prove to be a morphism) from the blow up of \(A\) at the origin to \((| 2 \Theta |_{00})^*\). We check that when \((A, \Theta)\) is generic, the threefold \(T\) is isomorphic to the abstract cubic threefold which \textit{R. Donagi} associated to \((A, \Theta)\) [in: curves, Jacobians, and abelian varieties, Proc. AMS.-IMS-SIAM Jt. Math. 136, 55-125 (1992; Zbl 0783.14025)].
As an application we prove two conjectures of \textit{B. van Geemen} and \textit{G. van der Geer} [Am. J. Math. 108, 615-641 91986; Zbl 0612.14044)] which give geometric characterizations of the locus of jacobians in terms of the linear system \(| 2 \Theta |_{00}\). The first of these conjectures asserts that, if \((A, \Theta)\) is not a jacobian, then the base locus of \(| 2 \Theta |_{00}\) does not contain any points besides the origin (it has dimension 2 for a jacobian by a result of Welters). principally polarized abelian variety; theta divisor; divisors; Clemens' double solids; Prym map; blow up; locus of jacobians E. Izadi, The geometric structure of \?\(_{4}\), the structure of the Prym map, double solids and \Gamma \(_{0}\)\(_{0}\)-divisors, J. Reine Angew. Math. 462 (1995), 93 -- 158. Theta functions and abelian varieties, Picard schemes, higher Jacobians The geometric structure of \(\mathcal{A}_ 4\), the structure of the Prym map, double solids and \(\Gamma_{00}\)-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 0518.00005.]
Let k be an algebraically closed field of characteristic zero, and let \(\sigma_{n,d}\) be a maximal irreducible system, defined over k, of plane curves of degree n such that the general curve of the system has d nodes. An irreducible curve of degree n has no more than \((n-1)(n-2)/2\) nodes. In Section 11 of Anhang F of the book by \textit{F. Severi} [''Vorlesungen über algebraische Geometrie'' (Leipzig-Berlin 1921)] it is claimed that if \(d\leq (n-1)(n-2)/2\) and the general curve \(C^*\) of \(\sigma_{n,d}\) is irreducible, then \(\sigma_{n,d}\) is unique. The argument which supports this claim is not convincing: this is the well- known gap in Anhang F. The author refers to the uniqueness of \(\sigma_{n,d}\), with the stated assumptions, as Basic Conjecture I.
In this interesting paper the author shows that Basic Conjecture I is implied by a quite different assertion, called Basic Conjecture II. Let \(f(X,Y)=\sum A_{ij}X^ iY^ j=0\)\ be the equation of \(C^*\), where \(C^*\) is assumed irreducible. Let R be the ring generated over k by the ratios of the \(A_{ij}\), and K the quotient field of R. It is clear that the coordinates of the d nodes \(Q^*_ 1,...,Q^*_ d\) of \(C^*\) are algebraic over K. Basic Conjecture II asserts that the d nodes \(Q^*_ i\) form a complete set of conjugate algebraic points over K.
The proof depends on a basic lemma, which has as a corollary that any system of curves of the above kind contains all the n-gons of the plane (curves consisting of n lines). Severi had pointed out that this implies Basic Conjecture I, but it is exactly here that the gap in Anhang F occurs.
A recent reference is a paper by the author [Am. J. Math. 104, 209-226 (1982; Zbl 0516.14023)]. A proof of Basic Conjecture II would imply, by a well-known argument of the author, that the fundamental group of the complement of a nodal plane curve is abelian. irreducibility of system of irreducible plane curves; abelian; fundamental group of the complement of a nodal plane curve; degree n Oscar Zariski, On the problem of irreducibility of the algebraic system of irreducible plane curves of a given order and having a given number of nodes, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 465 -- 481. Families, moduli of curves (algebraic), Singularities of curves, local rings, Enumerative problems (combinatorial problems) in algebraic geometry, Coverings in algebraic geometry On the problem of irreducibility of the algebraic system of irreducible plane curves of a given order and having a given number of nodes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0727.00013.]
The problem of giving an algorithm for the computation of the topological type of a real curve from its equation has been considered by several authors [\textit{P. Gianni} and \textit{C. Traverso}, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 29, 87-109 (1983; Zbl 0557.14011)]\ and more recently in the particular case of non singular curves by \textit{D. S. Arnon} and \textit{S. McCallum} [J. Symb. Comput. 5, No. 1/2, 213-236 (1988; Zbl 0664.14017)]. The approach presented here relies on a basic result in real algebraic geometry, Thom's lemma; it can be viewed as an illustration of the philosophy in the author's joint paper with \textit{M. Coste} [in J. Symb. Comput. 5, No. 1/2, 121-129 (1988; Zbl 0689.14006)]. Our algorithm runs in polynomial-time, needs no regularity hypothesis on the curve or on the projection and seems better adapted to situations where the connected components of the curve are small. algorithm for the computation of the topological type of a real curve; Thom's lemma Roy, M. -F.: Computation of the topology of a real curve. Astérisque 192, 17-33 (1990) Computational aspects of algebraic curves, Real algebraic sets Computation of the topology of a real curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a compact Lie group and \(\Omega\) an orthogonal representation of \(G\). We think of an orthogonal representation as an underlying Euclidean space \(\mathbb{R}^ n\) together with an action of \(G\) via orthogonal maps. A real algebraic \(G\) variety is the set of common zeros of polynomials \(p_ 1,\dots,p_ m: \Omega \to \mathbb{R}\),
\[
V = \{x\in \Omega\mid p_ 1(x) = \cdots = p_ m(x) = 0\},
\]
which is invariant under the action of \(G\). Examples of such \(G\) varieties are the Grassmannians of \(k\)-dimensional subspaces in an \(n\)-dimensional representation \(\Xi\) of \(G\)
\[
G_{\mathbb{R}}(\Xi,k) = \{L \in {\mathfrak M}^{n^ 2}_ \mathbb{R}\mid L^ 2 = L,\quad L^ t = L,\quad \text{trace }L = k\}.
\]
We chose a \(G\) invariant inner product and an orthonormal basis for \(\Xi\) so that we could identify the endomorphisms of \(\Xi\) with real \(n\times n\) matrices and subspaces of \(\Xi\) with orthogonal projections onto these subspaces. The action of \(G\) on \(G_ \mathbb{R}(\Xi,k)\) is given by conjugation on the endomorphisms of \(\Xi\). A smooth \(G\) manifold is said to be algebraically realized if there is an equivariant diffeomorphism \(\varphi: X\to M\) from a non-singular real algebraic variety \(X\) to \(M\). The set of all \(G\)-vector bundles over \(M\) is said to be algebraically realized by \(X\) if, for all \(\Xi\) and \(k\), every equivariant map \(\mu: X\to G_{\mathbb{R}}(\Xi,k)\) is equivariantly homotopic to an entire rational map, i.e., every \(G\) vector bundle over \(X\) is strongly algebraic. The main result of the paper is
Theorem B. The set of all real \(G\) vector bundles over a closed smooth \(G\) manifold is algebraically realized if one of the following assumptions holds:
(1) \(G\) is the product of a group of odd order and a 2-torus.
(2) The action of \(G\) on the manifold is semifree.
Let \(Y\) be a real algebraic \(G\) variety and \(f: M\to Y\) an equivariant map. We say that \((M,f)\) is algebraically realized by \((X,\varphi)\) if \(f\circ \varphi\) is equivariantly homotopic to an entire rational map. The cobordism class of \((M,f)\) has an algebraic representative if \((M,f)\) is equivariantly cobordant to a pair \((Z,\eta)\) where \(Z\) is a non- singular real algebraic \(G\) variety and \(\eta: Z\to Y\) is entire rational. The proof of theorem B is reduced to a bordism problem.
Theorem C. Let \(G\) be a compact Lie group. An equivariant map from a closed smooth \(G\) manifold to a non-singular real algebraic \(G\) variety is algebraically realized if and only if its equivariant bordism class has an algebraic representative.
Equivariant bordism results of \textit{R. E. Stong} are then generalized to obtain theorem B [Mem. Am. Math. Soc. 103 (1970; Zbl 0201.255)] and [Duke Math. J. 37, 779-785 (1970; Zbl 0204.236)]. The paper is motivated by the non-equivariant work of \textit{J. Nash} [Ann. Math., II. Ser. 56, 405-421 (1952; Zbl 0048.385)], \textit{A. Tognoli} [Ann. Scuola Norm. Sup. Pisa, Sci. fis. mat., III. Ser. 27, 167-185 (1973; Zbl 0263.57011)], \textit{S. Akbulut} and \textit{H. King} [Ann. Math., II. Ser. 113, 425-446 (1981; Zbl 0494.57004)], and of \textit{R. Benedetti} and \textit{A. Tognoli} [Bull. Sci. Math., II. Sér. 104, 89-112 (1980; Zbl 0421.58001)]. semifree action; algebraic realization; compact Lie group; orthogonal representation; real algebraic \(G\) variety; smooth \(G\) manifold; algebraically realized; equivariant diffeomorphism; \(G\)-vector bundles; product of a group of odd order and a 2-torus; equivariant map; equivariantly homotopic; equivariant bordism class Dovermann, K.H., Wasserman, A.G.: \textit{Algebraic Realization for Cyclic Group Actions with One Isotropy Type}, preprint (2005) Equivariant cobordism, Equivariant algebraic topology of manifolds, Real algebraic and real-analytic geometry, Equivariant fiber spaces and bundles in algebraic topology Algebraic realization of equivariant vector 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 Elliptic and hyperelliptic curves are widely studied due to their interest in public key cryptosystems. V. Miller and N. Koblitz initially proposed to use the discrete-log problem with the rational points of an elliptic curve over a finite field since it is an abelian group. Moreover, in contrary to what happen in multiplicative groups it is resistant to attacks like index-calculus and furthermore the key is shorter. The structure of the group is inherited from the divisor group of the Jacobian variety, so finding the group order is an interesting problem.
The author present in this article an algorithm for computing the number of points of the jacobian of an hyperelliptic curve of type \(y^2=x^3+a x^2\) over a finite field with \(p\equiv 1\pmod{6}\) and of type \(y^2=x^5+a\) over a finite field with \(p\equiv 1\pmod{5}\). The main step in that algorithm is to compute the characteristic polynomial of the Frobenius, because then the number of rational points is just the evaluation of the later polynomial at one. What he does is to give a closed formula for the characteristic polynomial of the Frobenius module \(p\) via the Hasse-Witt matrix. The algorithm ends because multiplying any rational point by the total number of rational points is equal to zero. So repeating it for many different rational points he can finally deduce the characteristic polynomial of the Frobenius. Jacobian variety; characteristic polynomial of the Frobenius; rational points Special algebraic curves and curves of low genus, Applications to coding theory and cryptography of arithmetic geometry, Number-theoretic algorithms; complexity Counting points for genus 2 hyperelliptic curves of two special types 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 Nach den bahnbrechenden Ergebnissen von \textit{A. S. Merkur'ev} und \textit{A. A. Suslin} [Math. USSR, Izv. 21, 307--340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011--1046 (1982; Zbl 0525.18008)] und \textit{A. A. Suslin} [``Torsion in \(K_2\) of fields'', LOMI preprint (1982); see also K-Theory 1, No. 1, 5--29 (1987; Zbl 0635.12015)] ist es möglich geworden, ein von Bloch stammendes Programm [\textit{S. Bloch}, Bull. Am. Math. Soc. 80, 941--945 (1974; Zbl 0289.14002) und ``Groupe de Brauer, Sémin., Les Plans-sur-Bex 1980'', Lect. Notes Math. 844, 76--102 (1981; Zbl 0467.12011)] weiterzuentwickeln. Es geht darum, aus den Beziehungen zwischen \(K\)-Theorie und Étalcohomologie, Schlüsse für die Struktur der Chowgruppen \(\mathrm{CH}^n(X) (= \) Zyklen) der Kodimension \(n\) auf einer glatten algebraischen Varietät \(X\), modulo rationaler Äquivalenz) zu ziehen. Genauer heißt es im Augenblick, die Torsionsuntergruppe von \(\mathrm{CH}^2(X)\) zu studieren. Zu diesem Thema, siehe die oben erwähnten Arbeiten, und \textit{S. Bloch}, Compos. Math. 39, 107--127 (1979; Zbl 0463.14002); ``Lectures on algebraic cycles'', Duke Univ. Math. Ser. IV (1980; Zbl 0436.14003); Ann. Sci. Ec. Norm. Supér., IV. Sér. 14, 41--59 (1981; Zbl 0524.14006); \textit{J.-L. Colliot-Thélène}, Invent. Math. 71, 1--20 (1983; Zbl 0527.14011); \textit{J.-L. Colliot-Thélène} and \textit{ J.-J. Sansuc} und \textit{C. Soulé}, Duke Math. J. 50, 763--801 (1983; Zbl 0574.14004).
Im folgenden sei \(k\) ein Körper, \(\operatorname{Char} k=0,\) \(\bar k\) ein algebraischer Abschluß von \(k\); sei \(X/k\) eine glatte vollständige Varietät, und \(\bar X=X\times_k\bar k\). Beim Studium der Gruppe \(\mathrm{CH}^2(X)=H^2_{\text{Zar}}(X,\underline K_2)\) (Bloch-Quillen Formel) erscheint es wichtig, Näheres über die Galoismoduln \(H^i(\bar X,\underline K_2)\) für \(i=0,1\) zu wissen, und dies ist das Hauptthema der vorliegenden Arbeit. Es wird gezeigt: diese beiden Gruppen sind Erweiterungen einer endlichen Gruppe mit Hilfe einer teilbaren Gruppe, deren Torsion als Étalcohomologiegruppe beschrieben wird. Dabei wird Rücksicht auf die Galoisstruktur genommen. Es wird auch die natürliche Abbildung
\[\mathrm{Pic}\, \bar X\otimes_Z\bar k^*\to H^1_{\text{Zar}}(\bar X,K_2)\]
untersucht. Ihr Kern ist torsionsfrei. Falls \(H^2(X,{\mathcal O}_X)=0\), ist der Kern eindeutig teilbar und der Cokern ist die Summe einer endlichen Gruppe und einer eindeutig teilbaren Gruppe. Bei den Beweisen spielen sowohl Ergebnisse von Suslin eine wesentliche Rolle als auch die von Deligne bewiesenen Weil-Vermutungen -- diese letzte Idee stammt von S. Bloch (vgl. die zitierte Arbeit in Compos. Math.).
Unter Benutzung der Galoiscohomologie-Version des Hilbertschen Satzes 90 für \(K_2\) werden dann Anwendungen auf die Gruppe \(\operatorname{Ker} [\mathrm{CH}^2(X)\to \mathrm{CH}^2(\bar X)]\) gegeben. Als typisches Beispiel erhalten wir: sei \(k\) ein \(p\)-adischer Körper; dann ist diese Gruppe in den folgenden Fällen endlich:
(a) \(H^1(X,\mathcal O_X) = H^2(X,\mathcal O_X) = 0\);
(b) \(H^2(X,\mathcal O_X)=0\) und \(X/k\) besitzt eine gute Reduktion. torsion subgroup of second Chow group; \(K_2\)-cohomology; Galois module structure; Bloch-Quillen-formula; Neron-Severi group; Picard variety Colliot-Thélène, J. -L.; Raskind, W., \(\mathcal{K}_2\)-cohomology and the second Chow group, Math. Ann., 270, 165-199, (1985) (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Galois cohomology, Parametrization (Chow and Hilbert schemes), Steinberg groups and \(K_2\), Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) \(K_2\)-cohomology and the second Chow group | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The problem considered in this article is the description of the points in the reduction modulo \(p\) of a Shimura variety. The description, which is conjectural, is based on the concept of an admissible homomorphism \(\phi: {\mathcal P}\to {\mathcal G}_ G\) of an explicitly defined gerb \({\mathcal P}\) into the neutral gerb associated to the reductive group G over \({\mathbb{Q}}\) which defines the Shimura variety. To each such homomorphism \(\phi\) there is associated a double coset space of the form \(X_{\phi}(K)=I_{\phi}\setminus [X^ p/K^ p\times X_ p]\), together with an operator \(\Phi_{\phi}\) such that the set of points in the reduction \(modulo\quad p\) of \(Sh(G,h)_ K\) together with the action of the Frobenius, is the disjoint union indexed by the equivalence classes of admissible \(\phi\) of \(X_{\phi}(K)\). Here \(K\subset G({\mathbb{A}}_ f)\) is assumed to be of the form \(K=K^ p\cdot K_ p\) and on the rational prime p there are severe restrictions which however include the case where \(K_ p\) is hyperspecial, which conjecturally implies that the Shimura variety has good reduction at the primes above p.
This conjecture generalizes and makes more precise the original conjecture in terms of Frobenius pairs of \textit{R. P. Langlands} [Math. Dev. Hilbert Probl., Proc. Symp. Pure Math. 28, De Kalb 1974, 401-418 (1976; Zbl 0345.14006)]. In fact, there is a bijective correspondence between equivalence classes of Frobenius pairs and local equivalence classes of admissible homomorphisms. Using concepts and results of \textit{R. E. Kottwitz}, some of them still unpublished [cf. Duke Math. J. 51, 611-650 (1984; Zbl 0576.22020), Math. Ann. 269, 287-300 (1984; Zbl 0547.14013), Compos. Math. 56, 201-220 (1985; Zbl 0597.20038)] it is shown that the conjecture yields in the case where \(K_ p\) is hyperspecial an explicit numerical formula for the number of points with values in a finite field of characteristic p of the Shimura variety.
The origin of the description in terms of gerbs is A. Grothendieck's conjectural theory of motives and of the description of their category in terms of gerbs [\textit{N. Saavedra Rivano}, ''Categories Tannakiennes'', Lect. Notes Math. 265 (1972; Zbl 0241.14008)]. It is shown that assuming the standard conjectures on algebraic cycles, the Tate conjecture and the Hodge conjecture for abelian varieties of CM-type, the conjecture is valid for the Shimura varieties which are moduli spaces of abelian varieties with additional structure. This uses results of \textit{T. Zink} [Math. Nachr. 112, 103-124 (1983; Zbl 0604.14029)]. reduction modulo p of a Shimura variety; gerb; motives; Tate conjecture; Hodge conjecture Langlands, R. P.; Rapoport, M., Shimuravarietäten und Gerben, J. Reine Angew. Math., 378, 113-220, (1987) Local ground fields in algebraic geometry, Generalizations (algebraic spaces, stacks), Cycles and subschemes, Nonabelian homological algebra (category-theoretic aspects) Shimuravarietäten und Gerben. (Shimura varieties and gerbs) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 problem of finding efficiently computable non-degenerate multilinear maps from \(G^n_1\) to \(G_2\), where \(G_1\) and \(G_2\) are groups of the same prime order, and where computing discrete logarithms in \(G_1\) is hard. We present several applications to cryptography, explore directions for building such maps, and give some reasons to believe that finding examples with \(n>2\) may be difficult. generalized Weil pairing; generalized Tate pairing; elliptic curve cryptography; abelian variety cryptography; multilinear maps D. Boneh and A. Silverberg, Applications of multilinear forms to cryptography, Topics in Algebraic and Noncommutative Geometry (Luminy/Annapolis 2001), Contemp. Math. 324, American Mathematical Society, Providence (2003), 71-90. Cryptography, Applications to coding theory and cryptography of arithmetic geometry Applications of multilinear forms to cryptography | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 detailed treatment of the result, announced by the authors in Journées géométrie algébrique, Angers 1979, 143-155 (1980; Zbl 0464.14015) that the degree of the Prym map \(P: {\mathcal R}_ 6\to {\mathcal A}_ 5\) is 27, where \({\mathcal R}_ g\) (for \(g=6)\) is a moduli space of unramified double covers of smooth genus g curves. The main idea is to investigate the fiber over a general Jacobian, which can be found explicitly. To compute the degree near a non-zerodimensional fiber the authors use the infinitesimal investigation, which was generously outlined for them by \textit{C. H. Clemens}. Since then \textit{V. Kanev} [Izv. Akad. Nauk SSSR, Ser. Mat. 46, 244-268 (1982)] has improved these methods and established that the degree of the Prym map \(P: {\mathcal R}_ g\to {\mathcal A}_{g-1}\) is one if \(g=9\); for \(g=7\) and 8 it is also one [\textit{R. Friedman} and \textit{R. Smith}, Invent. Math. 67, 473-490 (1982; Zbl 0506.14042)]. Jacobian variety; general Torelli theorem; moduli space of unramified double covers of smooth curves; degree of the Prym map Donagi, Ron; Smith, Roy Campbell, The structure of the Prym map, Acta Math., 146, 1\textendash2, 25-102, (1981) Families, moduli of curves (analytic), Jacobians, Prym varieties, Local structure of morphisms in algebraic geometry: étale, flat, etc., Picard schemes, higher Jacobians The structure of the Prym map | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\{\Gamma_ t | t \in \mathbb{P}^ 1\}\) be a linear pencil of projective plane curves of degree \(m\geq 3\) defined over an algebraically closed field of arbitrary characteristic. Assume that this linear pencil satisfies the following conditions:
(A1) Every member \(\Gamma_ t\) is irreducible, and the general member is smooth.
(A2) The \(m^ 2\) base points \(P_ 0,P_ 1,\ldots,P_{m^ 2-1}\) of the pencil are distinct.
Under these conditions, the generic member \(\Gamma\) of the pencil is a smooth plane curve of genus \((m-1)\cdot(m-2)/2\), defined over the rational function field \(K=k(t)\). The base point \(P_ 0\) (for example) defines an embedding of the general member into the Jacobian \(J\) of \(\Gamma\), and the remaining base points \(P_ 1,\ldots,P_{m^ 2-1}\) in \(\Gamma\) may be regarded as \(K\)-rational points in \(J\), i.e., as elements of the group \(J(K)\) of \(K\)-rational points of \(J\).
The theorem of Manin-Shafarevich states that, in the special case of \(m=3\) and under the conditions (A1) and (A2), the eight base points \(P_ 1,\ldots,P_ 8\) are independent and generate a subgroup of index 3 in the Mordell-Weil group of the elliptic curve \(\Gamma\). Recently, the author of the present paper has generalized the notion of Mordell-Weil lattices to the higher-genus case [cf. Proc. Japan Acad., Ser. A 68, 247- 250 (1992)] and, as an application of this concept, he provides a corresponding generalization of the Manin-Shafarevich theorem in this brief note under review. His generalization of the theorem of Manin- Shafarevich to linear pencils of plane curves of degree \(m\geq 3\), satisfying conditions (A1) and (A2), states that the group \(J(K)\) of \(K\)- rational points in the Jacobian \(J\) is a torsion-free abelian group of rank \(m^ 2-1\), and the points \(P_ 1,\ldots,P_{m^ 2-1}\) are independent and generate a lattice of index \(m\) in \(J(K)\).
In fact, this generalized version of the theorem of Manin-Shafarevich is an immediate corollary of a more general result on Mordell-Weil lattices, which is also proved in the present note. More precisely, the author proves the following theorem: The Mordell-Weil lattice \(J(K)\) is an integral unimodular lattice of rank \(m^ 2-1\), positive-definite with respect to the height pairing. It is even if and only if \(m\) is odd. Moreover, the points \(P_ 1, \ldots,P_{m^ 2-1}\) generate a sublattice of index \(m\), and there is a unique point \(Q\in J(K)\) such that \(mQ=P_ 1+\cdots+P_{m^ 2-1}\) and \(J(K)\) is freely generated by \(\{P_ 1, \ldots, P_{m^ 2-1},Q\}\). -- At the end, the author points out that his proof can be generalized to Lefschetz pencils of hyperplane sections of smooth algebraic surfaces of degree \(d\) in a projective space \(\mathbb{P}^ N\) with trivial Picard variety. The proof will be published elsewhere, but the result is illustrated by the instructive example of the Fermat surface of degree 4 in \(\mathbb{P}^ 3\). Jacobian variety; rational points; linear pencil of projective plane curves; Mordell-Weil lattices; Manin-Shafarevich theorem; height pairing; Lefschetz pencils of hyperplane sections Shioda, T.: Generalization of a theorem of Manin-Shafarevich. Proc. Japan acad. 69A, 10-12 (1993) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Divisors, linear systems, invertible sheaves, Special surfaces Generalization of a theorem of Manin-Shafarevich | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Soit \(X\) une courbe irréducible et projective sur un corps algébriquement clos \(k\) qui n'a qu'un seul point double ordinaire \(x\) comme singularité. Soit \(Y\) sa normalisation avec \(y_1,y_2\) les deux points de \(Y\) au-dessus de \(x\). Notons \(J\) la jacobienne de \(Y\), et \(J_{y_1+y_2}\) la jacobienne généralisée de \(Y\) par rapport à \(y_1+ y_2\). Les jacobiennes généralisées du type \(J_{y_1+y_2}\) sont intéressantes puisque par exemple elles engendrent le groupe \(\text{Ext}(J,\mathbb{G}_m)\) des classes d'isomorphisme des extensions de \(J\) par \(\mathbb{G}_m\). \(J_{y_1+y_2}\) correspond à un couple d'éléments de \(\text{Ext} (J,\mathbb{G}_m)\), l'un inverse de l'autre. Par l'isomorphisme naturel \(\text{Ext} (J, \mathbb{G}_m) \cong \text{Pic}^0(J) \cong \text{Pic}^0(Y)\), alors ce couple d'éléments correspond au couple \(\pm(y_1-y_2)\). Autrement dit on peut décrire la jacobienne \(\text{Pic}^0_{X/k} = J_{y_1 + y_2}\) de \(X\) à l'aide des deux points de la courbe normalisée \(Y\) qui sont au-dessus du point singulier de \(X\).
Le but de ce papier est de donner une description analogue de la jacobienne \(\text{Pic}^0_{X/k}\) d'une courbe \(X\) (éventuellement réductible) à singularités ordinaires sur un corps algébriquement clos \(k\). generalized Jacobian; ordinary curve singularities Zhang B. (1997). Sur les jacobiennes des courbes à singularités ordinaires. Manuscripta math. 92: 1--12 Jacobians, Prym varieties, Singularities of curves, local rings On the Jacobians of curves with ordinary 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 The author introduces the Hamburger Noether (HN for short) matrices over rings as a way to study equisingular deformations of curve singularities. In two previous papers \textit{A. Campillo} and the author in [Algebraic geometry, Proc. Int. Conf., La Rabida 1981, Lect. Notes Math. 961, 22--31 (1982; Zbl 0497.14012)] and the author in [J. Pure Appl. Algebra 43, 119--127 (1986; Zbl 0612.14008)] defined the HN matrices (over the definition field) as a useful tool in order to describe the sequence of infinitely near points related with a resolution procedure of a curve singularity (not necessarily plane). On the other hand \textit{A. Campillo} [Trans. Am. Math. Soc. 279, 377--388 (1983; Zbl 0559.14020)] extended the HN expansions to the case in which the coefficients are taken over rings and study the applications to the equisingular deformation theory of plane curve singularities.
The relationship between the HN matrices and the HN expansions permits to give a parametrization \(\Phi:A[[X_1,\ldots,X_N]]\to A[[t]]\) with coefficients over the base local ring \(A\). When \(A=k[[V_1,\ldots,V_r]]\), the quotient ring \(R=A[[X_1,\ldots,X_ N]]/(\operatorname{ker} \Phi)\) can be seen as a deformation (HN deformation) of the algebroid curve \(R_0\) induced over the residual field \(k\) of \(A\) with the section \(s\) given by the maximal ideal of \(A\). In characteristic zero the generic fiber is well defined, but in general, the deformation is not flat and \(R_ 0\) is not reduced. The author proves that this deformation is an equisingular deformation along \(s\) in the sense given by \textit{O. Zariski} [Am. J. Math. 87, 972--1006 (1965; Zbl 0146.42502)] (essentially, the generic fibre and \((R_0)_{red}\) have the same multiplicity sequence) and also in the sense introduced by \textit{J. Becker} and \textit{J. Stutz} [Rice. Univ. Stud. 59, No. 2, Proc. Conf. Complex Analysis 1972, part I, 1--9 (1973; Zbl 0286.32009)]. In some restricted cases all the equisingular deformations in the sense of Stutz and Becker can be reached by HN deformations. When one considers the special case of Arf matrices - -- his definition is based on the description of HN matrices for the Arf curves introduced by \textit{J. Lipman} [Am. J. Math. 93, 649--685 (1973; Zbl 0228.13008)] --- , the HN deformation becomes flat, with reduced special fibre and has a good support by monoidal transformations. Taking into account that the Arf curve associated to a given space curve reproduces its sequence of multiplicities, the HN deformation of Arf matrices seems to provide a way to construct equisingular flat deformations for reduced space curves. Hamburger-Noether matrices; equisingular deformations of curve singularities; infinitely near points; resolution procedure of a curve singularity; HN matrices; HN expansions; multiplicity sequence; HN deformations; Arf matrices; monoidal transformations Castellanos, J.: ''Hamburger-Noether matrices over rings'' J.P.P.A. 64 (1990) 7--19 Singularities of curves, local rings, Rational and birational maps, Formal methods and deformations in algebraic geometry, Infinitesimal methods in algebraic geometry, Equisingularity (topological and analytic), Local deformation theory, Artin approximation, etc. Hamburger-Noether matrices over 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 We give an algorithm to compute \((\ell,\ell,\ell)\)-isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves over a finite field of characteristic different from 2 in time \(\tilde{O}(\ell^3)\), where \(\ell\) is an odd prime which is coprime to the characteristic. An important application is to reduce the discrete logarithm problem in the Jacobian of a hyperelliptic curve to the corresponding problem in the Jacobian of a non-hyperelliptic curve. Jacobian variety; isogeny; theta group; theta function; discrete logarithm problem; non-hyperelliptic curve Isogeny, Jacobians, Prym varieties, Theta functions and abelian varieties, Cryptography, Algebraic moduli of abelian varieties, classification, Applications to coding theory and cryptography of arithmetic geometry, Algebraic functions and function fields in algebraic geometry, Authentication, digital signatures and secret sharing, Algebraic coding theory; cryptography (number-theoretic aspects) Translating the discrete logarithm problem on Jacobians of genus 3 hyperelliptic curves with \((\ell ,\ell ,\ell)\)-isogenies | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 prove the following theorem: If \(g\) belongs to the set \(S=(1\),\dots,29, 31, 33, 37, 40, 41, 43, 45, 47, 49, 50, 53, 55, 57, 61, 65, 73, 82, 97, 109, 121, 129, 145, 163, 217, 257, 325, 433, 649, 1297) then there exists a curve of genus \(g\) whose Jacobian is isogenous to a product of elliptic curves. These curves are constructed either as modular curves or as coverings of curves of genus 2 or 3. The theorem includes all well known examples of algebraic curves with decomposable Jacobians and many new ones in addition. genus; Jacobian; isogenous to a product of elliptic curves Torsten Ekedahl and Jean-Pierre Serre, Exemples de courbes algébriques à jacobienne complètement décomposable, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 5, 509 -- 513 (French, with English and French summaries). Jacobians, Prym varieties Examples of algebraic curves with totally split Jacobian | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 introduces a meromorphic function M(s,f,g) depending bilinearly on two cusp forms f, g of weight 2 for \(\Gamma_ 0(N)\). This is not a Dirichlet series (and so does not have an Euler product) but it does satisfy a functional equation. This expresses \(M(s,f,g)+M(2-s,f| W_ N,g| W_ N)\) in terms of the Mellin transforms of f and g. In some cases M(s,f,g) can be expressed as a Rankin type convolution. In general it is defined as an iterated integral so that M(1,f,g) is the generalized period investigated by the author in several previous papers. Its significance is that it determines the image of X-i(X) \((i(x)=-x\) in J(X)) in the intermediate Jacobian of J(X) where \(X=\Gamma_ 0(N)\setminus {\mathbb{H}}\) and J(X) is the Jacobian variety of X. meromorphic function; cusp forms; functional equation; Mellin transforms; Rankin type convolution; generalized period; Jacobian variety Holomorphic modular forms of integral weight, Jacobians, Prym varieties An analytic function and iterated integrals | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A construction of a foliation of a toric Fano variety by Lagrangian tori is presented; it is based on linear subsystems of divisor systems of various degrees invariant under the Hamiltonian action of distinguished function-symbols. It is shown that known examples of foliations (such as the Clifford foliation and D. Auroux's example) are special cases of this construction. As an application, nontoric Lagrangian foliations by tori of two-dimensional quadrics and projective space are constructed. foliation; toric Fano variety; Lagrangian torus; Auroux foliation; Berezin symbol; Hamiltonian action of a symbol; moment map; geometric quantization S. A. Belev and N. A. Tyurin, Math. Notes, 87, 43--51 (2010). Toric varieties, Newton polyhedra, Okounkov bodies, Lagrangian submanifolds; Maslov index, Foliations (differential geometric aspects), Fano varieties Nontoric foliations by Lagrangian tori of toric Fano varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies the scheme \(M_ d(X;p,n)\) parametrizing morphisms of degree \( d\) from an elliptic curve X to a Grassmann variety Gr(p,n). He proves that \(M_ d(X;p,n)\) is smooth if and only if \(p=1\), or \(p=n-1\), or \(d=2\); it is irreducible of dimension nd if \(d\geq n\); if \(d>n\), it is irreducible if and only if it is smooth. More precisely, if \(d>n\), the author shows that the irreducible components are the Zariski closures of certain locally closed subschemes \(M_ d^{a,b}(X;p,n)\), for \(0\leq a<p\), \(0\leq b<n-p\) and \(a+b=n-d\), of dimension \(nd+ab\). These subschemes are defined as follows: If \(f\in M_ d(X;p,n)\) is a closed point, let \(0\to V_ f\to E_ X\to Q_ f\to 0\) denote the pullback of the universal sequence on Gr(p,n). Then set \(M_ d^{a,b}(X;p,n)=\{f;\quad h^ 0(V_ f)=a,\quad h^ 0(Q_ f)=b\}\).
The last part of the paper gives sufficient conditions for three sheaves on X to fit into an exact sequence. If the sheaves are locally free and the sheaf of highest rank is trivial, this is related to the existence of maps from X into a corresponding Grassmann variety, with fixed pullbacks of the trivial bundles. unidecomposable sheaf; morphisms from an elliptic curve to a Grassmann variety Bruguières, A., The scheme of morphisms from an elliptic curve to a Grassmannian, Compositio Math., 63, 1, 15-40, (1987) Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special algebraic curves and curves of low genus, Elliptic curves The scheme of morphisms from an elliptic curve to a Grassmannian | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a smooth real affine curve \(\gamma\) that is semi-algebraically connected and semi-algebraically compact, we show that the Witt group \(WP\) of the ring \(P\) of polynomial functions is a direct summand of the Witt group \(WR\) of the ring \(R\) of regular functions on \(\gamma\). In particular, this means that the canonical injection \(WP\hookrightarrow WK\) splits, where \(K\) is the field of rational functions on \(\gamma\). Witt ring of a ring; real algebraic curve; Knebusch-Milnor sequence Algebraic theory of quadratic forms; Witt groups and rings, Witt groups of rings, Forms over real fields, Algebraic functions and function fields in algebraic geometry Splitting natural injection of Witt rings of geometric 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 Similar to elliptic modular curves, Drinfeld modular curves are analytically constructed as quotients of the Drinfeld upper half-plane \(\Omega\) by arithmetic subgroups \(\Gamma\) of \(\text{GL}(2, A)\). Here \(A= {\mathcal O}_{\mathcal C}({\mathcal C}- \{\infty\})\) is the affine ring of a smooth complete geometrically connected curve \({\mathcal C}/ \mathbb F_q\) minus a closed point \(\infty\in {\mathcal C}\) and \(\Omega= C- K_\infty\), where \(K_\infty=\) completion of \(K= \text{Quot}(A)\) and \(C=\) completed algebraic closure of \(K_\infty\). There is a well-known formal analogy between the data \(\mathbb Z\), \(\mathbb Q\), \(\mathbb R\), \(\mathbb C\), \(H=\) complex upper half-plane, \(\text{SL}(2, \mathbb Z)\), elliptic curves \(E/\mathbb C\), elliptic modular curves\dots (the ``classical side'') and \(A\), \(K\), \(K_\infty\), \(C\), \(\Omega\), \(\text{GL}(2, A)\), rank-two Drinfeld \(A\)-modules \(\phi/C\), Drinfeld modular curves\dots (the ``Drinfeld side''). In particular, the \(C\)-points \(M_\Gamma(C)\) of a Drinfeld modular curve are in \(1-1\) correspondence with the isomorphism classes of \(\phi/C\) as above (with some level structure depending on \(\Gamma\)).
Far deeper than these formal similarities is Drinfeld's reciprocity law [Theorem 2 in \textit{V. G. Drinfeld}, Math. USSR, Sb. 23, 561--592 (1976); translation from Mat. Sb., New Ser. 94, 594--627 (1974; Zbl 0321.14014)]. It expresses the Galois representation associated with \(J_\Gamma\), the Jacobian of the compactification \(\overline M_\Gamma\) of \(M_\Gamma\), through automorphic data on \(\text{GL}(2, {\mathfrak A}_K)\), where \({\mathfrak A}_K\) is the adele ring of the ground field \(K\).
As follows from Drinfeld's work (although it is nowhere explicitly stated), the analogue of the Shimura-Taniyama-Weil conjecture on the uniformization of elliptic curves through modular curves holds in our case:
STW/K: Each elliptic curve \(E/K\) with split multiplicative reduction at \(\infty\) is the quotient of a suitable Drinfeld modular curve \(\overline M_\Gamma\), or equivalently, appears up to isogeny in the Jacobian \(J_\Gamma\).
However, there are several important problems left open by Drinfeld's work.
(A) The above results STW/K is a sheer existence statement. Of course, one would like to dispose of a construction that, given \(E/K\), produces a ``Weil uniformization'' \(p_E: \overline M_\Gamma\to E\). Equivalently, one would like to construct \(E\) (or some curve isogenous with \(E\)) out of the automorphic Hecke newform \(\varphi_E\), and to understand how properties of \(E\) are reflected in \(\varphi_E\) and vice versa.
(B) In the Drinfeld modular curve context, there are two different concepts that generalize classical modular forms, viz, automorphic forms, which are \(\mathbb C\)- or \(\mathbb Q_\ell\)- or \(\mathbb Q\)-valued functions on some adele groups, and Drinfeld modular forms, which are \(C\)-valued holomorphic functions on \(\Omega\). Both of these are needed for a full understanding of the curves \(\overline M_\Gamma\), so the question of their relationship arises.
These problems are closely related with the main result of the paper, the description (given in section 7) of \(J_\Gamma\) as a torus divided by some lattice. Satisfactory answers to both questions are given: See section 9, notably (9.6.1) for (A) (where an elliptic curve \(E\) is constructed from its newform \(\varphi_E\) by specifying the Tate period) and section 6, notably (6.5) for (B) (roughly speaking, Drinfeld modular forms of a certain type ``are'' the reductions mod \(p\) of \(\mathbb Z\)-valued automorphic forms).
The basic tool for the construction of \(J_\Gamma\) is the theory of theta functions for \(\Gamma\), i.e., of meromorphic functions on \(\Omega\) behaving nicely under \(\Gamma\) and at the ``cusps'' of \(\Gamma\backslash \Omega\). In the context of Schottky groups, these have been introduced by Manin-Drinfeld and studied by Gerritzen-van der Put and \textit{M. van der Put} [Groupe Étude Anal. Ultramétrique 1981/82, Exp. No. 10 (1983; Zbl 0515.14027)].
For arithmetic groups \(\Gamma\subset \text{GL}(2, K)\) as above, several new problems arise, due to the non-compactness of \(\Gamma\backslash \Omega\) and the existence of torsion in \(\Gamma\). These are dealt with by a careful analysis of the relationship between modular data (on \(\Omega\) or \(\Gamma\backslash \Omega\)) and automorphic data (on \(\mathcal T\) or \(\Gamma\backslash {\mathcal T}\), where \(T\) is the Bruhat-Tits tree \(\mathcal T\) of \(\text{PGL}(2, K_\infty)\)). E.g. the absolute value of the theta pairing \((. , .)\) on \(\overline \Gamma= \Gamma^{ab}/\text{tor} (\Gamma^{ab})\) (which is a lattice in the \(\mathbb C\)-vector space \(\underline H_!({\mathcal T}, \mathbb C)^\Gamma\) of automorphic forms) turns out to agree with a conveniently normalized Petersson product (Theorem 5.7.1).
The result is as follows: For each \(\alpha\in \Gamma\), there exists a holomorphic theta function \(u_\alpha: \Omega\to C^*\) that satisfies \(u_\alpha(\beta z)= c_\alpha(\beta) u_\alpha(z)\). The pairing \(\Gamma\times \Gamma\to C^*\), \((\alpha, \beta)\mapsto c_\alpha(\beta)\) takes its values in \(K^*_\infty\) and induces via \(v_\infty: K^*_\infty\to \mathbb Z\) a symmetric pairing \((.,.): \overline\Gamma\times \overline\Gamma\to \mathbb Z\), which is positive definite by the above. Therefore
\[
\begin{aligned} \overline c: \overline\Gamma\quad &\to \quad \Hom(\overline\Gamma, C^*):= T_\Gamma(C)\\ \text{class of } \alpha\quad &\mapsto c_\alpha\end{aligned}
\]
is injective, and the torus \(T_\Gamma\) divided by \(\overline c(\overline\Gamma)\) is an Abelian variety defined over \(K_\infty\), which happens to agree with \(J_\Gamma/K_\infty\) (Theorem 7.4.1).
As an application of the construction, it is shown how to obtain the strong Weil curve \(E_\varphi\) of a normalized rational Hecke eigenform \(\varphi\in \overline\Gamma \hookrightarrow \underline H_!({\mathcal T}, \mathbb C)^\Gamma\). In the commutative diagram
\[
\begin{matrix} 1 & \rightarrow & \overline\Gamma & \rightarrow & T_\Gamma(C) & \rightarrow & J_\Gamma(C) & \rightarrow & 0\\ && \downarrow && \downarrow ev && \downarrow pr_\varphi\\ 1 & \rightarrow & \Lambda & \rightarrow & C^* & \rightarrow & C^*/\Lambda & \rightarrow & 0,\end{matrix}
\]
let the middle vertical arrow be defined by \(f\mapsto f(\varphi)\) and \(\Lambda:= ev(\overline\Gamma)\). Then \(\Lambda\subset K^*_\infty\) is a lattice, \(C^*/\Lambda= E_\varphi(C)\), and \(pr_\varphi\) is the strong Weil uniformization searched for.
As is shown in subsequent work of the first author [Analytical construction of Weil curves over function fields, J. Théor. Nombres Bordx 7, No. 1, 27--49 (1995; Zbl 0846.11037)], the degree of \(pr_\varphi\) and the valuation of the invariant \(j(E_\varphi)\) may be read off from the position of \(\varphi\) in the lattice \((\overline\Gamma, (. ,.))\), at least if the base ring \(A\) is a polynomial ring \(\mathbb F_q[T]\). construction of elliptic curves from automorphic Hecke newforms; Drinfeld modular curves; Shimura-Taniyama-Weil conjecture; Jacobian; strong Weil curve; strong Weil uniformization Gekeler, E-U; Reversat, M, Jacobians of Drinfeld modular curves, J. für die reine und angewandte Mathematik, 476, 27-93, (1996) Drinfel'd modules; higher-dimensional motives, etc., Modular and Shimura varieties, Jacobians, Prym varieties, Elliptic curves over global fields Jacobians of Drinfeld 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 The polar invariants (also called polar quotients) of isolated hypersurface singularities are, by definition, the contact orders between the hypersurface and the branches of its generic polar curve. For this type of hypersurfaces, its Teissier collection \(\{(q, m_q)\}\), \(q\) being the polar invariants and \(m_q\) its multiplicities, constitutes an analytic invariant.
In this paper, the authors provide an overview of a number of recent results on the polar invariants of plane curve singularities that complete the well-known classical results on this subject. The authors collect their own results and others with Garcia-Barroso, being the most interesting facts the use of Newton diagrams and some applications to pencils of plane curve singularities. plane curve singularity; polar invariant; Jacobian Newton polygon; pencil of plane curve singularities J. Gwoździewicz, A. Lenarcik and A. Płoski, Polar invariants of plane curve singularities: intersection theoretical approach, \textit{Demonstratio Math}. \textbf{43} (2010), 303-323. Singularities of curves, local rings, Milnor fibration; relations with knot theory, Plane and space curves Polar invariants of plane curve singularities: intersection theoretical approach | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Fix a one-dimensional group variety \(G\) with Euler-characteristic \(\chi(G)=0\), and a quasi-projective variety \(Y\), both defined over \(\mathbb{C}\). For any \(f\in \text{Hom}(Y,G)\) and constructible sheaf \({\mathcal F}\) on \(Y\), we construct an invariant \(c_{{\mathcal F}}(f)\in G\), which provides substantial information about the topology of the fiber-structure of \(f\) and the structure of \({\mathcal F}\) along the fibers of \(f\). Moreover, \(c_{{\mathcal F}}:\text{Hom}(Y,G)\to G\) is a group homomorphism. generalized Weil's reciprocity law; one-dimensional group variety; topology of the fiber-structure; invariant of homomorphism; Riemann surface Topological properties in algebraic geometry, Compact Riemann surfaces and uniformization, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic groups, Birational geometry Generalized Weil's reciprocity law and multiplicativity theorems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 in the paper for computing the topological type of a nonsingular real-algebraic curve on a projective plane. The topological type is a structure including \((1)\quad the\) parity of the degree of the curve; \((2)\quad the\) number of ovals to which the curve splits; \((3)\quad partial\) 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 coefficients length. cylindrical decomposition; CAD; computing the topological type of a nonsingular real-algebraic curve; ovals S. Arnborg and H. Feng, Algebraic decomposition of regular curves, J. Symbolic Comput. 5 (1988) 131--140. Families, moduli of curves (algebraic), Symbolic computation and algebraic computation Algebraic decomposition of regular 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 \(\mathfrak{g}\) be a split finite dimensional semisimple Lie algebra over a field of chracteristic zero. Let \(q\) an element in the ground field of \(\mathfrak{g}\) that is transcendental and let \(\mathcal{U}_q(\mathfrak{g})\) denote the quantized universal enveloping algebra of \(\mathfrak{g}\) at \(q\) with standard generators \(X_1^\pm,\dots,X_r^\pm\) and \(K_1^{\pm 1},\dots,K_r^{\pm 1}\). \textit{C. De Concini,V. G. Kac} and \textit{C. Procesi} [Some quantum analogues of solvable Lie groups. in: Geometry and analysis. Papers presented at the Bombay colloquium, India, January 6--14, 1992. Oxford: Oxford University Press. Stud.\ Math., Tata Inst.\ Fundam.\ Res.\ 13, 41--65 (1995; Zbl 0878.17014)] defined for a given element \(w\) of the Weyl group of \(\mathfrak{g}\) certain subalgebras \(\mathcal{U}_\pm^w\) of \(\mathcal{U}_q(\mathfrak{g})\) generated by the Lusztig root vectors obtained from reduced expressions of \(w\) and showed that these subalgebras are independent of the choice of the reduced expression.
The goal of the paper under review is to study the prime ideals of \(\mathcal{U}_-^w\) invariant under the conjugation action of the group-like elements \(H:=\langle K_1^{\pm 1}, \dots,K_r^{\pm 1}\rangle\) of \(\mathcal{U}_q(\mathfrak{g})\). The main results are as follows: 1) an explicit description of these invariant prime ideals using Demazure modules, 2) a construction of a small generating set for each invariant prime ideal, and 3) an identification of the poset structure on invariant prime ideals with a Bruhat interval. Even in the special case of quantum matrices 1) and 2) are new. The proof uses \textit{M. Gorelik}'s investigation of the spectra of quantum Bruhat cell translates [J.\ Algebra 227, No.\ 1, 211--253 (2000; Zbl 1038.17006)], \textit{A. Joseph}'s results on generating sets for ideals of the quantized coordinate ring of the simply connected semisimple algebraic group \(G\) with Lie algebra \(\mathfrak{g}\) [C.\ R.\ Acad.\ Sci.\ Paris ,Sér.\ I Math.\ 321, No.\ 2, 135--140 (1995; Zbl 0837.17007)], and an interpretation of \(\mathcal{U}_-^w\) as a quantized function algebra on the Schubert cell \(B_+w\cdot B_+\). Similar results are also obtained for vanishing ideals of torus orbit closures of symplectic leaves of related Poisson structures on Schubert cells in flag varieties. The results of the paper under review play a crucial role in the author's recent classification of the \(H\)-invariant prime ideals of arbitrary quantum partial flag varieties of \(G\) [Proc.\ Am.\ Math.\ Soc.\ 138, No.\ 4, 1249--1261 (2010; Zbl 1245.16030)]. Subalgebra of a quantized universal enveloping algebra; invariant prime ideal; Schubert cell; Demazure module; generating set; Bruhat interval; torus orbit; flag variety; Poisson structure; symplectic leaf; quantum matrix M. Yakimov, \textit{Invariant prime ideals in quantizations of nilpotent Lie algebras}, Proc. Lond. Math. Soc. (3) \textbf{101} (2010), no. 2, 454-476. Quantum groups (quantized enveloping algebras) and related deformations, Ring-theoretic aspects of quantum groups, Poisson manifolds; Poisson groupoids and algebroids, Grassmannians, Schubert varieties, flag manifolds Invariant prime ideals in quantizations of nilpotent Lie algebras | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth projective curve defined over a finite field \(\mathbb F _ q\). Then the action of the absolute Galois group \(\mathcal{G}(\overline {\mathbb F _ q}/\mathbb F _ q)\) on the algebraic closure \(\overline {\mathbb F _ q}\) of \(\mathbb F _ q\) gives rise to an action on the set \(C(\overline {\mathbb F _ q})\) of \(\overline {\mathbb F _ q}\)-rational points on \(C\). Identifying \(\mathcal{G}(\overline {\mathbb F _ q}/\mathbb F _ q)\) with the profinite completion \(\widehat Z\) of the additive group \(\mathbb Z\) of integers, the paper under review introduces the notion of a divisor on a submodule \(M\) of \(C\) (in the sense that \(M(\overline {\mathbb F _ q})\) is a subset of \(C(\overline {\mathbb F _ q})\) which is closed under the action of \(\widehat Z\)). The main result of the paper concerns the case in which \(M(\mathbb F _ q) = C(\mathbb F _ q) \cap M(\overline {\mathbb F _ q})\) is properly included in \(C(\mathbb F _ q)\). It shows that then the linear equivalence class \(\mathbb P(L _ M ^ C(D))\) of any fixed divisor \(D\) on \(M\) does not contain a projective space of positive dimension. The author also considers smooth projective curves \(C _ 1\) and \(C _ 2\) defined over \(\mathbb F _ q\), whose intersection does not include \(C _ 1(\mathbb F _ q)\). Assuming that \(O = O _ {C _ 1 \setminus C _ 2}\) is the ring of regular functions on \(C _ 1 \setminus C _ 2\), and \(O ^ {\ast }\) is the unit group of \(O\), she deduces from the main result of the paper that the segments \(S _ f = \{g \in O ^ {\ast }\colon \;0 \leq (g) _ {\infty } \leq (f) _ {\infty }\}\), \(f \in O ^ {\ast }\), do not contain an \(\mathbb F _ q\)-linear space of dimension \(\geq 2\). divisor on a \(\widehat Z\)-submodule \(M\) of a curve over a finite field; effective divisor; linear equivalence of a divisor on \(M\) Algebraic functions and function fields in algebraic geometry, Finite ground fields in algebraic geometry \(\widehat Z\)-submodules 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 On a uniruled projective complex manifold \(X^n\), a VMRT-structure \(C \subset \mathbb{P}TX\) is the Zariski closure of the collection of tangent lines of all deformations of a given minimal degree rational curve. The problem addressed in this paper is whether a VMRT-structure is \textit{locally flat}, i.e. admits, near a generic point, coordinates in which the VMRT is invariant under translation. If this happens, then the VMRT has generic fibers \(C_x=C \cap \mathbb{P}T_x X\) isomorphic to one another as projective varieties. A \(Z\)-VMRT, for some projective variety \(Z \subset \mathbb{P}^{n-1}\), is one for which every \(C_x\) is carried to \(Z\) by some projective transformation \(\mathbb{P}T_x X \cong \mathbb{P}^{n-1}\). The main theorem: if \(Z\) is smooth, nondegenerate, and a complete intersection, then (except for certain possible choices of \(Z\) made explicit in the paper), every \(Z\)-VMRT is locally flat. The authors conjecture that the result continues to hold for some of those exceptions, but they give explicit counterexamples for certain \(Z\): general hyperplane sections of (a) spin varieties or (b) Lagrangian Grassmannians, of high enough dimension. The proofs use Cartan's approach to differential geometry, adapting connections to the VMRT, and some new results on cohomological invariants of complete intersection projective varieties. equivalence problem; minimal rational curve; complete intersection; variety of minimal rational tangents (VMRT) Exterior differential systems (Cartan theory), Complete intersections, Local differential geometry Isotrivial VMRT-structures of complete intersection type | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(\mathbb{K}\) be an algebraically closed field of characteristic zero. We say that a polynomial automorphism \(f:\mathbb{K}^n\to\mathbb{K}^n\) is special if the Jacobian of \(f\) is equal to 1. We show that every \((n-1)\)-dimensional component \(H\) of the set \(\mathrm{Fix}(f)\) of fixed points of a non-trivial special polynomial automorphism \(f:\mathbb{K}^n\to\mathbb{K}^n\) is uniruled. Moreover, we show that if \(f\) is non-special and \(H\) is an \((n-1)\)-dimensional component of the set \(\mathrm{Fix}(f)\), then \(H\) is smooth, irreducible and \(H=\mathrm{Fix}(f)\). Moreover, for \(\mathbb{K}=\mathbb{C}\) if \(f\) is non-special and \(\mathrm{Jac}(f)\) has an infinite order in \(\mathbb{C}^\ast\), then the Euler characteristic of \(H\) is equal to 1. affine variety; group of automorphisms; fixed point of a polynomial automorphism Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Group actions on affine varieties On the set of fixed points of a polynomial automorphism | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(T_g\subseteq M_g\), \(g\geq 4\), be the moduli scheme of all smooth genus \(g\) trigonal curves and \(T_{g,n}\), \((g-1)/3\leq n\leq g/2\), the subset of \(T_g\) parametrizing trigonal curves, \(C\), whose trigonal pencil, \(D\), is of kind \(n\), i.e. \(h^0(C,{\mathcal O}_C(nD))= n+1\) and \(h^0(C,{\mathcal O}_C((n+ 1)D))> n+2\). For any \(\alpha\geq 0\) let \(T_{g,n}(\alpha)\subseteq T_{g,n}\) be the set of all trigonal curves with a trigonal pencil with at least \(\alpha\) total ramification points. It is known that \(T_g\) is irreducible and unirational and that for certain \(g\) it is rational. Here the authors prove that if \(g\geq 4\) and \(\alpha\leq 2\), then \(T_{g,n}(\alpha)\) is irreducible and unirational and that \(T_{g,[g/2]}(\alpha)\) is rational. The authors also prove the rationality of some subloci of \(M_4\). trigonal curve; rationality; unirationality; total ramification point; moduli scheme of trigonal curves; rational variety Casnati G., Del Centina A.: On certain loci of curves of genus g 4 with Weierstrass points whose first non-gap is three. Math. Proc. Cambridge Philos. Soc. 132, 395--407 (2002) Riemann surfaces; Weierstrass points; gap sequences, Rational and unirational varieties On certain loci of curves of genus \(g\geq 4\) with Weierstrass points whose first non-gap is three | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The theorem of \textit{J. Ax} [Ann. Math. (2) 88, 239-271 (1968; Zbl 0195.05701)] says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and is investigated in the present paper in the category of proregular mappings of proalgebraic spaces. We show that such maps are surjunctive if they commute with sufficiently large automorphism groups. Of particular interest is the case of proalgebraic varieties over infinite graphs. The paper intends to bring out relations between model theory, algebraic geometry, and symbolic dynamics. regular selfmapping of a complex algebraic variety; surjunctivity; proregular mappings; proalgebraic spaces; proalgebraic varieties; model theory; symbolic dynamics M. Gromov, ``Endomorphisms of symbolic algebraic varieties'', J. Eur. Math. Soc. (JEMS)1 (1999) no. 2, p. 109-197 Varieties and morphisms, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Graphs and abstract algebra (groups, rings, fields, etc.), Model-theoretic algebra, Rational and birational maps Endomorphisms of symbolic algebraic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(A\) be a commutative noetherian ring. We investigate a class of functors from \(\langle \langle\)commutative \(A\)-algebras\(\rangle \rangle\) to \(\langle \langle\)sets\(\rangle \rangle\), which we call coherent. When such a functor \(F\) in fact takes its values in \(\langle \langle\)abelian groups\(\rangle \rangle\), we show that there are only finitely many prime numbers \(p\) such that \({}_pF(A)\) is infinite, and that none of these primes are invertible in \(A\). This (and related statements) yield information about torsion in \(\operatorname{Pic} (A)\). For example, if \(A\) is of finite type over \(\mathbb{Z}\), we prove that the torsion in \(\operatorname{Pic} (A)\) is supported at a finite set of primes, and if \({}_p\operatorname{Pic} (A)\) is infinite, then the prime \(p\) is not invertible in \(A\). These results use the (already known) fact that if such an \(A\) is normal, then \(\operatorname{Pic} (A)\) is finitely generated. We obtain a parallel result for a reduced scheme \(X\) of finite type over \(\mathbb{Z}\). We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field. coherent functor; representable functor; torsion; Picard group of a scheme of finite type over a finite field Jaffe, DB, Coherent functors, with application to torsion in the Picard group, Trans. Am. Math. Soc., 349, 481-527, (1997) Picard groups, Special properties of functors (faithful, full, etc.), Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.), Picard schemes, higher Jacobians Coherent functors, with application to torsion in the Picard group | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 variety defined over a field \(K\). The index \(\mathrm{ind}(X)\) of \(X\) is the smallest degree of extension \(L/K\) for which \(X\) has a \(L\)-rational point. In the article under review, the authors prove the following theorem: if \(X\) is a variety defined over the quotient field \(K\) of a Henselian DVR \(R\) with algebraically closed residual field \(k\) and suppose that either \(X\) is \(K\)-smooth or \(R\) is excellent, then \(\mathrm{ind}(X)\) (resp. the prime-to-\(p\) part of \(\mathrm{ind}(X)\)) divides \(\chi(X,E)\) whenever \(\mathrm{char}(k) = 0\) or \(\mathrm{char}(k) = p > \dim(X) + 1\) (resp. \(\mathrm{char}(k) = p > 0\)). A short and beautiful proof of this result is presented in Section 2 and 3. This theorem has in particular the following applications:
(1) A smooth proper variety \(X\) over \(K=\mathbb{C}((t))\) or \(\mathbb{Q}_p^{\mathrm{nr}}\) with \(p > \dim X + 1\) such that \(H^i(X,\mathcal{O}_X) = 0\) for all \(i > 0\) (e.g. when \(X\) is rationally connected) has index \(1\). This result is already known when \(K=\mathbb{C}((t))\) (see [Duke Math. J. 161, No. 5, 735--801 (2012), Zbl 1244.14010]). Note that since \(\mathbb{Q}_p^{\mathrm{nr}}\) is a \((C_1)\) field, a separably rationally connected variety over \(\mathbb{Q}_p^{\mathrm{nr}}\) has a \(\mathbb{Q}_p^{\mathrm{nr}}\)-point according to the \((C_1)\)-conjecture.
(2) Let \(X\) be a smooth proper variety of general type over \(\mathbb{C}((t))\) or \(\mathbb{Q}_p^{\mathrm{nr}}\) with \(p > \dim X + 1\). The index of \(X\) divides the pluri-genus \(h^0(X,\mathcal{O}_X(nK_X))\) provided \(n \geq 2\).
(3) Let \(d\) be an positive integer and \(d = \prod_{i = 1}^{n} p_i^{\alpha_i}\) its prime factorization. If \(N \geq \max_{1 \leq i \leq n} p_i^{\alpha_i}\), then every hypersurface \(X \subset \mathbb{P}^N\) over \(\mathbb{C}((t))\) of degree \(d\) has index \(1\). The reader is referred to Section 4 of the article under review for the whole discussion on the applications of their main theorem to hypersurfaces.
In Section 5, the authors prove the same divisibility result with \(\chi(X,E)\) replaced by the class of \(X\) in the cobordism ring of \(\mathrm{Spec}(K)\) under the same assumptions. Using this, the authors provide examples of characteristic numbers other than the Euler characteristic having the same divisibility property. index of a variety; rational points; \((C_1)\)-conjecture; cobordism ring; Euler characteristic Hélène Esnault, Marc Levine, and Olivier Wittenberg, Index of varieties over Henselian fields and Euler characteristic of coherent sheaves, J. Algebraic Geom. 24 (2015), no. 4, 693 -- 718. Rational points, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Rationally connected varieties, Fano varieties Index of varieties over Henselian fields and Euler characteristic of coherent sheaves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0745.00034.]
Let \(X\) be a projective threefold over an algebraically closed field of characteristic zero, with only \(\mathbb{Q}\)-factorial terminal singularities, whose canonical class is not pseudoeffective. The aim of the paper is to study the geometry of the cone \(NE^ 1(X)\) of effective divisors and its dual cone \(NM_ 1(X)\) of numerically effective curves. The author introduces the notion of coextremal ray in \(NM_ 1(X)\) which appears from some class of covering family of rational curves and proves the decomposition theorem for \(NM_ 1(X)\) using results from Mori's theory for threefolds.
Let \(H\) be an ample Cartier divisor on a projective variety, \(X\), of any dimension. Let \(u(X,H)\) be the (unnormalized) spectral value of \((X,H)\) that is \(u(X,H):=\sup \bigl\{ t\in \mathbb{Q} | \kappa(K_ X+tH)=-\infty \bigr\}\). Note that \(u(X,H) <+ \infty\) since \(H\) is ample. This number, \(u(X,H)\), turns out to be very useful to classify polarized pairs and it has been studied in several papers by Sommese, by Sommese and the reviewer, and by Fujita. It is an open question whether \(u(X,H)\) is rational. In the paper it is also proved that \(u(X,H)\in \mathbb{Q}\) if \(\dim X=3\). Note that the rationality is trivial in dimension 1, while for surfaces was previously proved by \textit{Sakai}. factorial terminal singularities; cone of effective divisors; cone of numerically effective curves; rationality of spectral value; threefold; coextremal ray; Mori's theory; ample Cartier divisor Victor V. Batyrev, The cone of effective divisors of threefolds, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 337 -- 352. \(3\)-folds, Divisors, linear systems, invertible sheaves The cone of effective divisors of threefolds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth curve of degree \(n\) and genus \(g\) in the projective space \(\mathbb P^{2k-2}\) defined over the complex field \(\mathbb C\), where \(k \in \mathbb N\). We shall use \(\nu_k\) to denote the number of linear subspaces of dimension \(k-2\) in \(\mathbb P^{2k-2}\), which are \(k\)-secant to \(C\). Concerning \(\nu_k\), there are two problems:
i) provide a formula for the number \(\nu_k\);
ii) find the so-called generating function of the series \(\sum_{k \geq 0}\nu_kt^k \in \mathbb Z[[t]]\).
The solution to problem i) was given by Castelnuovo in 1889; cf. also \textit{E. Arbarello} et al. [Geometry of algebraic curves. Volume I. Grundlehren der mathematischen Wissenschaften, 267. New York: Springer-Verlag, (1985; Zbl 0559.14017)]. The purpose of the paper under review is to solve problem ii) without using the result by Castelnuovo. More precisely, in the paper it is shown that the generating function \(S(t)\) of the series \(\sum_{k \geq 0}\nu_kt^k\) is given by
\[
S(t)= \left ( \frac{1+\sqrt{1+4t}}{2} \right )^n \left ( \frac{-1-4t+(1+2t)\sqrt{1+4t}}{2t^2} \right )^{g-1}.
\]
By way of example, we make the development in series explicit: \(\frac{1+\sqrt{1+4t}}{2}=1+t-t^2+2t^3-5t^4+14t^5+\cdots\). projective space; linear subspaces; degree and genus of a projective curve; \(k\)-secant a curve Barz, P., Sur une formule de Castelnuovo pour LES espaces multisécants, Bollettino dell unione matematica italiana. Sezione B: articoli di ricerca matematica, 10, 381-388, (2007) Classical problems, Schubert calculus, Configurations and arrangements of linear subspaces, Curves in algebraic geometry On Castelnuovo's formula for multisecant 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 Let \(F=\mathbb{F}_ q(t)\) be the rational function field with finite constant field \(\mathbb{F}_ q\), let \(f_ i(t)\in\mathbb{F}_ q[t]\) for \(i=1,\dots,n\), and take \(E=F(f^{1/r}_ 1,\dots,f^{1/r}_ n)\), where \(r\) is a prime dividing \(q-1\). The author investigates the splitting of prime divisors in the extension \(E/F\) and computes the genus of \(E\). As an application he shows that \(A(3)\geq 1/3\) and \(A(5)\geq 1/2\), where \(A(q)\) is defined as follows: Let \(X\) be a smooth projective curve of genus \(g_ X\) over \(\mathbb{F}_ q\) and let \(X(\mathbb{F}_ q)\) denote the number of points of \(X\) defined over \(\mathbb{F}_ q\). Take \(q\) fixed and let \(g_ X\) tend to infinity. Then \(A(q)=\limsup X(\mathbb{F}_ q)/g_ X\). Kummer extension; rational function field; splitting of prime divisors; genus; smooth projective curve Xing, C. P.: Multiple Kummer Extensions and the Number of Prime Divisors of Degree One in Function Fields. J. of Pure and Appl. Algebra84, 85--93 (1993) Arithmetic theory of algebraic function fields, Curves over finite and local fields, Algebraic coding theory; cryptography (number-theoretic aspects), Algebraic functions and function fields in algebraic geometry Multiple Kummer extension and the number of prime divisors of degree one in function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0607.00004.]
The results in this paper yield an explicit description of the quadratic intersection form on the Néron-Severi-group of a compact complex non- algebraic surface. The authors prove in particular that this form is negative-definite (modulo torsion), if the algebraic dimension of the surface is zero. Finally several applications to 2-vector bundles on non- algebraic surfaces are given. Picard group; quadratic intersection form on the Néron-Severi-group of a compact complex non-algebraic surface; algebraic dimension; 2-vector bundles Brînzânescu, V., Flondor, P.: Quadratic intersection form and -vector bundles on nonalgebraic surfaces, Proc. Conf. on Alg. Geometry, Berlin 1985, Teubner: Band 92, 1986 Picard groups, Compact complex surfaces, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Holomorphic bundles and generalizations Quadratic intersection form and 2-vector bundles on nonalgebraic 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 Math. Ann. 277, 453-468 (1987; see the preceding review) [see also the correction thereof, ibid. 280, No.4, 697-698 (1988)], the author established an exact sequence
\[
0\quad \to \quad W(X)\quad \to \quad W(K)\quad \to \coprod_{co\dim (x)=1}W(k(x)),
\]
where X is a smooth surface over a field k (char(k)\(\neq 2)\), K is the function field, k(x) is the residue field at \(x\in X\) and \(W(\quad)\) is the Witt ring functor. This sequence is used to determine W(X), where X is a smooth surface over \({\mathbb{C}}\). Relations with K-theory and étale cohomology are discussed. Witt group of a smooth complex curve; Witt group of a smooth complex surface; K-theory; étale cohomology Fernández-Carmena, Fernando: The Witt group of a smooth complex surface, Math. ann. 277, No. 3, 469-481 (1987) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], General binary quadratic forms, Schemes and morphisms, Curves in algebraic geometry The Witt group of a smooth complex 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 Given d,g,n, what is the maximal integer m such that m general points of \({\mathbb{P}}^ n\) are contained in a smooth, connected, curve of degree \(d\) and genus \(g\)? The aim of this research monograph is to give the answer to this question in many interesting cases, to give lower or upper bounds for m in other cases and to apply the information obtained; essentially the author is interested in the case \(n=3\). The proofs are infinitesimal (compute the tangent space to an appropriate functor); this computation often boils down to the proof of existence of space curves C with \(h^ 1(N_ C(-2)=0\) \((N_ C\) normal bundle); thus the paper by \textit{G. Ellingsrud} and \textit{A. Hirschowitz} [C. R. Acad. Sci., Paris, Sér. I 299, 245-248 (1984; Zbl 0572.14007)] is extremely useful; this monograph under review is based also on Kleppe's Oslo thesis, part of which (or generalizations of part of it) are at last appearing in print: \textit{J. O. Kleppe}, ``Non-reduced components of the Hilbert scheme of smooth space curves'' in: Space curves, Proc. Conf., Rocca di Papa/Italy 1985, Lect. Notes Math. 1266, 187-207 (1987; Zbl 0631.14022) and ``Liaison of families of subschemes in \({\mathbb{P}}^ n\)'' (to appear in: Algebraic curves and projective geometry, Proc. Trento Conf.).
The reviewer found this monograph very useful for his research. Hilbert scheme; liaison; linkage; number of general points on a curve; space curves; normal bundle Perrin D., Courbes passant par m points généraux de \({\mathbb{P}^{3}}\), Mém. Soc. Math. France 28-29 (1987). Projective techniques in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Curves in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Courbes passant par m points généraux de \(P^ 3\). (Curves passing through m general points of \(P^ 3)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\tilde{C}\) be a non-singular plane curve of degree \(d\geq 8\) with an involution \(\sigma \) over an algebraically closed field of characteristic 0 and \(\tilde{P}\) a point of \(\tilde{C}\) fixed by \(\sigma \). Let \(\pi : \tilde{C}\rightarrow C = \tilde{C} / \langle\sigma\rangle \) be the double covering. We set \(P = \pi (\tilde{P})\). When the intersection multiplicity at \(\tilde{P}\) of the curve \(\tilde{C}\) and the tangent line at \(\tilde{P}\) is equal to \(d - 3\) or \(d - 4\), we determine the Weierstrass gap sequence at \(P\) on \(C\) using blowing-ups and blowing-downs of some rational surfaces. Weierstrass gap sequence; Weierstrass semigroup; smooth plane curve; double covering of a curve; blowing-up of a rational surface Riemann surfaces; Weierstrass points; gap sequences, Plane and space curves, Coverings of curves, fundamental group, Rational and ruled surfaces Weierstrass gap sequences at points of curves on some rational 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 Here we focus on the geometry of \({\bar P}_{d,g}\), the compactification of the universal Picard variety constructed by L. Caporaso. In particular, we show that the moduli space of spin curves constructed by M. Cornalba naturally injects into \({\bar P}_{d,g}\) and we give generators and relations of the rational Picard group of \({\bar P}_{d,g}\), extending previous work by \textit{A. Kouvidakis} [J. Differ. Geom. 34, No. 3, 839--850 (1991; Zbl 0780.14004)]. universal Picard variety; geometric invariant theory; spin curve; stable curve Fontanari, C., On the geometry of moduli of curves and line bundles, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16, No. 1 (2005), 45-59. Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Jacobians, Prym varieties, Geometric invariant theory On the geometry of moduli of curves and line 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 In this note we state a decomposition theorem into bunches of branches (i.e. analytically irreducible germs) for the generic polar curves of a reduced germ of a plane analytic curve, with equation \(f(x,y)=0\). These are the curves with equation \({\partial f(x,y)\over\partial y}+ \tau{\partial f(x,y) \over\partial x}=0\) with generic \(\tau\). All the branches of the same bunch have the same contact with each branch of \(C\). A number of the first terms of the Puiseux expansion of each branch of the polar is therefore independent of \(\tau\); this number depends only on the bunch to which the branch belongs.
This generalizes results of \textit{H. J. S. Smith} [Proc. Lond. Math. Soc. 6, 153-182 (1876; JFM 08.0433.01)], \textit{M. Merle} [Invent. Math. 41, 103-111 (1977; Zbl 0371.14003)] (where \(C\) is a branch), \textit{E. Casas-Alvero} [Ann. Inst. Fourier 41, No. 1, 1-10 (1991; Zbl 0707.14024)], \textit{F. Delgado de la Mata} [Compos. Math. 92, No. 3, 327-375 (1994; Zbl 0816.14017)]. We show by an example that it is also optimal. We have shown elsewhere that it implies the results of \textit{Le Dung Trang}, \textit{F. Michel} and \textit{C. Weber} [Compos. Math. 72, No. 1, 87-113 (1989; Zbl 0705.32021)]. polar curves; germ of a plane analytic curve E. Garcia Barroso, Un théorème de décomposition pour les polaires génériques d'une courbe plane, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 59--62. Plane and space curves, Germs of analytic sets, local parametrization A decomposition theorem for the generic polars of a plane curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a complex smooth projective threefold of general type. Suppose that the canonical system is composed of pencils, let \(f: X \to C\) be an induced fibration over a curve \(C\) of genus \(b\), let \(F\) be a general fiber of \(f\), and let \(q\), \(p_g\) denote irregularity and geometric genus.
Then, the authors show that \(p_g(F)\geq 1\), and that one of the following holds in case \(b\geq 2\):
(1) \(p_g(X)\geq b-1\) and \(p_g(F)=1\), or
(2) \(b=p_g(F)=p_g(X)=2\).
The proof is based on some computation of cohomology groups of sheaves on \(C\). By a similar method the authors show \(q(X)\leq b+q(F)\), and discuss about examples where the equality holds.
\{Reviewer's remark: By using the well-known Hodge theory one can easily prove \(q(X)\leq q(S)+q(F)\) for any algebraic fiber space \(X \to S\) of arbitrary dimensions, where \(F\) is a general fiber\}. threefold of general type; canonical system; fibration over a curve; genus; irregularity; geometric genus; cohomology groups DOI: 10.1017/S0305004198002734 \(3\)-folds, Special algebraic curves and curves of low genus, Fibrations, degenerations in algebraic geometry Irregularity of canonical pencils for a threefold of general type | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(A\) be an abelian variety over a function field \(K\) in one variable over a finite field \(k\). Let \(v\) be a place of \(K\). In this paper we study the topology induced on \(A(K)\) by the \(v\)-adic topology on \(A(K_v)\). In many cases this will lead to bounds for the \(v\)-adic distance between points in \(A(K)\) in terms of their height and to abelian analogues of Leopoldt's conjecture. This paper also studies the question of integral points on affine open subsets of abelian varieties in positive characteristics. In the classical case of number fields, Lang has conjectured and \textit{G. Faltings} [Ann. Math., II. Ser. 133, No. 3, 549-576 (1991; Zbl 0734.14007)] proved that for \(A\) an abelian variety over the number field \(K\), if \(V\) is an affine open subset of \(A\) and \(S\) is a finite set of places of \(K\), then the set of \(S\)-integral points of \(V\) is finite. Faltings also has a noneffective bound. \textit{A. Buium} and the author [Math. Ann. 297, No. 2, 303-307 (1993; Zbl 0789.14017)] obtained a similar result for function fields of characteristic zero, but in positive characteristic the problem has not been studied, except in dimension 1, i.e., for elliptic curves. In this case we have obtained [Compos. Math. 74, No. 3, 247-258 (1990; Zbl 0715.14027)] results on this problem.
In this paper we have obtained a result under restrictive, but quite general, hypotheses for abelian varieties of arbitrary dimension and deduce the finiteness of integral points on affine subsets under these hypotheses. Our strategy is similar to that of the author's cited joint paper with A. Buium. finiteness of integral points; prime characteristic; abelian variety over a function field Voloch, J.F., Diophantine approximation on abelian varieties in characteristic \textit{p}, Amer. J. math., 117, 4, 1089-1095, (1995) Varieties over global fields, Algebraic theory of abelian varieties, Abelian varieties of dimension \(> 1\), Algebraic functions and function fields in algebraic geometry, Diophantine approximation, transcendental number theory Diophantine approximation on abelian varieties 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 The authors give an algorithm for finding \textit{\(n\)-contact curves} for a given elliptic curve \(E\). Given a plane curve \(C\), an \(n\)-contact (plane) curve \(D\) of \(C\), is one such the intersection multiplicity at each point in \(C\cap D\) is \(n\). If \(E=C\), the curve \(D\) is also known as a curve of type \((d,n)\), where \(d\) is the degree of \(D\). For the algorithm, one can assume that all points of \(E\cap D\) are affine, the authors then require the determination of an \(n\)-torsion affine point \(P\in E\) that can be written as the sum of \(k\) affine points of \(E\), where 3 divides \(nk\). The algorithm first provides a candidate weak \(n\)-contact curve i.e. a curve that intersects \(E\) at these \(k\) points, each with multiplicity \(n\), and possibly intersects the point at infinity. This candidate weak \(n\)-contact curve is obtained via the Mumford-representation of the divisor consisting of only these \(k\) points. The algorithm then gives a procedure to obtain the \(n\)-contact curve from the weak \(n\)-contact curve.
After some examples in Section 4, the authors discuss its application in obtaining Zariski \(k\)-plets [\textit{T. Shirane}, Proc. Am. Math. Soc. 145, No. 3, 1009--1017 (2017; Zbl 1358.14018)]. \(n\)-contact curve; elliptic curves; representation of divisors; Zariski tuple Topological properties in algebraic geometry, Plane and space curves, Elliptic curves, Computational aspects of algebraic curves An explicit construction for \(n\)-contact curves to a smooth cubic via divisions of polynomials and Zariski tuples | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 concerned with \(p\)-adic heights of rational points on an elliptic curve defined over a number field. The author first compares three different approaches to construct \(p\)-adic analogs of the Weierstraß \(\sigma\)-function, which lead to the definition of a \(p\)-adic height. She then introduces a ``naive'' \(p\)-adic height and shows that it is in fact a quadratic form. The relation between these \(p\)-adic heights has been investigated in a different paper of the author [C. R. Acad. Sci., Paris, Sér. I 296, 291--294 (1983; Zbl 0532.14012)].
For the entire collection see [Zbl 0541.00003]. \(p\)-adic heights of rational points on an elliptic curve defined over a number field; Weierstraß \(\sigma\)-function; \(p\)-adic heights of rational points on an elliptic curve over a number field Elliptic curves over local fields, Heights, Global ground fields in algebraic geometry \(p\)-adic heights | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a \(\mathbb{Q}\)-factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group \(\mathrm{Pic}(X)\) in the group \(\mathrm{Cl}(X)\) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of \(\mathrm{Pic}(X)\) in \(\mathrm{Cl}(X)\) is contained in a free part of the latter group. \(\mathbb{Q}\)-factorial complete toric varieties; Cartier and Weil divisors; pure modules; free and torsion subgroups; localization; completion of fans Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Torsion-free groups, finite rank, Torsion groups, primary groups and generalized primary groups Embedding the Picard group inside the class group: the case of \(\mathbb{Q}\)-factorial complete 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 Liaison techniques have been intensively used in the classification of curves in \(\mathbb{P}^ 3\) and, more generally, of two-codimensional subschemes of \(\mathbb{P}^ n\). But already since the fundamental paper of \textit{P. Rao} [Math. Ann. 258, 169-173 (1981; Zbl 0493.14009)] liaison is considered in a more general setting, in particular when the variety where we work is a smooth arithmetically Gorenstein subscheme \(X\) of \(\mathbb{P}^ n\). The main purpose of this paper is to furnish in this relevant situation (e.g., hypersurfaces of \(\mathbb{P}^ n\), the Grassmann variety Gr(1,3), etc.) all the machinery of liaison in codimension two as developed in the projective situation. So, many of the results here are just an adaptation of results known in the projective case, and hence our discussion will sometimes simply describe the connections with the known cases and sketch the proofs.
We define the notion of basic double link, the notion of shift and of minimal shift of an even liaison class, and we prove a structure theorem for even liaison classes, which generalizes the Lazarsfeld-Rao property (LR-property) known in \(\mathbb{P}^ n\). -- We devote \S2 to the licci (linkage class of a complete intersection) liaison class, characterizing the possible locally free resolution of licci (in \(X)\) two-codimensional subschemes of \(X\) and, when \(\dim X \leq 4\), of the smooth ones. We hope that this will help in the study of licci subscheme of \(\mathbb{P}^ n\) in codimension \(>2\). -- The LR-property gives a description of all possible locally free resolutions of elements of a fixed liaison class, and also geometric consequences, such as deformation to reducible subschemes with nice irreducible components. -- In the last section we sketch some applications of these results to the Grassmann variety Gr(1,3). arithmetically Gorenstein variety; linkage class of a complete intersection; liaison in codimension two; basic double link; Lazarsfeld- Rao property; licci; Grassmann variety Bolondi, G.; Migliore, J. C.: The lazarsfeld--Rao property on an arithmetically Gorenstein variety. Manuscripta math. 78, 347-368 (1993) Linkage, Low codimension problems in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Complete intersections The Lazarsfeld-Rao property on an arithmetically Gorenstein 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 \(E\) be a torsion free sheaf on an integral scheme \(X\) and \(V \subset H^0 (X,E)\) a finite dimensional vector space; set \(r:= \text{rank} (E)\) and \(m(B,V) : = \dim (H^0 (B,E) \cap V)/ \text{rank} (B)\) for every subsheaf \(B\) of \(E\), \(B\neq 0\); \(m(B,V)\) will be called the slope of \(B\) (with respect to \(V)\). A subsheaf \(B\) of \(E\) is said to be saturated if \(E/B\) has no torsion. We will say that \(E\) is \(h^0\)-\(V\)-stable (resp. \(h^0\)-\(V\)-semistable) if for every proper subsheaf \(B\) of \(E\) we have \(m(B,V) < m(E,V)\) (resp. \(m(B,V) \leq m(A,V))\). We will say that \(E\) is \(h^0\)-\(V\)-unstable if it is not \(h^0\)-\(V\)-semistable. If \(V = H^0 (X,A)\), we will drop \(V\) from the notations and write \(m(B)\), \(h^0\)-stable \(h^0\)-semistable, \(h^0\)-unstable.
The main results of this note are the existence of a maximal filtration related to this notion of stability, the semicontinuity properties of this filtration and the properties of \(h^0\)-stability and \(h^0\)-semistability. geometry of the morphisms from a projective variety to a Grassmannian; dimensions of sections of subsheaves; \(h^ 0\)-stability; \(h^ 0\)-semistability; torsion free sheaf Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the global sections of a vector bundle over a projective 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 \(\varphi:(S,O)\to (T,O')\) be an analytic morphism between smooth complex surfaces. The set of singular points is defined by the Jacobian of \(\varphi\), its image is the discriminant of \(\varphi\). Although some components of the singular set map onto some branches of the discriminant, some other components are mapped to \(O'\). Nevertheless, these latter ones also define some infinitesimal structures in the target, the shadows.
The article shows that the images of irreducible germs of plane curves by \(\varphi\) have a certain contact either with branches of the discriminant or with certain shadows of \(\varphi\). plane curve singularities; analytic morphism; discriminant; Jacobian; shadow; clusters; germs of pencils; Puiseux series Singularities in algebraic geometry, Local complex singularities, Infinitesimal methods in algebraic geometry Shadows, discriminant and direct images 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 For a symmetric closed category \({\mathcal C}\), the author considers the Picard group \(\text{Pic} {\mathcal C}\) and the Brauer group \(\text{Br} {\mathcal C}\) of \({\mathcal C}\). Then for a functor \(F : {\mathcal C} \to {\mathcal D}\) a category \(\Phi (F)\) with a product is associated and an exact sequence
\[
\text{Pic} {\mathcal C} \to \text{Pic} {\mathcal D} \to K_1 A(F) \to \text{Br} {\mathcal C} \to \text{Br} {\mathcal D}
\]
is derived, where \(K_1 A(F)\) is the Grothendieck group of the category \(\Phi (F)\).
For an indempotent kernel functor \(\sigma\) on \({\mathcal C}\) the relative Picard group \(\text{Pic} ({\mathcal C}, \sigma)\) and the relative Brauer group \(\text{Br} ({\mathcal C}, \sigma)\) [cf. \textit{A. Verschoren}, Commun. Algebra 8, 1169-1194 (1980; Zbl 0465.18005), Lect. Notes Math. 917, 260- 278 (1982; Zbl 0484.16002)] are defined such that \(\sigma\leq\tau\) determines an exact sequence
\[
\text{Pic} ({\mathcal C}, \sigma) \to \text{Pic} ({\mathcal C}, \tau) \to K_1 A(Q^\sigma_\tau) \to \text{Br} ({\mathcal C}, \sigma) \to \text{Br} ({\mathcal C}, \tau),
\]
for some functor \(Q^\sigma_\tau : ({\mathcal C}, \sigma) \to ({\mathcal C}, \tau)\).
Then some applications to the category \({\mathcal C} = {\mathcal O}_X\)-Mod of sheaves of modules are presented. Grothendieck category; symmetric closed category; Picard group; Brauer group; Grothendieck group; relative Brauer group; bundles of moduli Relative homological algebra, projective classes (category-theoretic aspects), Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Grothendieck groups (category-theoretic aspects), Grothendieck groups, \(K\)-theory and commutative rings, Closed categories (closed monoidal and Cartesian closed categories, etc.), Picard groups The exact sequence \(Pic(C)\to Pic(D)\to K_ 1A(F)\to Br(C)\to Br(D)\) in closed categories. Applications to the theory of relative invariants of bundles of moduli. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is a report on the current status of attempts to generalize, to the Prym setting, two famous approaches to the Torelli problem for Jacobians: the ``base locus of quadric tangent cones'' method of \textit{A. Andreotti} and \textit{A. Mayer} [Ann. Sc. Norm. Supér. Pisa, Sci. Fis. Mat. (3) 21, 189--238 (1967; Zbl 0222.14024)], \textit{D. Mumford} [Curves and their Jacobians, Univ. Mich. Press (1975; Zbl 0316.14010)], and \textit{M. Green} [Invent. Math. 75, 85--104 (1984; Zbl 0542.14018)] and the ``branch divisor of the Gauss map'' method of \textit{A. Andreotti} [Am. J. Math. 80, 801--828 (1958; Zbl 0084.17304)], both of which refer to construction on the theta divisor. We will also describe for Prym varieties the ``infinitesimal variation of Hodge structures'' (IVHS) approach to Torelli problems of \textit{J. A. Carlson} and \textit{P. A. Griffiths} [Journées de geometrie algébrique, Angers/Fr. 1979, 51--76 (1980; Zbl 0479.14007)] and an analog of a result of \textit{G. Kempf} [Math. Centrum, Amsterdam, Afd. zuivere Wisk. ZW6/71 (1971; Zbl 0223.14018), p. 16; and Ann. Math. (2) 110, 243--273 (1979; Zbl 0452.14011), Cor. 4.4, p. 253] linking the Torelli problem to properties of ``Picard sheaves'', i.e. (higher) direct images of Poincaré bundles on Abelian varieties.
Although the article's goal is primarily expository, much of the material discussed is very recent, some arguments (such as the IVHS argument for degree one Torelli for Pryms) seem not to have occurred in print before and some results (such as the density of double points in the stable singular locus of Prym theta divisors and the requisite Riemann singularities theorem for double points which are both stable and exception are new. base locus of quadrics; Prym varieties; Jacobian Gauss divisors; Prym Gauss divisors; birational Donagi conjecture; double covers R. Smith and R. Varley, ''The Prym Torelli problem: An update and a reformulation as a question in birational geometry,'' in Symposium in Honor of C. H. Clemens, Bertram, A., Carlson, J. A., and Kley, H., Eds., Providence, RI: Amer. Math. Soc., 2002, pp. 235-264. Torelli problem, Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Divisors, linear systems, invertible sheaves, Picard schemes, higher Jacobians The Prym Torelli problem: An update and a reformulation as a question in birational geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We correct the proof of Theorem 3.8 in the author's paper [J. Math. Soc. Japan 62, No. 3, 975--1004 (2010; Zbl 1258.14039)]. approximate roots; deformations of a plane curve Singularities of surfaces or higher-dimensional varieties, Invariants of analytic local rings, Toric varieties, Newton polyhedra, Okounkov bodies Correction of a proof in the paper ``Approximate roots, toric resolutions and deformations of a plane branch'' | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 structured as follows: Section 1 is introduction and motivational examples. In Section 2, the authors introduce a natural compactification of the linear space \({\mathcal L}\) which they denote by \(\overline{\mathcal L}\). In order to prove the main theorem they show that one has a characteristic map \(\chi\) defined on a Zariski open set of the variety \(\overline{\mathcal L}\). As a consequence the number of solutions in the critical dimension is equal to \(\deg\,\overline{\mathcal L}\) when counted with multiplicities and when some possible ``infinite solutions'' are taken into account. The results in Section 2 generalize mathematical ideas which have been developed for the static pole placement problem by \textit{R.W. Brockett} and \textit{C.I. Byrnes} [IEEE Trans. Autom. Control 26, 271--284 (1981; Zbl 0462.93026)] and for the dynamic pole placement problem by \textit{M.S.
Ravi, J. Rosenthal} and \textit{X. Wang} [SIAM J. Control Optimization 32, No. 1, 279--296 (1994; Zbl 0797.93018); ibid. 34, No.3, 813--832 (1996; Zbl 0856.93043)].
In Section 3 the authors compute the degree of \(\overline{\mathcal L}\) in many special cases. As a corollary they rediscover several matrix completion results as they were derived earlier.
Section 4 is concerned with the value of the ``generic degree''; this is the largest possible degree of a variety \(\overline{\mathcal L}\) of fixed dimension can have. The authors determine an upper bound for the generic degree in the case when \(d= n\) and prove that this bound is reached when \(m= n< 5\). inverse eigenvalue problems; Grassmann varieties; degree of a projective variety Kim, M., Rosenthal, J., and Wang, X., Pole Placement and Matrix Extension Problems: a Common Point of View, SIAM J. Control Optim., 2004, vol. 42, no. 6, pp. 2078--2093. Grassmannians, Schubert varieties, flag manifolds, Eigenvalues, singular values, and eigenvectors, Pole and zero placement problems, Eigenvalue problems Pole placement and matrix extension problems: a common point of view | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 rational curves on the Tian-Yau complete intersection Calabi-Yau threefold (CICY) in \(\mathbb{P}^3\times\mathbb{P}^3\). Existence of positive dimensional families of nonsingular rational curves is proved for every degree \(\geq 4\). The number of nonsingular rational curves of degree \(1,2,3\) on a general Tian-Yau CICY is finite and enumerated. The number of curves of these degrees are also enumerated for the special Tian-Yau CICY. There are two 1-dimensional families of singular rational curves of degree 3 on a general Tian-Yau CICY, making this degree a turning point between finite and infinite number of curves. We also introduce a notion of equivalence of a family of rational curves, and determine the equivalences of the two 1-dimensional families on the Tian-Yau CICY. The equivalences equal the predicted numbers of curves obtained by a power series expansion of the solution of a Picard-Fuchs equation that arises in superconformal field theory. supersymmetric theories for a 10-dimensional universe; Tian-Yau complete intersection Calabi-Yau threefold; number of nonsingular rational curves; Picard-Fuchs equation DOI: 10.2140/pjm.2000.192.415 Calabi-Yau manifolds (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Relationships between surfaces, higher-dimensional varieties, and physics, Rational and unirational varieties, Complete intersections Rational curves on a complete intersection Calabi-Yau variety in \({\mathbb{P}}^3\times{\mathbb{P}}^3\) | 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 paper studies the Hilbert scheme of a generic complete intersection X of type \((m_ 1,...,m_ k)\) in the Grassmann variety G of r-planes in an \((n+1)\)-dimensional vector space \(V.\) If \(H_ 0\) is an open irreducible set of the Hilbert scheme, which parametrizes smooth irreducible subvarieties of X, let \({\mathcal Z}_ 0\to H_ 0\) be the corresponding universal family of subvarieties, let F: \({\mathcal Z}_ 0\to X\) be the natural map, and let \(Y=F({\mathcal Z}_ 0)\). The main result of the paper shows for a general member of the family \(H_ 0\) that \(text{codim}_ XY\geq m_ 0+m_ 1+...+m_ k-n-1,\) where \(m_ 0\) is the least integer such that \(H^ 0(K_ Z\otimes {\mathcal O}_ Z(m_ 0))\neq 0\), and that \(N_{Z/X}\otimes {\mathcal O}_ Z(1)\) is generated by its sections. A version of the argument establishes the following theorem:
If \(H_ 0\) parametrizes smooth curves of degree \(d\) and genus \(g\) in X, and C is a general curve of \(H_ 0\) such that C spans an \(n_ 0\)- dimensional space in \(P(\bigwedge^ rV)\), then \(co\dim_ XY\geq (2- 2g+(m-n-1)d)/(d-n_ 0+1).\) This generalizes a result of \textit{H. Clemens} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 629-636 (1986) and 20, 281 (1987; Zbl 0611.14024 and Zbl 0625.14012)] which treats the case of curves on generic hypersurfaces. Koszul resolution; Hilbert scheme of a generic complete intersection; Grassmann variety Ein, L., \textit{subvarieties of generic complete intersections}, Invent. Math., 94, 163-169, (1988) Parametrization (Chow and Hilbert schemes), Complete intersections, Grassmannians, Schubert varieties, flag manifolds Subvarieties of generic complete intersections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Associated to a toric variety \(X\) of dimension \(r\) over a field \(k\) is a fan \(\Delta\) on \(\mathbb{R}^r\). The fan \(\Delta\) is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on \(X\). The fan \(\Delta\) inherits the Zariski topology from \(X\). In this article some cohomological invariants of \(X\) are studied in terms of whether or not they depend only on \(\Delta\) and not \(k\). Secondly some numerical invariants of \(X\) are studied in terms of whether or not they are topological invariants of the fan \(\Delta\). That is, whether or not they depend only on the finite topological space defined on \(\Delta\). The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the étale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan all of whose maximal cones are nonsimplicial. toric variety; fan; torus action; Weil divisors; Picard group; Brauer group; étale cohomology T. J. Ford, Topological invariants of a fan associated to a toric variety, Comm. Algebra 23 (1995), 4031--4045. Toric varieties, Newton polyhedra, Okounkov bodies, Étale and other Grothendieck topologies and (co)homologies, Brauer groups of schemes, Picard groups, Divisors, linear systems, invertible sheaves Topological invariants of a fan associated to a toric 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 X be a normal algebraic variety over an algebraically closed field of characteristic zero and G a reductive connected algebraic group acting on X. X is called a spheric variety if there is a Borel subgroup of G with a dense orbit in X. This notion generalizes that of symmetric varieties [see \textit{C. De Concini} and \textit{C. Procesi} in Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1-44 (1983; Zbl 0581.14041)].
The paper under review describes the Picard group of spherical varieties, as well as the intersection numbers of their divisors. As customary, some applications to classical examples are given, and so, characteristic numbers for linear spaces, flags and plane and space conics are, once again, computed. spheric variety; Borel subgroup; dense orbit; symmetric varieties; Picard group of spherical varieties; intersection numbers; characteristic numbers Brion, M., Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math. J., 58, 397-424, (1989) Enumerative problems (combinatorial problems) in algebraic geometry, Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Picard groups Groupe de Picard et nombres caractéristiques des variétés sphériques. (Picard group and characteristic numbers of spheric 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 Consider a polynomial \(f\) in two variables with complex coefficients, of degree \(d\). Let \((O_1, \ldots, O_N)\) be the base points of the pencil given by \(C_\lambda:\alpha F + \beta X_0^d\), where \(F\) is the homogenization of \(f\) and \(\lambda =\beta / \alpha\). Furthermore consider the critical values of \(f\) at infinity, that is, those \(\lambda \) for which the family \((C_\lambda, O_i)\) is not topologically trivial for some \(i\). At one of the base points, say \(O\), one defines a weighted cluster \(( K, \nu)\) of base points. The authors give a bound for the number of critical values of \(f\) in terms of the tree structure of \(K\), the virtual multiplicities \(\nu_p\), the proximity relations between points of \(K\), and the number of points of \(K\) which lie on the line at infinity. base points of a pencil; polar curve; weighted cluster; number of critical values Cassas-Alvero, E.; Peraire, R.: A bound for the number of critical values at infinity. J. pure appl. Algebra 149, 35-47 (1999) Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves A bound for the number of critical values at infinity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(R = k[T_ 1, T_ 2, \dots, T_{n + 2}]\) be the ring of polynomials in \(n + 2\) variables over an algebraically closed field \(k\) and \(I\) an ideal of \(R\). If \(R[It]= \bigoplus_{i \geq 0} I^ i t^ i\) is the Rees algebra of \(I\) and \(M = (T_ 1, T_ 2, \dots, T_{n + 2})\) is the maximal ideal of the origin point in \(\mathbb{A}^{n + 2}\) then \(R[It]/MR [It]\) is the special fibre of the blowing-up of \(\mathbb{A}^{n + 2}\) along the subvariety defined by \(I\). The analytic spread of \(I\) is the dimension \(l(I)\) of the special fibre. It is known that \(l(I) \geq \text{ht} (I)\). In this paper it is proved the following result: If \(I\) is the ideal of a toric variety of codimension \(2\), \(T \subset \mathbb{A}^{n + 2}\), then \(l(I) = 2\) if \(T\) is a complete intersection and \(l(I) = 3\) in other cases. The method used by the authors consists in using the combinatorial properties of the block-arithmetic matrices (APB), introduced by \textit{M. Morales}. Then a result of M. Morales concerning the characterisation of ideals of toric varieties asserts that \(I\) is generated by the \(2 \times 2\)-minors of an APB matrix. In such way, the Plücker relations give the possibility to obtain a new presentation of the Rees algebra \(R[It]\).
As usually, the articles published in C. R. Acad. Sci., Paris, do not give much details concerning proves of the announced results. For this article, two subsequent papers of \textit{Ph. Gimenez} and \textit{M. Morales} are announced. Rees algebra; analytic spread; ideal of a toric variety; block-arithmetic matrices Gimenez, Ph.; Morales, M.; Simis, A.: L'analytic spread de l'idéal de définition d'une variété monomiale de codimension deux est toujours inférieur ou égal à trois. C. R. Acad. sci. Paris 319, 703-706 (1994) Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Linkage, complete intersections and determinantal ideals, Relevant commutative algebra, Toric varieties, Newton polyhedra, Okounkov bodies The analytic spread of the ideal of definition of a codimension 2 monomial variety is always less or equal to 3 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an integral projective non-hyperelliptic curve of arithmetic genus \(g:=p_a(X)\geq 3\) and \(f:Y\to X\) its normalization. A point \(P\in Y\) will be called a pseudoweierstrass point of \(Y\) if it is a ramification point for the linear system \(|V|:=\mathbb P(f^* (H^0(X,\omega_X)^*)\) on \(X\). Theorem. Assume either \(\text{char} (\mathbb{K})=0\) or \(\text{char}(\mathbb{K})>2g-4\). Let \(X\) be an integral non-Gorenstein projective curve with \(g:=p_a(X)\geq 3\). Assume that \(X\) has at least one pseudoweierstrass point, i.e., assume that Rosenlicht's canonical model of \(X\) is not a rational normal curve. Assume \(X\) not hyperelliptic, \(\#(\text{Sing}(X))=1\), \(X\) unibranch at its unique singular point. Then \(X\) has a smooth pseudoweierstrass point. Weierstrass point; canonical model of a singular curve Singularities of curves, local rings, Riemann surfaces; Weierstrass points; gap sequences On the existence of smooth pseudoweierstrass points on a non-Gorenstein curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We give an overview on various constructions of algebraic minimal surfaces in Euclidean three-space. Especially, low degree examples shall be studied. For that purpose, we use the different representations given by Weierstrass including the so-called Björling formula. An odd result by Lie dealing with the evolution of space curves can also be used to construct minimal surfaces with rational parametrizations. We describe a one-parameter family of rational minimal surfaces which touch orthogonal hyperbolic paraboloids along their curves of constant Gaussian curvature. Furthermore, we find a new class of algebraic and even rationally parametrizable minimal surfaces and call them cycloidal minimal surfaces. minimal surface; algebraic surface; rational parametrization; polynomial parametrization; meromorphic function; isotropic curve; Weierstraß representation; Björling formula; evolute of a space curve; curve of constant slope Odehnal, B., On algebraic minimal surfaces, KoG, 20, 61-78, (2017) Minimal surfaces in differential geometry, surfaces with prescribed mean curvature, Differential geometry of immersions (minimal, prescribed curvature, tight, etc.), Minimal surfaces and optimization, Rational and ruled surfaces On algebraic minimal 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 Let \(Z\subset X\subset\mathbb P^n\) be real projective schemes with \(X\) integral and \(Z\) of co dimension at least 2 in \(X\). Here we find a real hypersurface of \(X\) containing \(Z\), with bounded degree and with other properties. real algebraic variety; real divisor; degree of a hypersurface Real algebraic sets, Nash functions and manifolds Real divisors of a projective variety containing a given scheme | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 III of this paper see ibid. 313-332 (1994; see the preceding review).]
This fourth part of the authors' series makes essential use of the algorithms introduced in the first two points [part I: Contemp. Math. 78, 425-555 (1988; Zbl 0674.14019); part II: Lect. Notes Math. 1479, 131-179 (1989; Zbl 0764.14014)] together with the projective degenerations of the Veronese surface \(V_ 3\) constructed in the third part (loc. cit.) to compute the braid monodromy of \(S_ 3\), the branch locus of a generic projection \(V_ 3 \to \mathbb{P}^ 2\). These computations are needed to study \(\pi_ 1 (\mathbb{C} \mathbb{P}^ 2 - S_ 3)\). fundamental group of complement of a plane curve; projective degenerations of the Veronese surface; braid monodromy B Moishezon, M Teicher, Braid group techniques in complex geometry IV: Braid monodromy of the branch curve \(S_3\) of \(V_3{\rightarrow}\mathbbC\mathrmP^2\) and application to \(\pi_1(\mathbbC\mathrmP^2-S_3,*)\), Contemp. Math. 162, Amer. Math. Soc. (1994) 333 Families, moduli, classification: algebraic theory, Braid groups; Artin groups, Projective techniques in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Braid group techniques in complex geometry. IV: Braid monodromy of the branch curve \(S_ 3\) of \(V_ 3\to\mathbb{C}\mathbb{P}^ 2\) and application to \(\pi_ 1(\mathbb{C}\mathbb{P}^ 2-S_ 3,\ast)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an affine variety over a field \(K\) and let \(G\) be a reductive group of automorphisms of \(X\). A categorical quotient of a variety is a pair \((\pi,Y)\), where \(Y= \text{Spec}(K[X]^G)\) is an invariant affine space and \(\pi\) is a morphism dual to the embedding \(K[X]^G\rightarrow K[X]\). In case when the field \(K\) is of zero characteristic, the structure of the quotient variety \(Y\) is well known. In case the field \(K\) is of nonzero characteristic, the question concerning the properties of this variety is open.
The article is devoted to studying the quotient of the variety of \(m\)-tuples of matrices \(X=M(n)^m\) with the natural action of some reductive subgroups of the general linear group \(\text{GL}(n)\) on this variety. For the field \(K\) of zero characteristic and \(H=\text{Sp}(n)\) or \(H=\text{SO}(n)\), the quotient was calculated by \textit{C.~Procesi} [Adv. Math. 19, 306-381 (1976; Zbl 0331.15021)].
The author gives a description of the invariant algebra \(K[M(n)^m]^H\) for an arbitrary infinite field \(K\) and \(H=\text{Sp}(n)\) or \(H=\text{O}(n)\). group of automorphisms; categorical quotient of a variety; invariant of a linear group; variety of \(m\)-typles Zubkov, A.N.: Invariants of an adjoint action of classical groups. Algebra Logic 38(5), 299--318 (1999) Geometric invariant theory, Actions of groups on commutative rings; invariant theory, Group actions on varieties or schemes (quotients), Classical groups (algebro-geometric aspects), Linear algebraic groups over arbitrary fields Invariants of an adjoint action of classical 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 Hurwitz numbers count the number of connected coverings of the Riemann sphere of given genus and degree, with prescribed ramification over \(\infty\) and simple branching at all other branch points. Their study goes back to \textit{A. Hurwitz} [Math. Ann. XXXIX. 1--61 (1891; JFM 23.0429.01)]. The most important advance in the recent study of Hurwitz numbers has been the so-called ELSV formula (proved in [\textit{T. Ekedahl, S. Lando, M. Shapiro} and \textit{A. Vainstein}, Invent. Math. 146, No. 2, 297--327 (2001; Zbl 1073.14041)]), which expresses Hurwitz numbers as integrals of tautological classes over the moduli space of stable curves with marked points.
The paper under review goes a step further and deals with connected coverings with prescribed ramification over two points. Specifically, consider all possible connected coverings of genus \(g\) and degree \(d\) of the Riemann sphere, with fixed branch locus containing \(0\) and \(\infty\). Suppose that the branching over \(0\) and \(\infty\) is specified, respectively, by a partition \(\alpha\) and a partition \(\beta\) of \(d\), and that the branching at all other branch points is simple. Then the authors define the double Hurwitz number \(H^g_{\alpha,\beta}\) as the number of such covers, together with a labelling of the branch points over \(0\) and \(\infty\) (hence, the integer \(H^g_{\alpha,\beta}\) is always a multiple of \(| \text{Aut}(\alpha)| | \text{Aut}(\beta)| \)). The aim of this paper is to investigate the structure of double Hurwitz numbers. In doing this, the authors are guided by the intuition that they should be related by an analogue of the ELSV formula to the intersection theory on a suitable compactification of the universal Picard variety, i.e., a moduli space of curves of genus \(g\) with \(n\) marked points and a given line bundle. This compactified Picard variety is expected to be related to the one constructed in [\textit{L. Caporaso}, J. Am. Math. Soc. 7, No. 3, 589--660 (1994; Zbl 0827.14014)]. Moreover, the authors announce that in a later paper they will exploit the structure of genus 0 double Hurwitz numbers to understand top intersections on the moduli space of smooth curves and to prove the intersection number conjecture in [\textit{C. Faber}, in: Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Aspects Math. E33, 109--129 (1999; Zbl 0978.14029)], for every genus and for up to three marked points.
If we fix the length \(m\) (respectively, \(n\)) of \(\alpha\), resp., \(\beta\), and we write \(\alpha=(\alpha_1,\dots,\alpha_m)\), \(\beta=(\beta_1,\dots,\beta_n)\), the double Hurwitz numbers \(H^g_{\alpha,\beta}\) define a function in the \(\alpha_i\), \(\beta_j\). Using ribbon graphs, the authors prove that the functions so obtained are piecewise polynomial of degree \(\leq 4g-3+m+n\). The proof is based on an interpretation of double Hurwitz numbers as the number of lattice points of certain polytopes, such that the position of the faces depends on \(\alpha\) and \(\beta\). In the special case \(g=0\), the authors prove that the degree of the polynomials is bounded below by \(2g-3+m+n\); they conjecture that this should hold also for positive genus. By considering the example \((g,m,n)=(0,2,2)\), the authors show that double Hurwitz numbers are not polynomial in general.
The central part of the paper deals with the case of covers with complete ramification over \(0\), i.e., double Hurwitz numbers of the form \(H^g_{(d),\beta}\), called here one-part double Hurwitz numbers. Using techniques from algebra, geometry and representation theory, the authors give explicit formulas (Theorem 3.1) for these Hurwitz numbers, which imply, among other things, that they are polynomials in the parts of the partition \(\beta\). Furthermore, the authors conjecture the form of an analogue of the ELSV formula for one-part Hurwitz numbers, expressing them as integrals of some (undefined as yet) ``geometrically natural classes'' on a suitable compactification of the Picard variety. The proprieties which the compactified Picard variety and the classes of it should satisfy are listed. For genus \(0\) and genus \(1\) the conjecture is proved, by choosing the compactified Picard variety to be, respectively, \(\overline{{\mathcal M}}_{0,n}\) and \({\overline{\mathcal M}_{1,n+1}}\). The authors also define a symbol \(\langle\langle \cdot \rangle\rangle_g\), which, in view of the conjecture, is expected to be the analogue of Witten's correlation function \(\langle\cdot\rangle_g\). Indeed, the symbol \(\langle\langle\cdot\rangle\rangle_g\) satisfies properties similar to those of \(\langle\cdot\rangle_g\). In particular, the authors prove that the generating series of \(\langle\langle\cdot\rangle\rangle_g\) satisfies the string and dilaton equation, and an analogue of \textit{Itzykson-Zuber}'s ansatz for intersection numbers on the moduli space of curves ((5.32) in [Int. J. Mod. Phys. A 7, No. 23, 5661--5705 (1992; Zbl 0972.14500)]). Moreover, they give analogues of the formulas in [\textit{C. Faber, R. Pandharipande}, Ann. Math. (2) 157, No. 1, 97--124 (2003; Zbl 1058.14046)].
Next, the authors investigate the generating function of double Hurwitz numbers and express it in terms of Schur symmetric functions. As a corollary, one gets formulas for \(H^g_{\alpha,\beta}\) as a function of \(g\), when \(\alpha\) and \(\beta\) are fixed. This extends results of Kuleshov and Shapiro for covers of degree \(3,4,5\). Finally, when the length \(m\) of the partition \(\alpha\) is fixed, the authors define a symmetrized generating function for these numbers, and prove that it satisfies a topological recursion relation. This leads to closed expressions for the generating function for small \((g,m)\). Hurwitz numbers; moduli of curves; Picard variety I. P. Goulden, D. M. Jackson, and R. Vakil, ''Towards the Geometry of Double Hurwitz Numbers,'' Adv. Math. 198(1), 43--92 (2005). Families, moduli of curves (algebraic), Symmetric functions and generalizations, Picard schemes, higher Jacobians, Coverings of curves, fundamental group, Enumerative problems (combinatorial problems) in algebraic geometry Towards the geometry of double Hurwitz numbers | 0 |
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