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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 \(C\) be a complete non-singular curve over \(\mathbb{C}\) of genus \(g\). We denote by \(W_d^r(C)\) the subscheme of the Picard variety \(\text{Pic}^d(C)\) whose support is the locus of complete linear series of degree \(d\) and dimension at least \(r\).
In case \(d>g+r-1\), \(W^r_d(C)=\text{Pic}^d(C)\) and if \(d=g+r-1\), \(W^r_d(C)\) has dimension \(d\). Therefore the dimension of \(W_d^r(C)\) is independent of \(C\) in the range \(d\geq g+r-1\). If \(d\leq g+r-2\), one knows that \(\dim W^r_d(C)\geq\rho (d,g,r):=g- (r+1)(g-d+r)\) for any curve \(C\) and is equal to \(\rho(d,g,r)\) for general curve \(C\). But the dimension of \(W_d^r(C)\) might be greater than \(\rho(d,g,r)\) for some special curve \(C\). Moreover, for curves \(C\) with \(\dim W^r_d(C)> \rho (d,g,r)\), \(C\) must be of some special type of curves. The results along this line can be summarized in the following form: Theorem C [Ballico, Coppens, G. Martens, Keem, Mukai and Ohbuchi] Let \(d\) and \(r\) be integers such that \(d\leq g+r-4\), \(r\geq 1\). If \(\dim W^r_d(C)\geq d-2r-\geq 0\), then \(C\) is either hyperelliptic, trigonal, bi-elliptic, tetragonal, a smooth plane sextic or a double covering of a curve of genus 2.
The numerical condition \(d\leq g+r-4\) in theorem \(C\) implies the condition \(d-2r-1> \rho(d,g,r)\) but not conversely. In fact \(d-2r-1>\rho(d,g,r)\) holds if and only if \(d\leq g+r-4\), or \((d,r)=(g-1,2)\) for which case theorem C fails to provide any information on the curve \(C\). In classifying smooth curves \(C\) with \(\dim W^2_{g-1}(C)=g-7\), one eventually ends up with curves of low gonality with some extra conditions. The main purpose of this short note is to estimate the dimension of \(W^2_{g-1} (C)\), especially when \(C\) is a tetragonal curve which is simultaneously a double covering of smooth plane quartic of genus \(g\).. Picard variety; tetragonal curve Special divisors on curves (gonality, Brill-Noether theory), Plane and space curves, Pencils, nets, webs in algebraic geometry, Coverings of curves, fundamental group Variety of special nets of degree \(g-1\) on double coverings of a smooth plane quartic of genus 9. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 studies a certain relation between coverings of a curve over a finite field of characteristic \(p>0\) and those of a system of curves over an algebraic closure of \({\mathbb{Q}}_ p\). This extends previous important work of Y. Ihara. p-adic field; liftings; coverings of a curve over a finite field Arithmetic ground fields for curves, Coverings of curves, fundamental group, Finite ground fields in algebraic geometry, Arithmetic theory of algebraic function fields, Local ground fields in algebraic geometry On the ramified congruence relations 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 [For the entire collection see Zbl 0527.00015.]
In this work a new method of construction of a real plane algebraical curve with prescribed topology is introduced. By this method the constructions of curves of degree 7 are made. The new method is based on a construction that builds a new algebraic curve from several other ones. The new curve is arranged as a result of gluing of the initial curves. The construction can be interpreted as a perturbation of a curve with complicated singularities. While the classical methods of construction of curves consists in small perturbation of a singular curve having only nondegenerate singularities. The author describes a new simple construction of curves of degree 6. It is proved that a nonsingular curve of degree 6 with any possible mutual position of its ovals can be obtained by a small perturbation of the union of three ellipses tangent to one another in two points. The construction of curves of degree 6 and 7 is based on small perturbations of singular points of type \(J_{10}\). Hilbert sixteenth problem; construction of a real plane algebraical curve with prescribed; topology; small perturbations of singular points; construction of a real plane algebraical curve with prescribed topology Viro, O. Ya., \textit{gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7}, Proc. int. conf. on topology, general and algebraic topology, and applications, Leningrad, 1982, 187-200, (1984), Springer, Berlin Special algebraic curves and curves of low genus, Real algebraic and real-analytic geometry, Topological properties in algebraic geometry Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Monomial curves in r-dimensional projective space \({\mathbb{P}}^ r_ k\) we shall call curves with generic zero \((t_ 0^{n_ r},t_ 0^{n_ r- n_ 1}t_ 1^{n_ 1},...,t_ 0^{n_ 1-n_{r-1}}t_ 1^{n_{r- 1}},t_ 1^{n_ r})\), where \(0<n_ 1<....<n_ r\) are integers with \(g.c.d.(n_ 1,...,n_ r)=1\). In this paper it is investigated another property of monomial curves namely the canonical module of a monomial curve. In particular it is described the canonical module for monomial curves in general and explicitly it is constructed a minimal generating set for the canonical module of space curves.
In the appendix there are proved some numerical criteria for the Cohen- Macaulayness and for the Gorenstein property of monomial space curves. canonical module of a monomial curve; minimal generating set; Cohen- Macaulayness; Gorenstein property DOI: 10.1080/00927878708823505 Special algebraic curves and curves of low genus, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the canonical module for monomial curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let R be a reduced Noetherian ring and let A,B,C be finitely generated torsionfree R-modules. This paper discusses conditions on the ring and the modules in order that, from \(A\oplus C\cong B\oplus C,\) one can conclude that \(A\cong B.\) When this is the case one says that torsionfree cancellation holds. It is shown that torsionfree cancellation holds when R is a regular integral domain of Krull dimension 2, finitely generated over an algebraically closed field of characteristic 0. Most of the paper concerns one-dimensional rings with finitely generated integral closure. It is shown that torsionfree cancellation fails when R is the coordinate ring of a singular affine curve over an algebraically closed field. The paper contains some general results that allow one to decide, for many quadratic orders, whether or not torsionfree cancellation holds; and there are algorithms for handling the orders not covered by these theorems. The last short section discusses integral group rings of finite abelian groups, and anwers the cancellation question for all abelian groups except the cyclic groups of orders 8 and 9. direct sum cancellation; quadratic order; Noetherian ring; torsionfree cancellation; regular integral domain; coordinate ring of a singular affine curve; quadratic orders; integral group rings DOI: 10.1016/0021-8693(84)90077-2 Structure, classification theorems for modules and ideals in commutative rings, Relevant commutative algebra, Quadratic extensions, Valuations, completions, formal power series and related constructions (associative rings and algebras), Integral domains, Group rings, Commutative Noetherian rings and modules Cancellation over commutative rings of dimension one and two | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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, on classical algebraic geometry, appears as a list of definitions, theorems, proofs, results, without any explanation. It is not written the introdution, the purpose, the motivations, the main results of it. In reading the paper the impression of the reviewer is that to open a new book and to read from page 205 to page 223. From the bibliography one can understand that this paper is neither the first and, maybe, nor the last one of the series, but for a single paper it is better to have the begin and the end.
Passing now to review the content of the paper, the reviewer finds nice arguments and interesting results about characterizations of the rank of the Jacobian matrix of a linear system of quadrics (hypersurfaces) in the projective space \({\mathbb{P}}^ r\). It is not possible here to give a precise idea of the work, anyhow the results are of the following type. We call \(L_{d_ 1},L_{d_ 2},...,L_{d_ s}^ a \)chain of linear systems of quadrics if they are not reducible and if \(L_{d_ 1}\) and \(L_{d_ 2}\) have in common at least a quadric, their union system \(L_ a\) has in common with \(L_{d_ 3}\) at least a quadric and the system \(L_ b\) union of \(L_ a\) and \(L_{d_ 3}\) has in common with \(L_{d_ 4}\) at least a quadric, and so on. Then a linear system of quadrics \(L_ d\) containing a chain of linear systems of ''first kind'' and no other quadric linearly independent, has the rank of the Jacobian matrix equal to r-k\(\leq d\) (k\(\geq 0)\) if and only if the quadrics of \(L_ d\), passing through a generic point of \({\mathbb{P}}^ r\), have in common a linear space \(S_{k+1}\). rank of the Jacobian matrix of a linear system of quadrics L. Degoli, Sur la caractéristique de la Jacobienne des systèmes linéaires de quadriques,Czechoslovak Math. J. 36 (1986), 476--484.Zbl 621: 14006 Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry Sur la caractéristique de la Jacobienne des systèmes linéaires de quadriques. (On the characteristic of the Jacobian of linear systems of quadrics) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We prove that the Grothendieck standard conjecture of Lefschetz type holds for a complex projective 3-dimensional variety fibred by curves (possibly with degeneracies) over a smooth projective surface provided that the endomorphism ring of the Jacobian variety of some smooth fibre coincides with the ring of integers and the corresponding Kodaira-Spencer map has rank 1 on some non-empty open subset of the surface. When the generic fibre of the structure morphism is of genus 2, the condition on the endomorphisms of the Jacobian may be omitted. Grothendieck standard conjecture of Lefschetz type; Kodaira-Spencer map; Jacobian variety Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects), Classical real and complex (co)homology in algebraic geometry, \(3\)-folds On the standard conjecture for a 3-dimensional variety fibred by curves with a non-injective Kodaira-Spencer 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 [For the entire collection see Zbl 0632.00004.]
Let S be the spectrum of a strictly local discrete valuation ring with algebraically closed residue field of characteristic p and let s be the closed point of S, let \(\eta\) (resp. \({\bar\eta}\)) be the (resp. geometric) generic point of S, let I be the inertia group of S, P be the wild ramification group of S and let \(\Lambda:={\mathbb{Q}}_{\ell}\) (where \(\ell \neq p\) is a prime number). A flat and separated S-scheme of finite type purely of relative dimension 1 will be called an S-curve. In the first paper under review the author studies the action of I and P. At first he gives a new proof of the stable reduction theorem of Deligne- Mumford and Grothendieck, using the theory of vanishing cycles and gives also a local version (these two results take care of the action of I). The action of P is described in the following result:
Suppose \(X_{\eta}\) is a proper smooth geometrically connected curve over \(\eta\) of genus \(\neq 1\), and X is a relative minimal regular normal crossing divisor model of \(X_{\eta}\). Then the action of P on \(H^ 1(X_{\eta},\Lambda)\) is trivial iff every irreducible component C of \(X_ s\) whose multiplicity in \(X_ s\) is divisible by p satisfies the following condition: C is isomorphic to \({\mathbb{P}}^ 1_ s\) and intersects with each other component of \(X_ s\) at exactly two points and these components have prime-to-p multiplicities in \(X_ s\). There is also a local version of this theorem.
In the second paper under review the author starts with the following conjecture: Suppose X is an S-scheme (not necessarily of relative dimension 1) and Z is a subscheme of \(X_ s\) such that Z is proper over s and X-Z is smooth over S. Then \(\dim tot(R\Gamma (Z,R\phi \Lambda))=- \)length\(_{{\mathcal O}_ S}R\Gamma (Z,\Omega^{\bullet}_{X/S,tors})\). (Here \(R\phi\Lambda\) is the complex of sheaves of the vanishing cycles, and dimtot is \(\dim_{\Lambda}+Sw\) where Sw is the Swan conductor.) This conjecture generalizes a conjecture of \textit{P. Deligne} [see Sémin. Géom. algébrique 1967-1969, SGA 7 II, Lect. Notes Math. 340, Exp. XVI, 197-211 (1973; Zbl 0266.14006)], which treats the case that Z consists of isolated singularities. The author mentions the cases hitherto known and proves the conjecture - in case the relative dimension of \(X/S\) is 1 - in some special cases [e.g. in the case that \(P\) acts trivially on \(R\Gamma(Z,R\psi\Lambda)\) \([R\psi\Lambda\): nearby cycles]. To prove the conjecture in the special cases considered the author introduces a new differential which he calls the relative canonical differential and derives a number of properties of it which suffice to give a proof of his results.
The aside results of this paper are too numerous to be mentioned.
Reviewer's remark: The author remarks that \textit{S. Bloch}, in Algebraic Geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, Part 2, Proc. Symp. Pure Math. 46, No.2, 421-450 (1987; Zbl 0654.14004) has given a formula which is equivalent to the above mentioned conjecture [the author; ``Self-intersection 0-cycles and coherent sheaves on arithmetic schemes'', Preprint (Univ. Tokyo 1987); see also Duke Math. J. 57, No.2, 555-578 (1988)]. stable reduction theorem; action of inertia group; action of ramification group; vanishing cycles; geometrically connected curve; relative canonical differential Saito, T.: Vanishing cycles and differentials of curves over a discrete valuation ring. Adv. stud. Pure math. 12 (1987) Arithmetic ground fields for curves, \(p\)-adic cohomology, crystalline cohomology Vanishing cycles and differentials of curves over a discrete valuation 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 Being able to explicitly determine a basis of the Riemann-Roch space associated with a divisor \(D\) on a curve \({\mathcal C}\) is extremely important for constructing and decoding algebraic geometric codes. Let \(G\) be a divisor on a curve \(C\) such that the dimension of the associated Riemann-Roch space \(L(G)\) is at least \(1\). In the first part of the paper under review, the authors introduce the notion of the floor of \(G\) as a divisor \(G'\) of minimum degree such that \(L(G)=L(G')\) and provide an algorithm to compute such divisor starting from \(G\). Then, the divisor \(G'\) is used to improve the estimate on the minimum distance of any algebraic-geometric code \(C(G,D)\), by replacing \(d\geq n-\deg G\) with \(d\geq n-\deg G'\).
The second part of the paper is focused on the hermitian function field with defining equation \(H:y^q+y=x^{q+1}\) and on determining explicit bases for large classes of Riemann--Roch spaces of this field. In particular, the authors investigate divisors of the form \( G=rQ_{\infty}+\sum_{\beta\in K_{\alpha}} k_{\beta}P_{\alpha,\beta}, \) where \(Q_{\infty}\) is the common pole of the functions \(x\) and \(y\); \(r\) and \(k_{\beta}\) are integers; \(K_{\alpha}=\{\beta:\beta^q+\beta=\alpha^{q+1}\}\) and \(P_{\alpha,\beta}\) is the unique place of \(H\) of degree \(1\) with \(x(P_{\alpha,\beta})=\alpha\) and \(y(P_{\alpha,\beta})=\beta\). In the final part of the paper, the results obtained about the hermitian function field are used in order to show how construct with a fast implementation low--discrepancy sequences. divisors; floor of a divisor Maharaj, H; Matthews, GL; Pirsic, G, Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences, J. Pure Appl. Algebra, 195, 261-280, (2005) Algebraic functions and function fields in algebraic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Well-distributed sequences and other variations, Applications to coding theory and cryptography of arithmetic geometry Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(C=\text{Spec} R\) be an affine reduced curve over a field \(k\). In this paper we want to compute the conductor \(I\) of \(R\) in its normalization \(\overline R\). In particular if \(C\) has only ordinary generic singularities we show that there is an algorithm to carry out this computation. If \(k=\mathbb{Z}_ p\) this algorithm can be easily implemented on personal computers using common languages like Basic, C, APL. This has been done by the second author. If \(k=\mathbb{Q}\), for the same goal, one can use computer algebra systems like Macsyma, Reduce, Maple. These programs allow to construct computations of the conductor over any field \(k\) (recall that any field contains \(\mathbb{Z}_ p\) or \(\mathbb{Q})\). In the course of the paper we extend also various theoretical results of the first named author [J. Lond. Math. Soc., II. Ser. 24, 85-96 (1981; Zbl 0492.14017)] and both authors [Manuscr. Math. 68, No. 1, 1-7 (1990; Zbl 0709.13010)]. computing the conductor of a reduced curve Computational aspects of algebraic curves, Curves in algebraic geometry On the computation of the conductor of an affine algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The book under review consists of two parts, in addition to the preface, introduction, appendices, a list of literature consisting of 10 pages, and an index. It should be noted that the bibliography includes 24 references to the original works of the authors and their collaborators (see, e.g., [\textit{D. Mond}, in: Singularities and foliations. Geometry, topology and applications. BMMS 2/NBMS 3, Salvador, Brazil, 2015. Proceedings of the 3rd singularity theory meeting, ENSINO, July 8--11, 2015 and the Brazil-Mexico 2nd meeting of singularities, July 13--17, 2015. Cham: Springer. 229--258 (2018; Zbl 1405.32048); \textit{J. J. Nuño-Ballesteros} et al., Collect. Math. 69, No. 1, 65--81 (2018; Zbl 1393.32014)]).
In the preface, the authors emphasize that the book ``is a monograph and not a textbook -- its shape reflects the subject, or rather our knowledge of it, rather than the structure of a course ...''. However, they hope that it can be used as a basis for a graduate course. In addition, the authors proclaim that the subject of their book ``is not algebraic geometry, but smooth (\(\mathcal C^\infty\)) and complex analytic geometry though the gap is not all that wide''. Then they provide a brief background to the subject based on a series of classical papers by M. Morse, H. Whitney, R. Thom, J. Mather, V. I. Arnold and their followers. More precisely, namely, the first part of the book concentrates at these works; it contains a brief description of the basics of the Thom-Mather theory and its relation with many other topics and various applications. All of them are discussed in detail in seven sections, including the theory of left-right equivalence, stability, contact equivalence, versal unfoldings, finite determinacy, classification of stable germs by their local algebras, etc.
In the second part, consisting of four sections, the authors focus mainly on the complex case. This part includes recent research results; it is devoted to the study of topology of stable perturbations, stable images and discriminants, the theory of multiple points, some important properties of bifurcation sets, the theory of knots in the framework of classical studies by \textit{K. Reidemeister} [Abh. Math. Semin. Univ. Hamb. 5, 24--32 (1926; JFM 52.0579.01)], and so on.
Five appendices on background material include a detailed description of necessary concepts and results from the theory of jet spaces and bundles, the theory of stratifications, commutative algebra, local analytic geometry and the theory of sheaves.
The book is written in a clear pedagogical style; it contains many examples, exercises, comments, remarks, nice pictures, very useful instructive and systematic references, computational algorithms with implementation in the computer algebra software systems \textit{Macaulay 2} and \textit{Mathematica}, etc.
At the end of some sections, the authors listed a number of open questions. Without a doubt, the book is understandable, interesting and useful for graduate students and can serve as a good starting point for those who are interested in various aspects of both pure and applied mathematics.
The variety of topics covered makes this book also extremely valuable to researchers, lecturers, and practitioners working in the field of general theory of singularities and catastrophe theory, algebraic geometry, real and complex analysis, commutative algebra, topology, and other areas of contemporary mathematics. singularities of mappings; Thom-Mather theory; nice dimensions; right-left equivalence; contact equivalence; stability; versal unfoldings; finite determinacy; vector fields and flows; local conical structure; Thom-Boardman singularities; topological stability; unstable map-germs; unipotent algebraic groups; critical space; discriminants; bifurcation sets; isosingular locus; logarithmic tangent space; logarithmic transversality; stable perturbations; disentanglement of a map; image Milnor numbers; discriminant Milnor numbers; free and almost free divisors; complete intersections; Fitting ideals; conductor ideals; multiple point spaces; knot theory; Reidemeister moves; rank condition; parameterised hypersurfaces; maximal Cohen-Macaulay modules; duality; Gorenstein rings; canonical module; triple points Research exposition (monographs, survey articles) pertaining to global analysis, Topological properties of mappings on manifolds, Classification; finite determinacy of map germs, Critical points of functions and mappings on manifolds, Deformation of singularities, Symmetries, equivariance on manifolds, Stability theory for manifolds, Local complex singularities, Complex surface and hypersurface singularities, Milnor fibration; relations with knot theory, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Knot theory, Singularities in algebraic geometry Singularities of mappings. The local behaviour of smooth and complex analytic mappings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a report on important recent results on resolution of singularities of algebraic varieties and the theory of semi-stable reduction, found in the paper ``Smoothness, semi-stability and alterations'', by \textit{A. J. de Jong} [Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996; Zbl 0916.14005)]. The key notion is ``alteration'': An alteration of an integral noetherian scheme \(X\) is a proper, surjective morphism \(\varphi:X'\to X\) (with \(X'\) integral, noetherian) such that, for a suitable open dense set \(U\subseteq X\), the induced morphism \(\varphi_U:\varphi^{-1}(U)\to U\) is finite. For instance, and working (to simplify) with projective varieties over a field \(k\), the author proves: Given an integral variety \(X\) and a closed subvariety \(Z\subset X\), there is an alteration \(\pi:X'\to X\) with \(X'\) regular, such that \(\pi^{-1} (Z)_{\text{red}}\) is a strict normal crossings divisor (i.e., the irreducible components are smooth and meet transversally). Note that there are no restrictions on the base field \(k\).
This is the method of proof: Obtain an alteration \(f:X_1\to X\) such that there is flat morphism \(g:X_1\to T\), with \(T\) regular, \(\dim(T)= \dim(X_1)-1\), such that the general fiber of \(g\) is regular and any fiber is a curve with, at worst, ordinary double points as singularities, moreover the set of points of \(T\) where the fiber of \(g\) is singular is contained in a strict normal crossings divisor. Then, to desingularize such a \(X_1\) by means of monoidal transformations is easy. An alteration as above is obtained by using some classical projective techniques, the theory of moduli for pointed semi-stable curves and an induction hypothesis (applied to \(T\), whose dimension is one less than that of \(X)\).
A. J. de Jong also proved (loc. cit.) a weak version of the general semi-stable reduction theorem where (essentially) one allows to substitute one of the relevant varieties involved by an alteration thereof. These results (of course, precisely stated) are discussed in this report. There is an essentially complete proof of the desingularization theorem and good sketch of the one for the reduction problem. The report is an excellent introduction to these topics.
But there is also a very useful section (the last one) on applications. P. Berthelot discusses three:
(1) O. Gabber's affirmative solution to Serre's problem on multiplicity of intersection for two modules over a local noetherian ring \(A\): ``\(\chi_A(M,N)\geq 0\)'' (no restrictions on the ring);
(2) a proof (due to Berthelot) showing that the Monsky-Washnitzer cohomology groups \(H^n_{MW} (X/K)\) (where \(X\) is a smooth affine scheme over a field \(k\) with \(\text{ch} (k)>0\), \(K\) the fractions of a Cohen ring of \(k)\) are finite dimensional vector spaces over \(K\);
(3) some recent work of Deligne on monodromy actions on étale cohomology groups (specially, an ``independence of \(l\)'' theorem).
All these results use the desingularization theorem of de Jong. semistable curve; moduli of curves; resolution of singularities; alteration; integral variety; monoidal transformations; semi-stable reduction theorem; multiplicity of intersection for two modules; Monsky-Washnitzer cohomology groups; monodromy actions on étale cohomology Berthelot, P., Altérations de variétés algébriques (d'après A.J. de jong), Séminaire Bourbaki, vol. 1995/96, Astérisque, 241, 273-311, (1997), Exp. No. 815, 5 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Families, moduli of curves (algebraic), \(p\)-adic cohomology, crystalline cohomology, Étale and other Grothendieck topologies and (co)homologies Alterations of algebraic varieties (after A. J. de Jong) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 abelian variety over an algebraically closed field \(k\). The author considers the functor of germs of formal curves on \(X\); on \(S\)-points, this is \({\mathcal C}X(S) = \text{ Hom}(\text{Spf}k[\![t]\!] \times S, X)\). For any \(k\)-scheme \(Y\), let \(Y[n] = Y\times \text{ Spec}k[t]/t^n\). The main theorem states that \({\mathcal C}X\) is represented by \(\text{Pic}^0(\lim_{\rightarrow n}X[n])\). The proof relies on a characterization of \(\ker (\text{Pic}(Y[n]) \rightarrow \text{ Pic}(Y))\) for a proper scheme \(Y\) and on the identification of \(\text{Hom}(S[n], \text{Pic}(\hat X))\) with \(\text{Pic}(\hat X[n])(S)\). (In fact, the proof of Theorem 3.4 shows that \({\mathcal C}X\) is represented by \(\text{Pic}^0(\lim_{\rightarrow n}\hat X[n])\); there is a missing duality in the last paragraph on page 101.) abelian variety; Picard scheme; formal curve Picard schemes, higher Jacobians, Algebraic theory of abelian varieties The scheme of formal curves on an Abelian 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 This article can be considered as an appendix to \textit{A. Beilinson} and the author [Proc. Symp. Pure Math. 55, Pt. 2, 123-190 (1994; Zbl 0817.14014)]. There, a general theory of elliptic polylogarithms is given. Here, exact formulas for motivic elliptic polylogarithms, i.e., involving higher algebraic \(K\)-theory, are presented.
Let \(B\) be a connected scheme and consider an elliptic curve \(p:X\to B\) over \(B\), i.e., a flat family \(X\to B\) of relative dimension one with geometrical fibers of genus one, and a zero section \(0:B\to X\). Write \(U =X \backslash 0(B)\), \(p_U:U\to B\) for the restriction of \(p\) to \(U\), and \(j_U:U \hookrightarrow X\) for the open embedding. Also, let \(i_0:0(B) \hookrightarrow X\) be the closed embedding. \(p_1: U\times_BU\backslash \Delta\to U\) (where \(\Delta\) is the (relative) diagonal) is the projection on the first factor. \({\mathcal H}\) will denote the Hodge \((\ell\)-adic, motivic,\dots) sheaf \(R^1p_* (\mathbb{Q}(1))\) on \(B\). It can be considered as the sheaf of relative homologies. \(G^{ (1)}\) will denote the sheaf \(R^1p_{1*} (\mathbb{Q}(1))\) on \(U\). It can be extended to a sheaf, also denoted \(G^{(1)}\), on \(X\). One has a short exact sequence
\[
0\to p^*{\mathcal H}\to G^{(1)}\to \mathbb{Q} \to 0
\]
on \(X\), which splits over \(0(B)\). \(G^{(1)}\) comes equipped with a weight filtration, an action of \(\mathbb{Q}\) and a unipotent action of \({\mathcal H}\). \({\mathcal H}\) can be considered as the (abelian) fundamental group of \(X/B\). Write \(G^{(n)}\) for the symmetric product \(S^n (G^{(1)})= \text{Sym}^n(G^{(1)})\). The weight filtration on \(G^{(1)}\) induces one on \(G^{ (n)}\). Also, an element \(\ell\in{\mathcal H}\) acts as the exponential \(\exp(\ell)\) on \(G^{(n)}\). One defines the logarithmic sheaf \(G=\varprojlim(G^{(n)})\). It has a weight filtration with successive quotients \(Gr^W_{-i} (G)=W_{-i}/W_{-i-1}(G)\) equal to \(S^i(p^*{\mathcal H})\), \(i\geq 0\). The fundamental group \({\mathcal H}\) acts again on \(G\) by multiplication with the exponential. Let \(X^{(n)}= X\times_B\times \cdots \times_BX\) be the (relative) \(n\)-th power of \(X\). On \(X^{(n)}\) one can define a set of divisors \(D_i^{(n)}\), \(i=1, \dots,n+1\), and \(\Delta^{(n)}_{i,j}\), \(i,j=1,2, \dots, n+1\), \(i\neq j\). One defines \(U_0^{(n+1)}= X^{(n+1)} \left\backslash \bigcup^{n+1}_{i=1} D_i^{(n+1)} \right.\). Then there is a natural map \(\Sigma= \Sigma^{(n+1)}: U_0^{(n+1)} \to U\). Of particular importance are the varieties \(Y^{(n)}=U \times_BU_0^{(n+1)}\) with projection \(\pi: Y^{(n)} \to U\) defined by \(p_U\times \Sigma:U \times_BU_0^{(n+1)} \to B\times_BU =U\). The elliptic polylogarithm \({\mathcal P}\) is defined as the extension
\[
0\to j^*_UG(1) \to{\mathcal P} \to p^*_U {\mathcal H} \to 0
\]
determined by the map (cf. loc. cit.) \({\mathcal H} \to I= R^1p_{U*} (j^*_UG(1))\), where \(I\) is the augmentation ideal of the sheaf of symmetric algebras \(\sum S^j ({\mathcal H})\) of \({\mathcal H}\). Letting \({\mathcal P}_n= {\mathcal P}/W_{-n-3} ({\mathcal P})\) one has a short exact sequence
\[
0\to j^*_U G^{(n)} (1)\to {\mathcal P}_n \to p^*_U {\mathcal H}\to 0.
\]
Using \(R^{n+1} \pi_*(\mathbb{Q}(n+1)) =p^*_U {\mathcal H} \otimes j^*_U G^{(n)}\), a spectral sequence argument leads to the existence of a canonical map \(\alpha_n\): \(\text{Ext}^{n+2}_{Y^{(n)}} (\mathbb{Q},\mathbb{Q} (n+1))\to \text{Ext}^1_U (\mathbb{Q}, R^{n+1} \pi_*(\mathbb{Q} (n+1)))\). Let \({\mathcal P}^{(n)}\in \text{Ext}^1_U (p^*_U {\mathcal H} ,j^*_U G^{(n)} (1))= \text{Ext}^1_U (\mathbb{Q},p^*_U {\mathcal H} \otimes j^*_U G^{(n)})= \text{Ext}^1_U (\mathbb{Q}, R^{n+1} \pi_*(\mathbb{Q} (n+1)))\) be the element corresponding to \({\mathcal P}_n\). Then an element \({\mathcal P}_{\mathcal M}^{(n)} \in K_n(Y^{(n)})\) is constructed such that \(\alpha_n (r({\mathcal P}_{\mathcal M}^{(n)})) = {\mathcal P}^{(n)}\), where \(r\) is the regulator map from \(K\)-theory to the group of extensions \(\text{Ext}^*(\mathbb{Q}, \mathbb{Q}(*))\). In agreement with Beilinson's conjectures, the \({\mathcal P}^{(n)}_{\mathcal M}\) come from pairs \((Z_q^{(n)}, S_q^{(n)})\) of divisors \(Z^{(n)}_q\) on \(Y^{ (n)}\) together with elements \(S_q^{(n)}\) of Milnor's \(K_{n,\mathbb{Q}}\)-groups.
This program is realized for \(B=\text{Spec}(k)\), \(k\) a field. First, using Tate's normal form of the curve \(X \to \text{Spec}(k)\), \(n\)-th order elliptic Vandermonde functions \(W_n\) on \(X^{(n)}\) are constructed as well as their \(i\)-th partial derivatives \(W_{n;i}'\). The divisors of \(W_n\) and \(W_{n;i}'\) can be expressed in terms of the \(D_i^{(n)}\) and the \(\Delta_{i,j}^{(n)}\). One also defines functions \(F_i^{(n)}= W_{n;i}'/W_n\), \(i=1,2, \dots,n\), and \(F_{n+1}^{(n)} =(W_n)^{-2} \prod^n_{i=1} F_i^{(n)}\). For \(k=\mathbb{C}\) the \(W\)'s and the \(F\)'s can be expressed in terms of theta functions. With the \(F\)'s one defines ``roots of functions'' \(\Phi_i^{(n)} =F_i^{(n+1)} \Delta^{-(n+3)/12}\), where \(\Delta\) is the discriminant of \(X\). Besides, one explicitly constructs divisors \(Z_i^{(n)}\), \(i=1,2, \dots, n+2\), on \(Y^{(n)}\). The \(\Phi\)'s define elements of the \(K_{ 1,\mathbb{Q}}\) of the generic point of \(Z_i^{(n)}\). Again, using the \(\Phi\)'s, one constructs explicit symbols \(S_i^{(n)}\) on the \(Z_i^{(n)}\). Let \({\mathcal P}_{\mathcal M}^{(n)} =(Z_i^{(n)}\), \(S_i^{(n) })\). Then the \({\mathcal P}^{(n)}_{\mathcal M}\) have nice properties and, most importantly, their image in \(\text{Ext}^1_U (\mathbb{Q}, R^{n+1} \pi_*(\mathbb{Q} (n+1)))\) is equal to the elliptic polylogarithm \({\mathcal P}^{(n)}\). connected scheme; elliptic curve; zero section; sheaf of relative homologies; weight filtration; logarithmic sheaf; elliptic polylogarithm; spectral sequence; regulator map; Beilinson's conjectures; Tate's normal form; elliptic Vandermonde functions; theta functions DOI: 10.1007/BF02362334 Higher symbols, Milnor \(K\)-theory, Elliptic curves over global fields, Algebraic number theory: global fields, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K\)-theory of schemes Elliptic polylogarithms in \(K\)-theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review is a survey on recent achievements on arithmetic algebraic geometry due especially to Vojta and Faltings, as well as their links to classical Diophantine approximation results of Roth and Schmidt.
The first chapter deals with Roth's Theorem. In 1954 Roth proved the following statement concerning rational approximation to algebraic numbers: let \(\alpha\) be an irrational real algebraic number, \(\varepsilon\) a positive real number. Then for all but finitely many pairs of integers \((p,q)\), \(|\alpha-p/q|>q^{-2-\varepsilon}\). The author gives the idea of the proof, stressing the main technical difficulty, i.e. a ``nonvanishing lemma'' for polynomials in several variables. Here two main approaches are possible: the original one (Roth's lemma) and the more modern ``Dyson's lemma'' in several variables, proved by \textit{H. Esnault} and \textit{E. Viehweg} [Invent. Math. 78, 445-490 (1984; Zbl 0545.10021)], revisited by \textit{M. Nakamaye} himself [Invent. Math. 121, 355-377 (1995; Zbl 0855.11036)]. The author gives a clear account of the latter, explaining its relation with Faltings' product theorem.
The second part deals with Mordell conjecture (now Faltings' theorem) asserting the finiteness of rational points on an algebraic curve of genus greater than one. The link with Roth's theorem is provided by \textit{P. Vojta}'s proof of Faltings' theorem [Ann. Math. (2) 133, 509-548 (1991; Zbl 0774.14019)], which follows the main steps of classical proofs in Diophantine approximation. The author emphasises this link through the presentation of Vojta's generalization of Dyson's lemma to products of curves of arbitrary genus. The author also gives an overview of the subsequent proof by Faltings of the Lang conjecture on rational points on algebraic subvarieties of abelian varieties.
Finally, a chapter is devoted to the new proof of the Subspace Theorem by Faltings and Wüstholz, leading to interesting generalizations. The Subspace Theorem, which is the natural generalization of Roth's theorem to higher dimension, provides a lower bound for the rational approximation to a hyperplane (or a family of hyperplanes) defined over the algebraic numbers. The new ideas of the Faltings-Wüstholz proof are clearly presented; they led to new results concerning approximations by rational points on an algebraic subvarieties as well as approximation to nonlinear subspaces. Since any clear account of these interesting new applications is lacking both in the Faltings-Wüstholz paper and in the article under review, the interested reader is referred to articles by \textit{J.-H. Evertse} and \textit{R. G. Ferretti}, especially [Int. Math. Res. Notes 25, 1295-1330 (2002)] and [\textit{R. G. Ferretti}, Compos. Math. 121, 247-262 (2000; Zbl 0989.11034)]. Diophantine approximation; rational points on algebraic varieties; arithmetic algebraic geometry; Roth's theorem; nonvanishing lemma for polynomials in several variables; Roth's lemma; Dyson's lemma; Mordell conjecture; Faltings' theorem; finiteness of rational points; algebraic curve of genus greater than one; Vojta's generalization of Dyson's lemma; products of curves of arbitrary genus; Lang conjecture; Subspace Theorem; lower bound for the rational approximation to a hyperplane Results involving abelian varieties, Varieties over global fields, Abelian varieties of dimension \(> 1\), Rational points Diophantine approximation on algebraic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let R be the complete local ring of a reduced curve singularity with algebraically closed residue field of characteristic 0. R is said to have finite Cohen-Macaulay (CM) type if there are only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules. The main result of the paper is that R has finite CM type if it dominates the local ring \(R'\) of a simple plane curve singularity in the sense of Arnol'd (i.e. of type \(A_ k\), \(D_ k\) or \(E_ 6\), \(E_ 7\), \(E_ 8)\). That \(R\) dominates \(R'\) means just that \(R'\subset R\subset \tilde R',\) where \(\tilde R'=\) normalization of \(R'\). As a corollary it is shown that if R itself is a plane curve singularity, then R is of finite CM type if R is simple. This result remains true for curves in arbitrary characteristic [\textit{K. Kiyek} and \textit{G. Steinke}, ''Einfache Kurvensingularitäten in beliebiger Charakteristik'', Arch. Math. (to appear; see the following review)] and for higher dimensional hypersurface singularities [cf. \textit{M. Artin} and \textit{J.-L. Verdier}, Math. Ann. 270, 79-82 (1985; Zbl 0553.14001), J. Auslander, Herzog and \textit{H. Esnault}, ''Reflexive modules on quotient surface singularities'', J. Reine Angew. Math. (to appear; Zbl 0553.14016) in dimension 2; H. Knörrer and R. Buchweitz, G.-M. Greuel and Schreyer (to appear) in higher dimensions]. torsionfree modules; finite Cohen-Macaulay type; complete local ring of a reduced curve singularity; finite CM type G.-M. Greuel and H. Knörrer, Einfache Kurvensingularitäten und torsionsfreie Moduln, Math. Ann. 270 (1985), no. 3, 417 -- 425 (German). Singularities of curves, local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Einfache Kurvensingularitäten und torsionsfreie Moduln | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the review of the English original (1980; Zbl 0456.14016). complex curve; Riemann surface; theta function; Schottky problem; Jacobian variety; quartics; quintics Jacobians, Prym varieties, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Elliptic curves, Theta functions and abelian varieties, Compact Riemann surfaces and uniformization, Non-Archimedean analysis A scrapbook of complex curve theory. (Mozaika teorii kompleksnykh krivykh). Transl. from the English | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 continues the study of deformation theory of singular Calabi-Yau threefolds. If \(\widetilde{X}\) is a non-singular Calabi-Yau threefold, we say \(\widetilde{X}\) is primitive if there is no birational contraction morphism \(\widetilde{X}\to X\) such that \(X\) is smoothable to a Calabi-Yau threefold not deformation equivalent to \(\widetilde{X}\). To study the properties of such threefolds, we first finish the analysis of smoothability of singular Calabi-Yau threefolds \(X\) obtained via a primitive contraction \(\pi:\widetilde{X}\to X\). The case that \(X\) has isolated singularities was considered previously; here we consider the case that \(X\) has a curve of singularities. We find that \(X\) is smoothable unless \(\pi\) contracts a rational surface \(E\) with \(E^3=7\) or 8. We then apply these results to study the structure of primitive Calabi-Yau threefolds. One sample result is the following: if \(\widetilde{X}\) is primitive, then either \(\widetilde{X}\) has Picard number 1, or \(\widetilde{X}\) possesses a small contraction, or there is a \(\text{nef }\mathbb{R}\)-divisor \(D\) with \(D^3=0\). deformation; smoothability; singular Calabi-Yau threefolds; curve of singularities; Picard number Gross, M.: Primitive Calabi-Yau threefolds. J. Differ. Geom. \textbf{45}(2), 288-318 (1997). arXiv:alg-geom/9512002 Deformations of complex singularities; vanishing cycles, Compact complex \(3\)-folds, Calabi-Yau manifolds (algebro-geometric aspects) Primitive Calabi-Yau 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 In his paper [Manuscr. Math. 139, No. 1--2, 71--89 (2012; Zbl 1252.11046)], \textit{Luis Dieulefait} gave a proof of Serre's modularity conjecture for the case of odd level and arbitrary weight. By means of an intricate inductive procedure he reduced the issue to the case of Galois representations of level 3 and weight 2, 4 or 6. As explained in his paper, these cases are taken care of by the following three theorems, respectively.
Theorem 1.1. There are no non-zero semi-stable abelian varieties over \(\mathbb Q\) with good reduction outside 3.
Theorem 1.2. There are no non-zero semi-stable abelian varieties over \(\mathbb Q(\sqrt 5)\) with good reduction outside 3.
Theorem 1.3. Every semi-stable abelian variety over \(\mathbb Q\) with good reduction outside 15 is isogenous, over \(\mathbb Q\), to a power of the Jacobian of the modular curve \(X_0(15)\).
Theorem 1.1 is due to \textit{A. Brumer} and \textit{K. Kramer} [Manuscr. Math. 106, No. 3, 291--304 (2001; Zbl 1073.14544)]. In this paper the author proves Theorems
1.2 and 1.3, each of which directly imply Theorem 1.1. First, the author discusses extensions of \(\mu_p\) and \(\mathbb Z/p\mathbb Z\) by one another that play an important role in this paper. powers of Jacobian of modular curve \(X_0(15)\) René Schoof, Semistable abelian varieties with good reduction outside 15, Manuscripta Math. 139 (2012), no. 1-2, 49 -- 70. Abelian varieties of dimension \(> 1\), Arithmetic ground fields for abelian varieties Semistable abelian varieties with good reduction outside 15 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, the authors study some aspects of complex affine surfaces that have a \({\mathbb{C}}^{*}\)-fibration. A surface \(X\) is said to have a \({\mathbb{C}}^{*}\)-fibration if there exists a surjective morphism \(\rho:X\to B\) where \(B\) is a smooth projective algebraic curve, and there is an open set of \(B\) over which the fibration is a trivial fiber bundle with fiber \({\mathbb{C}}^{*}\). The main topic of this article is the generalized Jacobian problem: If \(X\) is a smooth algebraic surface and \(\varphi:X\to X\) is an étale endomorphism, is \(\varphi\) an automorphism? This question is of particular interest for Platonic \({\mathbb{C}}^{*}\)-fiber spaces. These are surfaces which are isomorphic to the quotient of \({\mathbb{A}}^{2}\setminus \{0\}\), under the action of a non-cyclic finite subgroup \(G\) of \(GL(2,{\mathbb{C}})\) having no pseudo-reflections. It is shown that in this case, the generalized jacobian problem is equivalent to ask if an étale endomorphism of \({\mathbb{A}}^{2}\) commutes with the action of \(G\) is it necessarily an automorphism. A complete answer to the problem cannot be given, however, it is shown that if an étale endomorphism preserves the \({\mathbb{C}}^{*}\)-fibration, then it is an automorphism.
Next, the authors consider a surface \(X\) obtained by deleting the origin of a weighted hypersurface of \({\mathbb{A}}^{3}\) defined by an equation of the type \(x_{1}^{m_{1}}+x_{2}^{m_{2}}+x_{3}^{m_{3}}=0\). It is shown that if \(m_{1}\), \(m_{2}\) and \(m_{3}\) are pairwise coprime and all strictly larger than 1, then any étale endomorphism of \(X\) is an automorphism, with the possible exception of \(\{m_{1},m_{2},m_{3}\}=\{2,3,5\}\).
Finally, the authors consider the general case of a surface \(X=\Sigma\setminus\{0\}\) where \(\Sigma\) is an affine normal complex surface endowed with an unmixed action of \(G_{m}={\mathbb{C}}^{*}\), such that \(0\) is the unique fixed point, and it is in the closure of every orbit. In this case, \(B=X/G_{m}\) is a smooth projective curve, and the quotient map is a \({\mathbb{C}}^{*}\)-fibration. It is shown, for example, that if the genus of the curve \(B\) is strictly bigger than 1, then the generalized jacobian conjecture holds for \(X\). However, if the genus of \(B\) is 1, the authors find counter-examples. An extensive study of when the problem has a positive answer is given. generalized Jacobian problem; quotient of affine surfaces; Platonic fiber spaces; \(\mathbb{C}^*\)-fibration; étale endomorphism; automorphism Masuda, K.; Miyanishi, M.: Étale endomorphisms of algebraic surfaces with gm-actions. Math. ann. 319, No. 3, 493-516 (2001) Jacobian problem, Affine fibrations, Homogeneous spaces and generalizations Étale endomorphisms of algebraic surfaces with \(G_m\)-actions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 be given the centroid G, the orthocenter H, and the circumcenter O of some triangle ABC. Then there is a domain outside the circle on diameter GH, bounded by a closed quartic curve, in which no excenter of ABC lies. (The incenter lies inside the circle.)
Here the author gives a more simple proof of this fact than in a previous paper [Am. Math. Mon. 91, 290-300 (1984; Zbl 0539.51011)]. Use is made of the reflection point K (on the Euler line of ABC) of H in O, as the origin of a polar coordinate system with KO as axis.
A ruler-and-compass construction method for points on the boundary of the domain is derived. plane geometry; excenters of a triangle; incenter of a triangle; curve of degree 4; Euler line Euclidean geometries (general) and generalizations, Geometric constructions in real or complex geometry, Projective and enumerative algebraic geometry, Special algebraic curves and curves of low genus Incenters and excenters viewed from the Euler line | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Riemann surfaces of Parreau-Widom type (see, for instance, the reviewer's account ``Hardy classes on infinitely connected Riemann surfaces'' (1983; Zbl 0523.30028)) have been recognized to form a class of Riemann surfaces having good properties. In the paper under review, the authors point out that hyperelliptic Riemann surfaces of Parreau-Widom type play a role in connection with the study of Hill's equations.
Let \(E\) be a bounded closed set on the real line, which can be expressed as \([b_0, a_0]\backslash \bigcup_j (a_j, b_j)\), where \(-\infty< b_0< a_1< b_1<\cdots< a_n< b_n<\cdots< a_0< +\infty\), and consider the domain \(\Omega= \overline{{\mathbf C}}\backslash E\). Denote by \(\pi(\Omega)\) the fundamental group of the domain \(\Omega\), equipped with the discrete topology, and by \(\pi^*(\Omega)\) the dual group of \(\pi(\Omega)\), which is seen to be a compact torus. Suppose that \(E\) is regular in the sense of potential theory and let \(G(z)\) be the Green function for \(\Omega\) with pole at the point at infinity. Then, the authors say that the set \(E\) belongs to the class (RPW) of Parreau-Widom type if \(\sum G(c_j)< +\infty\), where the sum ranges over all the critical points \(c_j\) of \(G(z)\) with \(a_j< c_j< b_j\). Denote by \({\mathcal D}(E)\) the set of divisors \(D= \sum_j P_j\), where \(P_j= (x_j, \varepsilon_j)\) with \(x_j\in [a_j, b_j]\) and \(\varepsilon= \pm 1\). Here, \((x_j, +1)\) and \((x_j, -1)\) are identified when \(x_j\) is equal to \(a_j\) or \(b_j\). Let \(\omega(z, E_k)\) be the harmonic measure for the set \(E_k\) with respect to the domain \(\Omega\), where \(E_k= E\cap [b_k, a_0]\) for \(k= 1,2,\dots\), and define \(w_k= {1\over 2}[\omega(z, E_k)+ i^*\omega(z, E_k)]\). The Abel correspondence is then defined by the following:
\[
{\mathcal A}: D\mapsto \Biggl( \sum_j \varepsilon_j \int^{b_j}_{x_j} dw_1\text{ mod } \mathbb{Z},\dots, \sum_j \varepsilon_j \int^{b_j}_{x_j} dw_k\text{ mod } \mathbb{Z},\dots\Biggr).
\]
When the system \(\{(a_j, b_j)\}^n_{j= 1}\) is finite, the Abel correspondence \(\mathcal A\) gives a homeomorphism between \({\mathcal D}(E)\) and the real torus \(\mathbb{R}^n/\mathbb{Z}^n\). Assuming that \(E\) belongs to (RPW), the authors show that the Abel correspondence \(\mathcal A\) is well-defined.
Theorem 1 states, in fact, that the Abel correspondence gives a continuous surjection to \(\pi^*(\Omega)\). In order to obtain further results, they assume a stronger condition of homogeneity: Namely, \(E\) is said to be homogeneous if there exists a positive number \(\varepsilon\) such that the inequality \(|(x- \delta, x+ \delta)\cap E|\geq \varepsilon\delta\) holds for any \(x\in E\) and any positive \(\delta\) which is less than the diameter of \(E\).
Theorem 2: If \(E\) is a homogeneous compact set, then the Abel map \(\mathcal A\) is a homeomorphism of \({\mathcal D}(E)\) to \(\pi^*(\Omega)\). The results may be extended to the case of unbounded \(E\). Special instances of such cases have been studied in connection with Hill's equations: see \textit{H. P. McKean} and \textit{E. Trubowitz} [Commun. Pure Appl. Math. 29, 143-226 (1976; Zbl 0339.34024); Bull. Am. Math. Soc. 84, 1042-1085 (1978; Zbl 0428.34026)]. These theorems are proved by constructing an inverse map \(\mathcal L\) from \(\pi^*(\Omega)\) into \({\mathcal D}(E)\) in an explicit way. In the construction Hardy spaces of character-automorphic functions are used effectively. Abelian integrals; Jacobian variety; Jacobi inversion problem; Hardy spaces of character-automorphic functions; hyperelliptic Riemann surfaces of Parreau-Widom type Ideal boundary theory for Riemann surfaces, \({\mathit H}^ p\)-classes, Analytic theory of abelian varieties; abelian integrals and differentials The infinite-dimensional real Jacobi inversion problem, and Hardy spaces of character-automorphic functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is an old and important problem to represent a given closed subvariety \(X\) of a given projective space (over the complex numbers) by coordinates. One solution was proposed by Bertini. To \(X\), associate its dual variety \(X^*\), the locus of tangent hyperplanes. Then \(X\) can be recovered as the dual of \(X^*\). Moreover, in general, \(X^*\) is a hypersurface, so defined by a form, whose coefficients are then suitable coordinates for \(X\). However, if \(X^*\) has codimension \(\delta+1\), then a general point of \(X^*\) represents a hyperplane that is tangent to \(X\) along a \(\delta\)-plane. The various such planes define a closed subvariety \(B(X)\) of the Grassmannian, which lies in another projective space. If the dual of \(B(X)\) is a hypersurface, then the process terminates; if not, then it is repeated. Does it terminate eventually? Bertini, Severi, and B. Segre thought so, it seems. However, the author provides a class of counterexamples, notably including the C. Segre variety \(\mathbb{P}^1 \times \mathbb{P}^s\) with \(s>1\). These counterexamples are given as part of a rudimentary general study of \(B(X)\) for \(X\) that carry a family of linear spaces, specially scrolls, quadrics, Grassmannians, and Segre varieties. The methods are completely elementary and familiar. coordinates of a given subvariety; dual variety; codimension; Segre varieties Varieties and morphisms, Projective techniques in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds A note on a construction suggested by Eugenio Bertini | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 closed \(n\)-dimensional subvariety of the projective \(N\)-dimensional space \(\mathbb P^ N\) over a field. Denote by \(\mathrm{Sec}(X)\) the secant variety of \(X\), that is, the subvariety of \(\mathbb P^ N\) which is the union of the secant lines to \(X\). Let \(\delta\) be the number of secants to \(X\) (counted with multiplicity) that pass through a general point of \(\mathrm{Sec}(X)\). When \(N=2n+1\) and \(X\) is a non-singular curve or surface there exist classical formulas for \(\delta\) in terms of the Segre classes of \(X\). These formulas have been reproven and generalized by Peters-Simonis, Holme, Catanese, Roberts, the reviewer,...\,. We now have formulas for \(\delta\cdot\mathrm{Sec}(X)\), that is the number of secants intersecting a general linear subspace of \(\mathbb P^ N\) of codimension \(N-(2n+1)\), for any variety \(X\) and any \(N\geq 2n+1\).
The main result of the present article is that if \(X\) is a curve, not contained in a hyperplane and \(N\geq 4\), then \(\delta >1\) if and only if all tangents to \(X\) pass through a fixed point, that is if \(X\) is strange (with respect to a point). From this result and the above mentioned formulas the author obtains, among other results, that if \(X\) is a non-singular curve of degree \(d\) and genus \(g\) that is not strange, then \(\deg (\mathrm{Sec}(X))=(d-1)(d-2)/2\).
The article also contains a very nice and simple proof of Terracini's lemma. strange curve; Terracini lemma; secant variety; number of secants; Segre classes M. Dale: ''Terracini's lemma and the secant variety of a curve'', Proc. London Math. Soc. (3), Vol. 49, (1984), pp. 329--339. Curves in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry Terracini's lemma and the secant variety of a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth, projective curve over \(\mathbb{C}\) of genus \(g\geq 2\) and let \([\mathcal{L}]\in \mathrm{Pic}^1(C)\). Let \(M=\mathcal{SU}_C(r,\mathcal{L})\) be the moduli space of stable vector bundles of rank \(r\) and determinant isomorphic to \(\mathcal{L}\). It is a smooth, projective variety of dimension \((r^2-1)(g-1)\). The present paper is devoted to the study of small rational curves on \(M\).
As shown by Drezet, Narasimhan and Ramanan, \( \mathrm{Pic} M=\mathbb{Z}\cdot\Theta\), where \(\Theta\) is an ample generator, and \(-K_M=2\Theta\). Hence the Fano index of \(M\), i.e., the largest integer \(d\geq 1\), such that \(-K_M=dD\), \(D\in \mathrm{Pic} M\), is two.
If \(X\) is a Fano variety of index \(d\) and \(l\subset X\) is a rational curve, we say that \(l\) \textit{has degree} \(k\) if \(-K_X\cdot l=kd\). We say that \(l\) is \textit{a line}, if it has degree one. In particular, for a rational curve \(\phi:\mathbb{P}^1\to M\), the degree is simply \(\deg (\phi^\ast\Theta)\).
\textit{M. S. Narasimhan} and \textit{S. Ramanan} [in: Tata Inst. fundam. Res., Stud. Math. 8, 291--345 (1978; Zbl 0427.14002)] showed that \(M\) is covered by rational curves of degree \(r\), the so-called \textit{Hecke curves}. \textit{S. Ramanan} [Math. Ann. 200, 69--84 (1973; Zbl 0239.14013)] found a family of lines contained in a proper closed subset of \(M\), and these were shown to be the only lines on \(M\) [\textit{X. Sun}, Math. Ann. 331, No. 4, 925--937 (2005; Zbl 1115.14027)].
Rational curves on \(M\) of degree smaller than \(r\) are known as \textit{small rational curves}. A result of \textit{X. Sun} (Theorem in [Zbl 1115.14027]) implies that curves of minimal degree passing through a general point of \(M\) are Hecke, and hence all small rational curves lie in a proper closed subset. This subset is known to have \((r-1)\) irreducible components [\textit{I. Choe}, Bull. Korean Math. Soc. 48, No. 2, 377--386 (2011; Zbl 1237.14040)].
The present paper gives an estimate for the codimension of the locus of small rational curves (Section 2) and an explicit construction of all small rational curves in the case \(r=3\) (Section 3).
In [Zbl 1115.14027], \textit{X. Sun} derived the following a formula for the degree of a rational curve \(\phi:\mathbb{P}^1\to M\). Suppose \(\phi\) is defined by a vector bundle \(E\) over \(X=C\times\mathbb{P}^1\). Let \(f= \mathrm{pr}_1\) and \(\pi= \mathrm{pr}_2\) be the two canonical projections and let the \textit{generic splitting type} of \(E\) be \(\alpha=(\alpha_1^{\oplus r_1},\dots, \alpha_n^{\oplus r_n})\in \bigoplus_i\mathbb{N}^{r_i}\). We may assume, after twisting, that \(\alpha_1>\dots >\alpha_n=0\). The bundle \(E\) admits a relative Harder-Narasimhan filtration \(E^{HN}_\bullet\), and the associated graded \(Gr^{HN}_\bullet(E)\) has torsion-free summands \(Gr^{HN}_i(E_\bullet):=E_i/E_{i-1}\) of generic splitting type \((\alpha_i^{\oplus r_i})\). Then Sun's formula states that
\[
\deg (\phi^\ast\Theta)= r\left(\sum_{i=1}^n c_2(F_i')+\sum_{i=1}^{n-1}(\mu(E)-\mu(E_i))(\alpha_i-\alpha_{i+1}) \mathrm{rk} E_i\right).
\]
Here \(F_i'=Gr^{HN}_i(E_\bullet)\otimes\pi^\ast\mathcal{O}(-\alpha_i)\), and \(\mu(E_i)\) is the slope of the restriction to a generic fibre of \(\pi\). Sun shows that for a small rational curve \(c_2(F_i')=0\) and \(F_i'=f^\ast V_i\), for some locally free sheaf \(V_i\) on \(C\).
In Section 2, Lemma 2.1 the author uses Sun's formula to show that \(V_i\) is semi-stable of degree zero for \(i\leq n-1\), whlie \(V_n\) is stable of degree one. The bundle \(E\) is then obtained by taking successive extensions of \(f^\ast V_i\otimes \pi^\ast \mathcal{O}_{\mathbb{P}^1}(\alpha_i)\).
Next the author goes on to estimate the codimension of the locus of small rational curves. In Theorem 2.4 it is proved that any small rational curve in \(M\) lies in a closed subset
\[
S = \bigcup_{0<r_1<\dots <r_n=r}S_{r_1\dots r_n}
\]
of codimension at least
\[
\min_{0<r_1<\dots <r_n=r}\left\{ \sum_{i=2}^n r_{i-1}(r_i-r_{i-1})(g-2)+ r_{n-1}(r_n-r_{n-1}-1) \right\}
\]
where \(0<r_1<\dots <r_n=r\) runs over \(n\) positive integers \(r_i\), satisfying \(\sum^{n-1}_{i=1}r_i(\alpha_i-\alpha_{i-1})<r\) for some \(n\) and some integers \(\alpha_1>\dots >\alpha_n\).
Section 3 is devoted to the description of all small rational curves in \(M=\mathcal{SU}_C(3,\mathcal{L})\) of degree two: a small rational curve must have degree \(1\) or \(2\), and the case of lines is treated in [Zbl 1115.14027], [\textit{N. Mok} and \textit{X. T. Sun}, Sci. China, Ser. A 52, No. 4, 617--630 (2009; Zbl 1194.14052)]. Hence Sun's formula reduces to
\[
2 =\sum_{i=1}^{n-1} \mathrm{rk}E_i (\alpha_i-\alpha_{i+1}),
\]
and there are only two possibilities for the bundle \(E\to C\times\mathbb{P}^1\):
(A) it fits in an extension of \(f^\ast V_2 \) by \( f^\ast V_1\otimes\pi^\ast\mathcal{O}_{\mathbb{P}^1}(2) \) with \(\mathrm{rk}V_1=1\), \(\mathrm{rk}V_2=2\), \(\deg V_1=0\), \(\deg V_2=1\)
(B) it fits in an extension of \(f^\ast V_2 \) by \(f^\ast V_1\otimes\pi^\ast\mathcal{O}_{\mathbb{P}^1}(1)\) with \(\mathrm{rk}V_1=2\), \( \mathrm{rk}V_2=1\), \(\deg V_1=0\), \(\deg V_2=1\).
To classify these one uses the description of universal extensions from [Zbl 0239.14013], Lemmas 2.3 and 2.4. See also [\textit{M. S. Narasimhan} and \textit{S. Ramanan}, Ann. Math. (2) 89, 14--51 (1969; Zbl 0186.54902), Proposition 3.1] and [\textit{M. S. Narasimhan} and \textit{C. S. Seshadri}, Ann. Math. (2) 82, 540--567 (1965; Zbl 0171.04803), Lemma 3.1]:
Proposition [Ramanan, Zbl 0239.14013]. Let \((W_t)_{t\in T}\), \((V)_{t\in T}\) be two families of vector bundles parametrised by a variety \(T\), such that\newline \(\dim H^1(C,\underline{Hom}(V_t,W_t))\) is independent of \(t\in T\). Let \(G=R^1p_{T\ast}\underline{Hom}(V,W)\), and let \(\pi:\mathbb{P}(G)\to T\) be the corresponding projective bundle. Assume that \(H^i(T, p_{T\ast}\underline{Hom}(V,W)\otimes G^\vee)=0\), \(i=1,2\). Then the extension
\[
0\longrightarrow \pi^\ast p_{C\times T}^\ast W\otimes p_{\mathbb{P}(G)}^\ast\mathcal{O}_{\mathbb{P}(G)}(1)\longrightarrow \mathcal{E}\longrightarrow \pi^\ast p_{C\times T}^\ast V\longrightarrow 0
\]
is a family of bundles on \(C\), parametrised by \(\mathbb{P}(G)\). For each \(t\in T\) the restriction of \(\mathcal{E}\) to \(C\times \mathbb{P}(G_t)\) has the property that for any \(x\in \mathbb{P}(G_t)\) \(\left. E\right|_{C\times \{x\}}\) is an extension of \(V_t\) by \(W_t\) with extension class in the line \(x\).
For dealing with vector bundles of type (A) or ones of type (B) with \(V_1\) unstable, one considers an extension \(E\) of \(V\) by \(\xi\) with extension class in the line \([e]\subset H^1(C,V^\vee\otimes C)\), where \([\xi]\in \mathrm{Pic}^1 C\) and \(\mathrm{rk}V=2\). The appropriate universal extension parametrising the data of all such \(x=([\xi],[V],[e])\) is constructed as follows.
Let \(U_C(2,1)\) be the moduli space of stable vector bundles of rank two and degree one on \(C\), and let \(\mathcal{V}\to C\times U_C(2,1)\) be the universal family. Let \(J_C=\mathrm{Pic}^0 C\), \(J^1_C=\mathrm{Pic}^1 C\) and let \(\mathfrak{L}\to C\times J_C\) be the Poincaré line bundle. Denote by \(\mathcal{R}\subset J_C\times U_C(2,1)\) the subvariety of pairs \(([\xi],[V])\) with \(\det V\otimes \xi\simeq \mathcal{L}\). We set \(G:=R^1p_\ast (\mathcal{V}^\vee\otimes\mathfrak{L})\), where \(p=\mathrm{pr}_2:C\times \mathcal{R}\to\mathcal{R}\), and \(q: \mathcal{P}=\mathbb{P}(G)\to \mathcal{R}\). Then the universal extension is Ramanan's bundle \(\mathcal{E}\to C\times \mathcal{P}\) from above (setting \(T=\mathcal{P}\) and \(\pi=q\)).
The bundle \(\mathcal{E}\) gives rise to a morphism \(\Phi: \mathcal{P}\to M=\mathcal{SU}_C(3,\mathcal{L})\).
Similarly, let \(U_C(2,0)\) be the coarse moduli space of semi-stable rank 2 bundles of degree zero, and \(U_C^{s}(2,0)\) the open set of stable bundles. Let \(\mathcal{R}'\subset U_C^{s}(2,0)\times J_C^1\) be the closed subvariety of pairs \(([V_1],[\xi'])\) with \(\det V_1\otimes \xi'=\mathcal{L}\). If \(\mathcal{V}\to C\times U_C(2,1)\) denotes the universal family, one constructs, using Hecke transformations, a family \(K(\mathcal{V})\to C\times \mathbb{P}(\mathcal{V}^\vee)\), giving a surjective morphism \(\theta: \mathbb{P}(\mathcal{V}^\vee)\to U_C(2,0)\). We also set \(\mathbb{P}(\mathcal{V}^\vee)^{s}:=\theta^{-1}(U_C^{s}(2,0))\) and \(T:= \left(\mathbb{P}(\mathcal{V}^\vee)^{s}\times J^1_C\right)\times_{U_C^{s}(2,0)\times J^1_C}\mathcal{R}'\). If \(p=\mathrm{pr}_2:C\times T\to T\) and \(\mathcal{F}=R^1p_\ast (\mathfrak{L}^\vee\otimes K(\mathcal{V}))\), let \(q':\mathcal{P}'=\mathbb{P}(\mathcal{F})\to T\). Then there exists a vector bundle \(\mathcal{E}\to C\times \mathcal{P}'\) which is an extension of \((1_C\times q')^\ast \mathfrak{L} \) by \((1_C\times q')^\ast K(\mathcal{V})\otimes \mathcal{O}_{\mathcal{P}'}(1) \) with the following property. Let \(x=\left([0\to V_1^\vee\to \mathcal{V}\to \mathcal{O}_p\to 0], [\xi], [\iota]\right)\in\mathcal{P}'\) so \(\det V_1\otimes \xi\simeq \mathcal{L}\), \([\iota]\in \mathbb{P}H^1(C,\xi^\vee\otimes V_1)\) and \([0\to V_1^\vee\to \mathcal{V}\to \mathcal{O}_p\to 0]\in \mathbb{P}(\mathcal{V}^\vee)^s\). Then the restriction \(\left. \mathcal{E}'\right|_{C\times \{x\}}\) is isomorphic to an extension \(E'\) of \(\xi\) by \(V_1\) with extension class in \([\iota]\).
The author shows (Lemma 3.5) that \(\mathcal{E}'\) gives rise to a family of stable rank 3 bundles with determinant \(\mathcal{L}\), and hence to a morphism \(\Psi: \mathcal{P'}=\mathbb{P}(\mathcal{F})\to M=\mathcal{SU}_C(3,\mathcal{L})\). Finally, in Theorem 3.7 the author proves that there exist small rational curves on \(M\), and that each of them can be obtained in one of four possible ways: as the image under \(\Phi\) of a 1) rational curve of degree 2 in the fibre of \(q\) 2) double cover of a line in the fibre of \(q\) 3) line which is not in the fibre of \(q\) and maps to a line in \(U_C(2,\mathcal{L}')\), \([\mathcal{L}']\in \mathrm{Pic}^1 C\) or 4) as the image under \(\Psi\) of a line in \(\mathcal{P}'\) in the fibre of \(q'\). small rational curves; moduli space of vector bundles over a curve Liu, M.: Small rational curves on the moduli space of stable bundles. Int. J. Math. 23, No. 8 (2012) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Small rational curves on the moduli space of stable 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 The Mordell-Lang conjecture says: Let \(X\) be a closed subvariety of a semiabelian variety \(A\) and let \(\Gamma\) be a subgroup of \(A\) of finite rank. Then \(X\cap \Gamma\) is contained in a finite union of translates of algebraic subgroups of \(X\).
This paper deals with the characteristic \(p\) analogue of this conjecture. Here the situation is different, the conjecture fails if \(X\) is isotrivial, i.e. can be defined over a finite field. The authors introduce the notion of weak isotriviality and state a characteristic \(p\) analogue of the Mordell-Lang conjecture in terms of this new notion. - In the main theorem they show that this conjecture is true under certain conditions on \(X\) and on the semiabelian variety \(A\). These conditions involve the \(p\)-rank of a quotient of \(A\). The paper extends the results of \textit{J. F. Voloch} in Invent. Math. 104, No. 3, 643-646 (1991; Zbl 0735.14019) where the conjecture is proved if \(X\) is a projective curve with ordinary jacobian variety \(A\). Mordell-Lang conjecture; semiabelian variety; characteristic \(p\); \(p\)- rank of a quotient Abramovich, D; Voloch, JF, Toward a proof of the Mordell-lang conjecture in characteristic \(p\), Int. Math. Res. Not., 5, 103-115, (1992) Arithmetic ground fields for abelian varieties, Global ground fields in algebraic geometry, Local ground fields in algebraic geometry Toward a proof of the Mordell-Lang conjecture 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 type of a line bundle \(L\) on an abelian variety \(X = \mathbb{C}^g/ \Lambda\) is defined as the type of the first Chern class of \(L\) considered as an alternating form on the lattice \(\Lambda\). If \(L'\) is a second line bundle on \(X\), it is not easy in general to determine the type of the tensor product \(L \otimes L'\) in terms of \(L\) and \(L'\).
The present paper gives an answer to this problem for abelian varieties of dimension 2 and 3. It turns out that the type of \(L \otimes L'\) not only depends on the types of \(L\) and \(L'\), but also on the intersection number \(L \cdot L'\) and the endomorphism ring \(\text{End} (X)\) of \(X\). Explicit formulas for these types are given. In order to apply these formulas one has to make sure that abelian varieties with given endomorphism ring exist. This is done for an arbitrary order in any totally real number field.
Applying this and formulas of \textit{V. Ennola} and \textit{R. Turunen} [Math. Comput. 44, 495-518 (1985; Zbl 0564.12006)] on units in totally real number fields many examples are given. In particular using a paper by \textit{C. Birkenhake}, \textit{H. Lange} and \textit{S. Ramanan} [Manuscr. Math. 81, No. 3-4, 299-310 (1993; Zbl 0807.14030)] it is shown that for any \(d \geq 13\) there is a three-dimensional family of abelian threefolds admitting two principal polarizations whose tensor product is very ample of type \((1,1,d)\). type of a line bundle on an abelian variety; family of abelian threefolds admitting two principal polarizations Birkenhake Ch., Complex Abelian Varieties., 2. ed. (2004) Algebraic theory of abelian varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves Tensor products of ample line bundles on abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We show that for any possible Weierstrass gap sequence \(L\) on a non-singular curve of genus 8 with twice the smallest positive non-gap is less than the largest gap there exists a pointed non-singular curve \((C, P)\) over an algebraically closed field of characteristic 0 such that the Weierstrass gap sequence at \(P\) is \(L\). Combining this with a result of the first author [J. Pure Appl. Algebra 97, No. 1, 51--71 (1994; Zbl 0849.14011)], we see that every possible Weierstrass gap sequence of genus 8 is attained by some pointed non-singular curve. Weierstrass semigroup of a point; double covering of a curve; cyclic covering of an elliptic curve DOI: 10.1007/s00574-008-0074-5 Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group, Divisors, linear systems, invertible sheaves Existence of the non-primitive Weierstrass gap sequences on curves of genus 8 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 elliptic curve over a field \(K\) of characteristic different from 2. The Witt ring \(W(X)\) of symmetric bilinear spaces over \(X\) is diagonalizable if a certain natural homomorphism from the \(W(K)\)-module \(\bigoplus_P W(K(T))\), where \(P\) runs through the rational closed points, into \(W(X)\) is an epimorphism. The authors completely determine when \(W(X)\) is diagonalizable for all elliptic curves \(X\) over local fields with the residue class field different from 2 and 4. Some sample results: Suppose \(X\) has no rational points of order two. Then \(W(X)\) is diagonalizable iff \(K\) is not dyadic. Suppose that \(X\) has only one rational point of order two. Then \(W(X)\) is diagonalizable iff there is an isogeny \(g: Y \rightarrow X\) of degree 2 such that \(g(Y(K))=2X(K)\). Some other results concerned with \(W(X)\) are also proved. elliptic curve; Witt ring over elliptic curve; étale cohomology of a curve Arason J. Elman R. Jacob B. On the Witt ring of an elliptic curve (in preparation) Algebraic theory of quadratic forms; Witt groups and rings, Elliptic curves over local fields, Vector bundles on curves and their moduli, Elliptic curves The Witt ring of an elliptic curve over a local field | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By Chevalley's theorem, if \(G\) is a finite reflection group acting on a vector space over \(\mathbb{Q}\) (with coordinates \(u_ 1, \dots, u_ n)\), then the ring of invariants in \(\mathbb{Q} [u_ 1, \dots, u_ n]\) is a graded polynomial ring with \(n\) fundamental invariants \(p_ 1, \dots, p_ n\): \(\mathbb{Q} [u_ 1, \dots, u_ n]^ G = \mathbb{Q} [p_ 1, \dots, p_ n]\). Let \(k_ 0 = \mathbb{Q} (p_ 1, \dots, p_ n)\) be the quotient field. -- An excellent family of elliptic curves with Galois group \(G\) is, roughly speaking, an elliptic curve \(E\) defined over \(k_ 0(t)\) such that its splitting field \({\mathcal K}\) is equal to \(\mathbb{Q} (u_ 1, \dots, u_ n)\). \(({\mathcal K}\) is defined as the smallest extension of \(k_ 0\) such that \(E({\mathcal K} (t)) = E(k(t))\), \(k\) being the algebraic closure of \(k_ 0\).)
The purpose of this paper is to construct excellent families of elliptic curves when \(G = W(L)\) is the Weyl group of a root lattice \(L\) which is a sublattice of \(E_ 8\) of relatively high rank. More precisely, we consider the Mordell-Weil lattice of a rational elliptic surface, the structure of which has been classified into 74 types [cf. \textit{K. Oguiso} and \textit{T. Shioda}, Comment. Math. Univ. St. Pauli 40, No. 1, 83-99 (1991; Zbl 0757.14011)]. Among them, there are exactly 31 ``admissible'' types for which the narrow Mordell-Weil lattice \(L\) is a root lattice of positive rank. In this paper, we treat about half of these admissible types; we construct an excellent family for each type where the rank of \(L\) is greater than 4 and for a few more types. This will extend a previous work [\textit{T. Shioda}, J. Math. Soc. Japan 43, No. 4, 673-719 (1991; Zbl 0751.14018)] for \(L = E_ 8, E_ 7, E_ 6, D_ 4, A_ 2\) and the recent one by \textit{H. Usui} [Math. Nachr. 161, 219-232 (1993; Zbl 0802.11022)] for \(D_ 5\). The remaining cases are to be treated in a forthcoming paper, where the configuration of singular fibres is more complicated as the rank of \(L\) gets smaller. Together with it, the existence of excellent families of elliptic curves (over the rational function field) will be established for all admissible types.
For more general situation, see the related article: ``Existence of a rational elliptic surface with a given Mordell-Weil lattice'' by \textit{T. Shioda}, Proc. Japan Acad., Ser. A 68, 251-255 (1992; Zbl 0785.14012). elliptic curve over the rational function field; ring of invariants; excellent families of elliptic curves; Weyl group; Mordell-Weil lattice of a rational elliptic surface Shioda, T; Usui, H, Fundamental invariants of Weyl groups and excellent families of elliptic curves, Comment. Math. Univ. St. Pauli, 41, 169-217, (1992) Elliptic surfaces, elliptic or Calabi-Yau fibrations, Algebraic functions and function fields in algebraic geometry, Actions of groups on commutative rings; invariant theory, Families, moduli of curves (algebraic) Fundamental invariants of Weyl groups and excellent families of elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(S\) be a smooth projective surface over \(\mathbb{C}\), with a fibration \(f: S\to C\) of genus \(g\geq 2\) onto a smooth curve \(C\). Let \(R_n= f_* \omega^{\otimes n}_{S/C}\) for \(n\geq 0\), where \(\omega_{S/C}= \omega_S \otimes f^* w_C^\vee\) is the relative canonical sheaf of \(f\). Then \(R(f)= \bigoplus_{n \geq 0} R_n\) is a graded \({\mathcal O}_C\)-algebra in a natural way. We call \(R(f)\) the relative canonical algebra of the fibration \(f\). The geometry of \(S\) can be partly recovered by the algebra \(R(f)\). \textit{M. Reid} conjectured that \(R(f)\) is finitely generated and usually generated in degree \(\leq 3\) and related in degrees \(\leq 6\). By a base change, this is equivalent to that the \(\mathbb{C}\)-algebra \(R(F, K_F)= \bigoplus_{n\geq 0} H^0 (F, nK_F)\) is generated in degrees \(\leq 3\) and related in degrees \(\leq 6\) for any fibre \(F\) of \(f\). For genus 2 and 3 fibrations, this was proved by \textit{M. Mendes Lopes}.
In this paper we prove that for any genus 4 fibre \(F\), \(R(F, K_F)\) is generated in degrees \(\leq 3\). relative canonical algebra of a fibration Structure of families (Picard-Lefschetz, monodromy, etc.), Curves in algebraic geometry The relative canonical algebra for genus 4 fibrations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 an extension of an abelian variety A by a torus T, everything over a complete valuation ring R with quotient field K, and suppose that A has good reduction. If \(\Lambda\) is a lattice in G(K) of the same rank as T, then \(G/\Lambda:=G\otimes_ RK/\Lambda\) is always a proper K- analytic group. The authors prove that if moreover G/\(\Lambda\) is an abelian variety then for every ample divisor E on G/\(\Lambda\), there exists an ample divisor D on A such that \(\pi^*D\) is linearly equivalent with \(g^*E\) (\(\pi\) : \(G\to A\) and g: \(G\to G/\Lambda\) the canonical maps). Furthermore, if f is a meromorphic function on G with divisor \(g^*E-\pi^*D\), then for each \(\lambda\in \Lambda\), \(h_{\lambda}(g):=f(g)f(\lambda g)^{-1}\) is meromorphic with divisor \(\lambda^{-1}\pi^*D-\pi^*D\), and for each \(t\in T\) we have \(h_{\lambda}(tg)=\phi_{\lambda}(t)h_{\lambda}(g)\) with a certain character \(\phi_{\lambda}\) on T. - The authors prove several properties of \(\phi\) and D, and show as their main result that the existence of \(\phi\) and D with these properties is equivalent with G/\(\Lambda\) being an abelian variety.
As an application the authors obtain a proof of the following result, announced by \textit{M. Raynaud} in 1970 [Actes Congr. Internat Math. 1970, part 1, 473-477 (1971; Zbl 0223.14021)]. If A is a proper rigid-analytic group with good reduction, and if there exists an extension G of A by a torus and a lattice \(\Lambda\) in G such that G/\(\Lambda\) is an abelian variety, then A is itself an abelian variety.
The paper gives a rigid analytic proof and a generalization to nondiscrete valuations of results of \textit{G. Faltings} [Arbeitstag. Bonn 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 321-383 (1985; Zbl 0597.14036)]. extension of an abelian variety by a torus; quotient by a lattice; proper rigid-analytic group M. van der PUT , M. Reversat . '' Construction analytique rigide de variétés abéliennes ''. A paraître. Bull. soc. math. France ( 1989 ). Numdam | MR 1042431 | Zbl 0715.14035 Arithmetic ground fields for abelian varieties, Local ground fields in algebraic geometry, Other algebraic groups (geometric aspects), Non-Archimedean analysis Construction analytique rigide de variétés abéliennes. (Rigid analytic constructions of abelian varieties) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0533.14016. anti-Kodaira dimension; rational surface; isolated rational singularities; ample Cartier divisor; minimal resolution; Zariski decomposition of divisors; pseudo effective divisor; dimension formula; anti-genus Sakai, F. Anticanonical models of rational surfaces,Math. Ann. 269(3), 389--410, (1984). Families, moduli, classification: algebraic theory, Special surfaces, Rational and unirational varieties, Divisors, linear systems, invertible sheaves, Singularities of surfaces or higher-dimensional varieties Anticanonical models of 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 [For the entire collection see Zbl 0561.00011.]
The essential statement of the paper is contained in theorem 2 which says that given a reduced hypersurface V in a domain \(U\subset {\mathbb{C}}^{n+1}\), a domain \(W\subset {\mathbb{C}}^ n\) and a holomorphic homeomorphism \(\phi\) : \(W\to V\), then if P is a smooth point of Sing(V), the given V is equisingular along Sing(V) at P. The proof of this theorem is reduced to the following problem: First it is easy to see that by hypothesis Sing(V) is of pure dimension n-1. Then shrinking V and W we may assume that: (1) W\(=W_ 1\times W_ 2\), where \(W_ 1\) is a ball in \({\mathbb{C}}^{n-1}\) (with coordinates \b{s}) and \(W_ 2\) is a disk in \({\mathbb{C}}\) (with coordinate t), both centered at 0; (2) V is a reduced hypersurface in \(U\subset W_ 1\times {\mathbb{C}}^ 2\) having a global defining equation \(f(s,x,y)=0\) in U; (3) Sing V\(=W_ 1\times \{0\}.\)
Let \(V_{\underline s}\) be the analytic subspace of \(U_{\underline s}=\{(x,y)\in {\mathbb{C}}^ 2| \quad (\underline s,x,y)\in U\},\) defined by \(f(\underline s,x,y).\) Then for all \b{s}\(\in W_ 1\), \(V_{\underline s}\) is a plane curve which is non-singular except at the origin. The high point of the proof is to show that the Milnor number of the singularity of \(V_{\underline s}\) at the origin is independent of \b{s}\(\in W_ 1\). Then the claim follows by a well-known result of Lê- Ramanujam.
Using his theorem 2 the author proves a generalized criterion for equisingularity (theorem 1): Let V be a reduced surface in a domain \(U\subset {\mathbb{C}}^{n+1}\), W be a domain in \({\mathbb{C}}^ n\) and let \(F: V\to W\) be a finite proper map which is smooth outside of \(F^{- 1}(\Delta)\), where \(\Delta\) is a proper submanifold of W. Then, either V is smooth, or Sing(V) is smooth of dimension n-1 and V is equisingular along Sing(V).
The reduction of theorem 1 to theorem 2 is using the fact that \(F: V\setminus F^{-1}(\Delta)\to W-\Delta\) is a finitely-sheeted covering map and \(Sing(V)\subset F^{-1}(\Delta)\). - The old discriminant criterion of Zariski has required the additional hypothesis that \(F: V\to W\) is locally the restriction of a projection map \({\mathbb{C}}^{n+1}\to {\mathbb{C}}^ n\). families of curve singularities; Milnor number of the singularity; generalized criterion for equisingularity Global theory and resolution of singularities (algebro-geometric aspects), Families, moduli of curves (analytic), Modifications; resolution of singularities (complex-analytic aspects), Singularities of curves, local rings A generalized criterion for equisingularity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The classical theory of algebraic differential equations has been significantly inspired by the deep work of P. Painlevé about the beginning of this century. In his celebrated ``Stockholm Lectures'' [cf. \textit{P. Painlevé}, Leçons de Stockholm, in: Oevres de Paul Painlevé, Tome I, Editions du Centre National de la Recherche Scientifique (Paris 1973), p. 199-818], Painlevé elaborated the link between algebraic differential equations, differential algebraic function fields, the algebraic varieties associated with them, and the structure of the birational automorphism groups of those varieties. Applying this interplay to integrating differential equations, he showed that all algebraic differential equations of the form \(y''=R(y',y,x)\) are solvable by the so far known functions, or those which are constructable from them by rational processes (including differentiation and integration), apart from six exceptions - the so-called Painlevé equations. The simplest among them is the equation \(y''=6y^ 2+x\), but its irreducibility (in the above sense), claimed by Painlevé, has - up to very recently [cf. the author, Nagoya Math. J. 117, 125-171 (1990; Zbl 0688.34006)] - never been proved rigorously.
The present paper is the (delayed) published version of the author's attempt, undertaken in 1984/85, to clarify Painlevé's work systematically and rigorously, just to find an access to prove the irreducibility of Painlevé's first equation thoroughly. This has been done, in the meantime (cf. the paper cited above), and the recent progress in studying Painlevé's first equation is, in fact, based upon the systematic account on the general Painlevé theory given in the present, already well-known treatise. Actually, this paper may be regarded as a fundamental source of the following recent, furthergoing articles dealing with Painlevé equations:
(1) \textit{H. Umemura}, On the irreducibility of the first differential equation of Painlevé, in: Algebraic geometry and commutative algebra, Vol. II, 771-789 (1988; Zbl 0704.12007); (2) \textit{K. Nishioka}, A note on the transcendency of Painlevé's first transcendent, Nagoya Math. J. 109, 63-67 (1988; Zbl 0613.34030); (3) \textit{K. Nishioka}, General solutions depending algebraically on arbitrary constants, ibid. 113, 1-6 (1989; Zbl 0702.12008); (4) \textit{K. Nishioka}, Differential algebraic function fields depending rationally on arbitrary constants, ibid. 113, 173-179 (1989; Zbl 0695.12016); (5) \textit{H. Umemura}, Second proof of the irreducibility of the first differential equation of Painlevé, ibid. 117, 125-171 (1990; Zbl 0688.34006).
As for the content of the present, finally published version of the contemporary interpretation of Painlevé's approach to algebraic differential equations, it consists of a thorough and consequent algebro- geometric foundation of Painlevé's ideas, concepts, and methods developed in his Stockholm lectures. The whole presentation is based upon the modern framework of algebraic and complex-analytic geometry, in the spirit of Grothendieck's E.G.A. and Serre's G.A.G.A., and as such highly self-contained, i.e., also accessible for non-algebraists. Part I of the paper is devoted to Painlevé's theorem on the relation between analytic subgroups and algebraic subgroups, respectively, of the birational automorphism group of a complex algebraic variety, whereas Part II provides a comprehensive and rigorous treatment of systems of Pfaffian differential equations over complex manifolds, their differential- algebraic aspects, and Painlevé's solvability theorems for algebraic differential equations. Altogether, the present work is certainly of fundamental importance in the field of algebraic analysis. Painlevé equations; Differential algebraic function fields; analytic subgroups; algebraic subgroups; birational automorphism group of a complex algebraic variety; Pfaffian differential equations over complex manifolds; algebraic differential equations N. N. Parfentiev, ''A review on the work by Prof. Schlesinger from Giessen,'' \textit{Izvestiya Fiz.-Mat. Obshchestva pri Imperat. Kazan. Universitete}, Ser. 2, \textbf{XVIII}, 4 (1912). Abstract differential equations, Birational automorphisms, Cremona group and generalizations, History of field theory, Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies Birational automorphism groups and differential equations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Suppose \(X\) is a smooth projective connected curve defined over an algebraically closed field of characteristic \(p>0\) and \(B \subset X\) is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the \(a\)-number of a \(\mathbb{Z}/p\mathbb{Z}\)-cover of \(X\) with branch locus \(B\). For odd primes \(p\), in most cases it is not known if this lower bound is realized. In this note, when \(X\) is ordinary, we use formal patching to reduce that question to a computational question about \(a\)-numbers of \(\mathbb{Z}/p\mathbb{Z}\)-covers of the affine line. As an application, when \(p=3\) or \(p=5\), for any ordinary curve \(X\) and any choice of \(B\), we prove that the lower bound is realized for Artin-Schreier covers of \(X\) with branch locus \(B\). Artin-Schreier cover; characteristic-\(p\); Cartier operator; \(p\)-rank; \(p\)-torsion; formal patching; wild ramification; \(a\)-number; curve; finite field; Jacobian Curves over finite and local fields, Polynomials over finite fields, Formal methods and deformations in algebraic geometry, Jacobians, Prym varieties, Linear transformations, semilinear transformations, Polynomials in number theory, Positive characteristic ground fields in algebraic geometry, Coverings of curves, fundamental group, Matrices over special rings (quaternions, finite fields, etc.) Realizing Artin-Schreier covers with minimal \(a\)-numbers in positive characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a curve, an embeddable noetherian scheme of pure dimension 1. First introduced by Hartshorne, the \textit{generalized divisors} are non-degenerate fractional ideals of \(\mathcal{O}_X\)-modules. Generalized divisors up to linear equivalence are equivalent to \textit{generalized line bundles}, i.e. pure coherent
sheaves which are locally free of rank 1 at each generic point. Further generalization is the notion of \textit{torsion-free sheaves of rank 1}.
Now, let \(\pi: X\rightarrow Y\) be a finite, flat morphism between noetherian curves. The goal of the paper under review is to define and study direct and inverse image for generalized divisors and for generalized line bundles on \( X \) and on \(Y\). Moreover, in the cases where \( X \) and \(Y\) are projective curves over a field (possibly
reducible, non-reduced) and the codomain curve is smooth, the author discusses the same notions for families of effective generalized divisors, parametrized by the Hilbert scheme. The same assumptions are required to introduce the notion of compactified Jacobians parametrizing torsion-free rank-1 sheaves and to study the Norm and the inverse image maps between them. Finally, the author consider the fibers of the Norm map and introduces the Prym stack as the fiber over the trivial sheaf. generalized divisors; generalized line bundles; norm map; compactified Jacobians; Prym variety Divisors, linear systems, invertible sheaves, Jacobians, Prym varieties, Algebraic moduli problems, moduli of vector bundles The direct image of generalized divisors and the norm map between compactified Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be an algebraically closed field of arbitrary characteristic. Let \(Y\) be a projective variety over \(k\) of dimension \(\geq 1\), and let \(L\) be an ample line bundle on \(Y\). The pair \((Y,L)\) will be referred to as a polarized variety. Let \(S(Y,L)\) be the graded \(k\)-algebra associated to \((Y,L)\) defined by \((Y,L)= \bigoplus^\infty_{i=0} H^0(Y, L^i)\). Then \(S=S(Y,L)\) is a finitely generated \(k\)-algebra such that \(S\) is normal if \(Y\) is so. The projective cone \(C(Y,L)\) over \((Y,L)\) is by definition \(\text{Proj} (S[T])\). The infinite section of \(C(Y,L)\) is the closed subvariety \(V_+(T)\) (which is isomorphic to \(Y)\). If \(Y\) is normal \(C(Y,L)\) is also normal. Consider the space \(T^1_S\) of first order infinitesimal deformations of the \(k\)-algebra \(S\). The graded structure of \(S\) yields a natural decomposition \(T^1_S =\bigoplus_{i\in\mathbb{Z}}T^1_S(i)\) such that \(T^1_S\) becomes a graded \(S\)-module. A basic role in this paper is going to be played by those polarized varieties \((Y,L)\) satisfying the following property:
\[
T^1_S(-i)=0 \quad\text{for every }i\geq 1.\tag{*}
\]
Such polarized varieties will be also called polarized varieties with no deformation of negative weights.
Let \(X\) be a normal projective variety. We shall say that \(X\) contains a given polarized variety \((Y,L)\) as an ample Cartier divisior if \(X\) contains \(Y\) as an effective ample Cartier divisor such that the normal bundle \(N_{Y,X}\) of \(Y\) in \(X\) is isomorphic to \(L\).
(**) If \(C\) is a class of normal projective varieties containing the given polarized variety \((Y,L)\) as an ample Cartier divisor (e.g. the class of \(D\) of all normal projective varieties with this property) then every \(X\in C\) is isomorphic to the cone \(C(Y,L)\) and \(Y\) is embedded in \(X\) as the infinite section. Note that the property (**) involves not only \((Y,L)\) but also a certain class of projective varieties containing \((Y,L)\) as an ample Cartier divisor. Then we can ask:
Problem 1: When (*) implies (**) for a given class of normal projective varieties containing \((Y,L)\) as an ample Cartier divisor?
It is easy to give counterexamples showing that (*) does not in general imply (**) for every \(X\in D\). A more classical (but closely related) problem is the following one:
Problem 2: Let \(Y\) be a closed subvariety of dimension \(\geq 1\) of the projective space \(\mathbb{P}^n\). Let \(X\subset\mathbb{P}^{n+1}\) be a closed subvariety of \(\mathbb{P}^{n+1}\) such that there is a hyperplane \(H\) of \(\mathbb{P}^{n+1}\) with the property that the (scheme-theoretic) intersection \(X\cup H\) coincides to \(Y\). When implies the property (*) for \((Y,{\mathcal O}_Y(1))\) that \(X\) is a cone over \(Y\)?
The starting point of this paper is the following result proved earlier [\textit{L. Bádescu} in: Algebraic geometry, Proc. Int. Conf., L'Aquila 1988, Lect. Notes Math. 1417, 1-22 (1990; Zbl 0727.14001)].
Theorem 0. Let \((Y,L)\) be a polarized variety of dimension \(\geq 1\), and let \(X\) be a normal projective variety containing \((Y,L)\) as an ample Cartier divisor. Let \(C\) denote the class of all normal projective varieties \(X\) with the property that the restriction map of graded \(k\)-algebras
\[
S\bigl(X,{\mathcal O}_X(Y)\bigr)= \bigoplus^\infty_{i=0} H^0\bigl(X,{\mathcal O}_X (iY)\bigr)\to S(Y,L)
\]
is surjective. Then (*) implies (**) for every \(X\in C\).
The aim of this paper is to apply this result in order to find several criteria in connection with problems 1 and 2. projective cone over a polarized variety; first order infinitesimal deformations; deformation of negative weights Formal methods and deformations in algebraic geometry, Formal neighborhoods in algebraic geometry, Deformations of singularities, Divisors, linear systems, invertible sheaves Polarized varieties with no deformations of negative weights | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathcal O\) denote the structure sheaf of a smooth, projective curve \(X\) over a finite field \(k\), and \(\text{Cl}(\mathcal O[G])\) the reduced Grothendieck group of \(\mathcal O[G]\)-vector bundles, where \(G\) is a finite abelian group. Any tamely ramified Galois covering \(f:Y \to X\) of smooth projective curves over \(k\) with Galois group \(G\) (``tame \(G\)-cover of \(X\)'') and any \(G\)-stable line bundle \(\mathcal A\) on \(Y\) yields a ``realizable'' class \((f_* \mathcal A) \in \text{Cl}(\mathcal O[G])\). The main aim of this paper is to give an explicit description of the set of classes arising from the structure sheaves of all tame (resp. étale) \(G\)-covers of \(X\), and of that subgroup of \(\text{Cl}(\mathcal O[G])\), which is generated by all realizable classes. Considering the natural Euler characteristic classes of \(\mathcal O[G]\)-vector bundles as introduced by \textit{T. Chinburg} [in: Sémin. Théor. Nombres Bordx., Sér. II 4, No. 1, 1-18 (1992; Zbl 0768.14008)], the authors apply their results to describe the realizable classes in \(\text{Cl} (k[G])\) as well.
To attack this problem, it suffices to consider the cases that \(G\) is a cyclic \(p\)-group, or \(p \nmid \#G\), resp., where \(p=\text{char}(k)\). In the former case, the proof uses Witt vectors and the cohomology groups of \textit{J.-P. Serre} [in: Sympos. int. Topol. Algebr. 24-53 (1958; Zbl 0098.13103)] to describe the realizable classes (theorem 2.5). In the latter one, the Hom-description of class groups -- originating from \textit{A. Fröhlich} [``Galois module structure of algebraic integers'' (1983; Zbl 0501.12012)] and adapted for function fields by \textit{R. J. Chapman} [in: The arithmetic of function fields, Proc. Workshop Ohio State Univ., Columbus 1991, Ohio State Univ. Math. Res. Inst. Publ. 2, 403-411 (1992; Zbl 0801.11047)] is employed to explicitly gain control of \(\text{Cl}(\mathcal O[G]) \simeq \text{Pic}(\mathcal O[G])\).
For the description of the realizable classes (theorem 2.8, in the paper referred to as theorem 2.9), the authors adapt the main ideas of \textit{L. R. McCulloh} [J. Reine Angew. Math. 375/376, 259-306 (1987; Zbl 0619.12008)], who used resolvents and Stickelberger maps to answer the corresponding question for abelian extensions of number fields, and of the second author [\textit{D. Burns}, Math. Proc. Camb. Philos. Soc. 118, No. 3, 383-392 (1995; Zbl 0863.11077)]. reduced Grothendieck group; equivariant line bundles on curves; curve over a finite field; resolvent; Stickelberger map; Euler-Poincaré characteristic map; Picard group; tamely ramified Galois covering; class groups 10.1353/ajm.1998.0045 Vector bundles on curves and their moduli, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Group actions on varieties or schemes (quotients) On the Galois structure of equivariant line bundles on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0688.00013.]
In constructing conformal field theories of free fermions on compact Riemann surfaces, a crucial step is to define and to interpret a reasonable bosonization rule. Among the many recent approaches to these problems, there is one proposed by the same authors of the present paper [Commun. Math. Phys. 121, No. 4, 603--627 (1988; Zbl 0672.14016)]. In that previous article, the authors have constructed a new bosonization over the integers and, concludingly, a conformal field theory over \(\mathbb Z\), in which the coordinate ring of the universal Witt scheme appears as a fundamental object.
The present paper may be regarded as a continuation and extension of the authors' foregoing investigations. Here they show that the formal group that they had derived from the universal Witt scheme, can also be obtained as a formal group naturally assigned to the Jacobian of an algebraic curve over a unitary commutative ring (e.g., the classical Jacobian of a compact Riemann surface), and that the coordinate ring of this formal group may be regarded as the charge zero sector of the boson Fock space, in this framework. Moreover, this interpretation, together with Cartier's isomorphism theorem for formal groups [cf. \textit{P. Cartier}, C. R. Acad. Sci., Paris, Sér. A 265, 49--52 (1967; Zbl 0168.27501)], allows to construct a Hirota tau function, which turns out to be a natural interpretation of the abstract tau function introduced in the approach of \textit{N. Kawamoto}, \textit{J. Namikawa}, \textit{A. Tsuchiya}, \textit{Y. Yamada} [Commun. Math. Phys. 116, No. 2, 247--308 (1988; Zbl 0648.35080)].
Finally, these tools are then used to introduce new operators on the fermion Fock space and its dual, which give rise to other operators of Hecke type and provide a systematic procedure for constructing various number-theoretic divisor functions as well as a suitable Riemann zeta function, including the functional equations satisfied by them.
In this vein, the present work is an important contribution to the explicit understanding of the different attempts (by mathematicians and physicists) to construct possible models for universal conformal field theories of free fermions. conformal field theories of free fermions; bosonization; Jacobian of a compact Riemann surface; formal groups; Hirota tau function; fermion Fock space Jacobians, Prym varieties, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Formal groups, \(p\)-divisible groups, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Formal groups and conformal field theory over \(\mathbb Z\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be the field of rational functions on a curve defined over \({\mathbb F}_ q\) and \(E_{\lambda}\) a finite extension of \({\mathbb Q}_{\ell}\). The author shows that for any continuous absolutely irreducible unramified representation \(\rho:\text{Gal}(\bar k/k)\to \mathrm{GL}_2(E_{\lambda})\) there exists a (unique) unramified cusp form
\[
\mathrm{GL}_2(A_ k)/\mathrm{GL}_2(k)\to {\overline{\mathbb Q}}_{\ell},
\]
which is an eigenfunction of the Hecke operators and whose associated Dirichlet series is equal to that of \(\rho\). When \(\rho\) is contained in a compatible system of \(\lambda\)-adic representations, an analogous result has been proved by Deligne without the assumption ``\(\rho\) is unramified''. fundamental group of a curve; \(\ell\)-adic representations; Langlands conjectures; field of rational functions on a curve; Dirichlet series Drinfed, V. G., \textit{two-dimensional \textit{\(\mathcal{l}\)}-adic representations of the fundamental group of a curve over a finite field and automorphic forms on \textbf{GL}(2)}, Amer. J. Math., 105, 85-114, (1983) Finite ground fields in algebraic geometry, Representation-theoretic methods; automorphic representations over local and global fields, Langlands-Weil conjectures, nonabelian class field theory, Holomorphic modular forms of integral weight, Arithmetic ground fields for curves, Coverings of curves, fundamental group Two-dimensional \(\ell\)-adic representations of the fundamental group of a curve over a finite field and automorphic forms on \(\mathrm{GL}(2)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper we prove that the Griffiths group of a general cubic sevenfold is not finitely generated, even when tensored with \(\mathbb{Q}\). Using this result and a theorem of Nori, we provide examples of varieties which have some Griffiths group not finitely generated but whose corresponding intermediate Jacobian is trivial. intermediate Jacobian; not finitely generated Griffiths group; Griffiths group of a general cubic sevenfold DOI: 10.1007/BF01459807 \(n\)-folds (\(n>4\)), Picard schemes, higher Jacobians, (Equivariant) Chow groups and rings; motives, Transcendental methods, Hodge theory (algebro-geometric aspects) On the Griffiths group of the cubic sevenfold | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 semi-simple algebraic group defined over an algebraically closed field \(k\), and let \(T\subset B \subset P\) be respectively a maximal torus, a Borel subgroup and a parabolic subgroup of \(G\). Inspired by a beautiful result of \textit{Dale Peterson} (unpublished) describing the singular locus of a Schubert variety in \(G/B\), we characterize the \(T\)-fixed points in the singular locus of an arbitrary irreducible \(T\)-stable subvariety of \(G/P\) (a \(T\)-variety for short).
Peterson's result (cf. \S1) says that if \(k=\mathbb C\), then a Schubert variety \(X \subset G/B\) is smooth at a \(T\)-fixed point \(x\) if and only if it is smooth at every \(T\)-fixed point \(y>x\) (in the Bruhat-Chevalley order on the fixed point set \(X^T\)) and all the limits \(\tau_C(X,x) = \lim_{z\to x} T_z(X)\) (\(z\in C\setminus C^T\)) of the Zariski tangent spaces \(T_z(X)\) of \(X\) coincide as \(C\) varies over the set of all \(T\)-stable curves in \(X\) with \(C^T=\{x,y\}\), where \(y>x\). Using this, Peterson showed that if \(G\) is simply laced (and defined over \(\mathbb C\)), then every rationally smooth point of a Schubert variety in \(G/B\) is smooth. More generally, the deformation \(\tau_C(X,x)\) is defined for any \(k\)-variety \(X\) with a \(T\)-action provided \(C\) is what we call good, i.e. \(C\) is a curve of the form \(C=\overline{Tz}\), where \(z\) is a smooth point of \(X\setminus X^T\) and \(x\in C^T\).
Our first main result (theorem 1.4) says that if \(x\in X\) is an attractive fixed point, then \(X\) is smooth at \(x\) if and only if there exist at least two good \(C\) containing \(x\) such that \(\tau_C(X,x) = \text{TE}(X,x)\), where \(\text{TE}(X,x)\) denotes the span of the tangent lines of the \(T\)-stable curves in \(X\) containing \(x\). In addition, if \(X\) is Cohen-Macaulay at \(x\) and \(\tau_C(X,x)= \text{TE}(X,x)\) for even one good \(C\), then \(X\) is smooth at \(x\).
Our second main result (theorem 1.6) says that if \(X\) is a \(T\)-variety in \(G/P\), where \(G\) is simply laced, then \(\tau_C(X,x) \subset \text{TE}(X,x)\) for each good \(C\). This is not true for general \(G\), but when \(G\) has no \(G_2\) factors, then \(\tau_C(X,x)\) is always contained in the linear span of the reduced tangent cone to \(X\) at \(x\). These results lead to several descriptions of the smooth fixed points of a \(T\)-variety in \(G/P\) and, in particular, they give simple proofs of Peterson's results valid for any algebraically closed field.
We also show (cf. example 7.1) that there can exist \(T\)-stable subvarieties in \(G/B\), where \(G\) is simply laced, which have rationally smooth \(T\)-fixed points in their singular loci. singular locus of a Schubert variety; Zariski tangent spaces; simply laced algebraic group; geometric quotient; quotient by maximal torus; descriptions of the smooth fixed points; rationally smooth fixed points Carrell, James B.; Kuttler, Jochen, Smooth points of \textit{T}-stable varieties in \(G / B\) and the Peterson map, Invent. Math., 151, 2, 353-379, (2003) Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Smooth points of \(T\)-stable varieties in \(G\)/\(B\) and the Peterson 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 X be a smooth complete algebraic \({\mathbb{C}}\)-variety and G a connected linear group acting on X and having a Zariski open orbit \(\Omega\subset X\). By work of Borel \(A=X\backslash \Omega\) has at most 2 connected components. When A is connected classification of such actions was performed by the author in Math. USSR, Izv. 11, 293-307 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 308-324 (1977; Zbl 0373.14016). In the present paper classification is performed when A is a connected divisor in X which is a homogeneous space for G; in this case \(X=\Omega \cup A\) is a two-orbit space. Main tool when G is semisimple is what the author calls the Mostow-Karpelevič fibration [see \textit{D. G. Mostow}, Am. J. Math. 77, 247-278 (1955; Zbl 0067.160); and 84, 466-474 (1962; Zbl 0123.163); \textit{F. I. Karpelevich}, Usp. Mat. Nauk 11, No.3, 131-138 (1956; Zbl 0072.182)]. The following geometric result is proved: any X as above can be fibered over a G-homogeneous complete variety with fibres isomorphic to a projective space, a product of projective spaces, a Grassmannian or a certain homogeneous space of \(E_ 6\); in particular such an X is projective and rational. Finally a classification of complete normal two orbit surfaces is provided. homogeneous divisors; group actions; almost homogeneous variety; two- orbit variety; classification of complete normal two orbit surfaces D. Ahiezer, ''Equivariant completions of homogeneous algebraic varieties by homogeneous divisors,''Ann. Glob. Anal. Geom.,1, 49--78 (1983). Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Families, moduli, classification: algebraic theory, Divisors, linear systems, invertible sheaves Equivariant completions of homogeneous algebraic varieties by homogeneous 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 Let \(F\) be a field, \(F_{\mathrm{sep}}\) a separable closure of \(F\), \(n\) a positive integer not divisible by \(\mathrm{char}(F)\), \(\mu_n\) the group of \(n\)-th roots of unity in \(F_{\text{sep}}\), and \(_n\text{Br}(F)=H^2(F,\mu_n)\) the maximal subgroup of the Brauer group \(\mathrm{Br}(F)\) of period dividing \(n\). We say that \(H^2(F,\mu_n)\) is generated by cyclic \(\mathbb Z/n\) classes (i.e. \(_n\text{Br}(F)\) is generated by Brauer equivalence classes of cyclic \(F\)-algebras of degrees dividing \(n\)), if the cup product map \(H^1(F,\mu_n)\otimes H^1(F,\mathbb Z/n)\to H^2(F,\mu_n)\) is surjective. This is the case when \(\mu_n\subset F\), by the Merkurjev-Suslin theorem [see \textit{A. S. Merkur'ev} and \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)]; the same holds in general, if \(n=3\) (Wedderburn's theorem) and in case \(n=5\) [\textit{E. Matzri}, Proc. Am. Math. Soc. 136, No. 6, 1925-1931 (2008; Zbl 1145.16008)].
Henceforth, we assume that \(F\) is a finitely-generated extension of transcendence degree one over the field \(\mathbb Q_p\) of \(p\)-adic numbers, and \(l\) is a fixed prime number different from \(p\). Let \(\Delta\) be a central division \(F\)-algebra of exponent \(l\). As shown by \textit{D. J. Saltman}, then the Schur index \(\mathrm{ind}(\Delta)\) divides \(l^2\), and in case \(\mathrm{ind}(\Delta)=l\), \(\Delta\) is a cyclic \(F\)-algebra [see J. Ramanujan Math. Soc. 12, No. 1, 25-47 (1997; Zbl 0902.16021), and J. Algebra 314, No. 2, 817-843 (2007; Zbl 1129.16014), respectively].
The paper under review shows that if \(\mathrm{ind}(\Delta)=l^2\), then \(\Delta\) decomposes into a tensor product of cyclic \(F\)-algebras of index \(l\). When \(F\) contains a primitive \(l\)-th root of unity, the same result has earlier been obtained by \textit{V. Suresh} [Comment. Math. Helv. 85, No. 2, 337-346 (2010; Zbl 1247.12010)]. Brauer groups; \(n\)-Brauer dimension; \(\mathbb Z/n\)-cyclic classes; \(\mathbb Z/n\)-lengths; connected regular projective relative curves; divisors; hot points; finitely-generated extensions of transcendence degree \(1\); Brauer equivalence classes of cyclic algebras; central division algebras E. Brussel and E. Tengan, Division algebras of prime period \( \ell \neq p\) over function fields of \( p\)-adic curves, Israel J. Math. (to appear). Finite-dimensional division rings, Curves over finite and local fields, Arithmetic theory of algebraic function fields, Skew fields, division rings, Local ground fields in algebraic geometry, Algebraic functions and function fields in algebraic geometry, Brauer groups (algebraic aspects) Tame division algebras of prime period over function fields of \(p\)-adic 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 \(J_5\) denote the Jacobian of the Fermat curve of exponent 5 and let \(K= \mathbb{Q} (\zeta_5)\). We compute the groups \(J_5(K)\), \(J_5(K^+)\), \(J_5(\mathbb{Q})\), where \(K^+\) is the unique quadratic subfield of \(K\). As an application, we present a new proof that there are no \(K\)-rational points on the 5-th Fermat curve, except the so-called ``points at infinity''. rational points; Jacobian of the Fermat curve Tzermias P.: Mordell-Weil groups of the Jacobian of the 5-th Fermat curve. Proc. Amer. Math. Soc. 125, 663--668 (1997) Jacobians, Prym varieties, Rational points, Arithmetic ground fields for curves, Higher degree equations; Fermat's equation Mordell-Weil groups of the Jacobian of the 5-th 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 We discuss various constructions which allow one to embed a principally polarized abelian variety into the Jacobian of a curve. Each of these gives representatives of multiples of the minimal cohomology class for curves which in turn produce subvarieties of higher dimension representing multiples of the minimal class. We then discuss the problem of producing curves representing multiples of the minimal class via deformation-theoretic methods. embedding abelian varieties into Jacobian variety; Prym varieties; deformation; multiples of the minimal class Izadi, E.: Subvarieties of Abelian varieties, Applications of Algebraic Geometry to Coding Theory, Physics and Computation (Eilat, 2001), NATO Sci. Ser. II Math. Phys. Chem., 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 207--214. Subvarieties of abelian varieties, Jacobians, Prym varieties, Theta functions and abelian varieties Subvarieties of abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0588.00014.]
In this paper, the conjectures of Bloch and Beilinson are partially proved by applying the results of Beilinson concerning modular curves. These conjectures relate special values of the associated L-functions to special regulators obtained by comparing different \({\mathbb{Q}}\)-structures.
In this case, one has a regulator map \(r: H^ 0(Sh,{\mathcal K}_ 2)\otimes {\mathbb{Q}}\to H^ 1_ B(Sh_{{\mathbb{R}}},{\mathbb{R}}(1))\), \(Sh=Sh(H)= the\) Shimura variety associated to the multiplicative group H of a quaternionic algebra and \(H^ 1_ B(Sh_{{\mathbb{R}}},{\mathbb{R}}(1))\) the Gal(\({\mathbb{C}}/{\mathbb{R}})\)-invariants of \(H^ 1_ B(Sh^{an},{\mathbb{R}}(1))\). You can decompose the motive \({\bar {\mathbb{Q}}}(Sh)\) into its automorphic components \(\oplus_{V}Sh_ V\otimes V_ f.\quad Let\ell_ 0(V)=(d/ds)L(V_ f,s)|_{s=0}\), then the author proves that there exists a \(H({\mathbb{A}}_ f)\)-submodule \({\mathfrak G}\subset H^ 0(Sh,{\mathcal K}_ 2)\otimes {\mathbb{Q}}\), \({\mathfrak G}=\oplus {\mathfrak G}_ V\), such that r(\({\mathfrak G}_ V)=\ell_ 0(V)\cdot H^ 1_ B((Sh_ v)_{{\mathbb{R}}},{\mathbb{Q}}(1))\subset H^ 1_ B((Sh_ V)_{{\mathbb{R}}},{\mathbb{R}}(1))\). In order to prove this theorem analog to the theorem of Beilinson for modular curves the author uses the correspondence between an automorphic form V of H and an automorphic form W of \(GL_ 2\) given by Jacquet-Langlands theory. By the isogeny theorem for abelian varieties of Faltings one obtains an isogeny between the corresponding parts of the Jacobian varieties. By the results of Soulé one gets isomorphisms of the corresponding K-cohomology groups that commute with the regulator maps, such that you can define the module \({\mathfrak G}_ V\) to be the image of the module \(B_ W\) given by Beilinson's theorem. L-functions; Beilinson conjecture; Bloch conjecture; Shimura variety; quaternionic algebra; Jacquet-Langlands theory; isogeny between the corresponding parts of the Jacobian varieties; K-cohomology groups; regulator maps D. Ramakrishnan, Higher regulators on quaternionic Shimura curves and values of \(L\)-functions , Applications of algebraic \(K\)-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 377-387. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arithmetic ground fields for curves, Quaternion and other division algebras: arithmetic, zeta functions, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Global ground fields in algebraic geometry, Representation-theoretic methods; automorphic representations over local and global fields Higher regulators on quaternionic Shimura curves and values of L- functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We give a new short proof of Bézout's Theorem for complex algebraic curves in \(\mathbb{P}^{2}\) which is local. degree of a curve; intersection multiplicity; plane algebraic curve Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Plane and space curves A short proof of Bezout's theorem in \(\mathbb{P}^{2}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper begins with an analysis of the divisor of a (higher order) differential on a Gorenstein curve. In particular the behaviour of such a divisor under desingularisation is investigated. Furthermore the notion of the ramification divisor for a possibly singular curve is introduced. Using this ramification divisor the authors find that the theory of Weierstrass points on a singular curve is a simple generalisation of the corresponding theory for non-singular curves. divisor of a differential; Gorenstein curve; desingularisation; ramification divisor; Weierstrass points on a singular curve De Carvalho, C. F.; Stöhr, K. -O.: Higher order differentials and Weierstrass points on Gorenstein curves. Manuscripta math. 85, 361-380 (1994) Riemann surfaces; Weierstrass points; gap sequences, Singularities of curves, local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Divisors, linear systems, invertible sheaves Higher order differentials and Weierstrass points on Gorenstein 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 families of smooth rational surfaces in \(\mathbb{P}^4\) have been classified in degree \(\leq 10\). All known rational surfaces in \(\mathbb{P}^4\) can be represented as blow-ups of the plane \(\mathbb{P}^2\). The fine classification of these surfaces consists of giving explicit open and closed conditions which determine the configurations of points corresponding to surfaces in a given family. Using a restriction argument originally due independently to Alexander and Bauer we achieve the fine classification in two cases, namely non-special rational surfaces of degree 9 and special rational surfaces of degree 8. The first case completes the fine classification of all non-special rational surfaces. In the second case we obtain a description of the moduli space as the quotient of a rational variety by the symmetric group \(S_5\). We also discuss in how far this method can be used to study other rational surfaces in \(\mathbb{P}^4\). rational surfaces; fine classification of non-special rational surfaces; quotient of a rational variety F.Catanese - K.Hulek,Rational surfaces in \(\mathbb{P}\)4 containing plane curves, to appear in Ann. Mat. Pura Appl. Rational and ruled surfaces, Plane and space curves, Projective techniques in algebraic geometry, Families, moduli, classification: algebraic theory Rational surfaces in \({\mathbb{P}}^4\) containing 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 The set \(\{p\in X\setminus I_ f; f(p)=c\}\) is called the level curve of a non-constant rational function f on a smooth algebraic surface X where \(I_ f\) denotes the set of all points of indeterminacy of f. An irreducible component of a level curve of f is called a prime curve of f. A smooth prime curve analytically isomorphic to the punctured Gaussian plane \({\mathbb{C}}^*\) is called of \({\mathbb{C}}^*\)-type. If all prime curves of f, except for a finite number of them, are of \({\mathbb{C}}^*\)-type, we say that f is of \({\mathbb{C}}^*\)-type.
In this paper, the author solves the problem of determining all the rational functions of \({\mathbb{C}}^*\)-type on \({\mathbb{P}}^ 2\). level curve of a rational function on a smooth algebraic surface; points of indeterminacy; smooth prime curve; rational functions of \({\mathbb{C}}^*\)-type Kizuka T. , Rational functions of C \ast -type on the two-dimensional complex projective space , Tohoku Math. J. (2) 38 ( 1 ) ( 1986 ) 123 - 178 . Article | Zbl 0577.14021 Families, moduli of curves (analytic), Special surfaces, Holomorphic functions of several complex variables Rational functions of \({\mathbb{C}}^*\)-type on the two-dimensional complex 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 The paper is devoted to present recent developments on the study and classification of the arcs of a curve traced on a singular germ of an algebraic variety \(V\). There are presented computations of the spaces of jets of arcs of the curve using Artin's theorem and they are applied to characterize the existence of smooth arcs traced on \(V\). Also some advances on the Nash problem, about relating the families of arcs and the desingularization of \(V\), are reported. curves through a singularity; jets; arcs of curve; Nash problem G. Gonzalez-Sprinberg and M. Lejeune-Jalabert, Sur l'espace des courbes trac\'{}ees sur une singularit\'{}e, Progress in Mathematics, 134 (1996), 9--32. Singularities in algebraic geometry, Families, moduli of curves (algebraic) On the space of curves through a singularity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 smooth projective family of minimal surfaces of general type parametrized by a complete curve \(C\) of genus \(< 2\) is proved to be a locally trivial fibre bundle. Though the result when \(g(C) = 1\) easily implies the case \(g(C) = 0\), the two cases are treated separately so as to point out the more geometrical aspects of the proof in the latter case. family of minimal surfaces of general type parametrized by a complete curve A. Weil, Généralisation des functions abéliennes, J. Math. Pures Appl. (9), \textbf{17} (1938), 47-87. Surfaces of general type, Families, moduli, classification: algebraic theory A smooth family of minimal surfaces of general type over a curve of genus at most one is trivial | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 huge variety of nonlinear integrable processes and phenomena in physics and mathematics can be described by a few nonlinear partial derivative equations: Korteweg-deVries (KdV) and Kadomtsev-Petviashvili (KP), 1D and 2D Toda, sine-Gordon, nonlinear Schrödinger. The present paper is a further contribution to the study of exact solutions to these equations. The author will be concerned with algebro-geometric solutions, doubly periodic in one variable. According to the well known Its-Matveev's formulae, the Jacobians of the corresponding spectral curves must contain an elliptic curve \(X\), satisfying suitable geometric properties. It turns out that the latter curves are in fact contained in a particular algebraic surface \(S^\bot\), projecting onto a rational surface \(\widetilde{S}\). Moreover, all spectral curves project onto a rational curve inside. The author is thus led to study all rational curves of \(\widetilde{S}\), having suitable numerical equivalence classes. At last he obtains \(d-1\)-dimensional of spectral curves, of arbitrary high genus, giving rise to KdV solutions doubly periodic with respect to the \(d\)-th KdV flow (\(d \geq 1\)). A completely analogous constructive approach can be worked out for the other three cases. Some results are presented, without proof, for the 1D Toda, nonlinear Schrödinger and sine-Gordon equations. elliptic and hyperelliptic curves; Jacobian variety; ruled and rational surfaces; exceptional curve; elliptic soliton Relationships between algebraic curves and integrable systems, Relationships between algebraic curves and physics, Soliton equations, KdV equations (Korteweg-de Vries equations), NLS equations (nonlinear Schrödinger equations), Coverings of curves, fundamental group, Jacobians, Prym varieties, Riemann surfaces; Weierstrass points; gap sequences, Divisors, linear systems, invertible sheaves, Soliton solutions Nonlinear evolution equations and hyperelliptic covers of elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a positive integer \(N\), let \(X_0(N)\) denote the modular curve associated with the modular group \(\Gamma_0(N)\), and let \(J_0(N)\) be the Jacobian variety of \(X_0(N)\). In this paper the author determines completely the set \(S\) of \(N\) for which all \(\mathbb{Q}\)-simple factors of \(J_0(N)\) are elliptic curves. The set \(S\) consists of 71 integers the minimum of which is 11 and the maximum 1200. For the proof he employs the trace formula of Hecke operators for \(\Gamma_0(N)\) together with the fact that, when \(N\) is odd, if all \(\mathbb{Q}\)-simple factors of \(J_0(N)\) are elliptic curves, then the trace of the polynomial \(T^2_2(T^2_2- T_1)(T^2_2- 4T_1)\) defined by the Hecke operators must vanish. A related problem was investigated by\textit{T. Ekedahl} and \textit{J.-P. Serre} [C. R. Acad. Sci., Paris, Sér. I 317, No. 5, 509--513 (1993; Zbl 0789.14026)], where the authors ask whether or not for every positive integer \(g\) there exists a curve of genus \(g\) whose Jacobian variety is completely decomposable. modular curve; Jacobian variety; Hecke operator Takuya Yamauchi, On \Bbb Q-simple factors of Jacobian varieties of modular curves, Yokohama Math. J. 53 (2007), no. 2, 149 -- 160. Jacobians, Prym varieties, Computational aspects of algebraic curves, Modular and Shimura varieties, Elliptic curves On \(\mathbb Q\)-simple factors of Jacobian varieties of 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 See the preview in Zbl 0536.14004. torsion subgroup of second Chow group; \(K_ 2\)-cohomology; Galois module structure; Bloch-Quillen-formula; Neron-Severi group; Picard variety Colliot-Thélène, J.-L. and Raskind, W.: K 2-Cohomology and the second Chow group, Math. Ann. 270 (1985), 165-199. (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Galois cohomology, Parametrization (Chow and Hilbert schemes), 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 We define a power series associated with a homogeneous ideal in a polynomial ring, encoding information on the Segre classes defined by extensions of the ideal in projective spaces of arbitrarily high dimension. We prove that this power series is rational, with poles corresponding to generators of the ideal, and with numerator of bounded degree and with nonnegative coefficients. We also prove that this `Segre zeta function' only depends on the integral closure of the ideal.
The results follow from good functoriality properties of the `shadows' of rational equivalence classes of projective bundles. More precise results can be given if all homogeneous generators have the same degree, and for monomial ideals.
In certain cases, the general description of the Segre zeta function given here leads to substantial improvements in the speed of algorithms for the computation of Segre classes. We also compute the projective ranks of a nonsingular variety in terms of the corresponding zeta function, and we discuss the Segre zeta function of a local complete intersection of low codimension in projective space. Segre class; zeta function; homogeneous ideal; ranks of a smooth variety; duality defect Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Other Dirichlet series and zeta functions, Projective techniques in algebraic geometry The Segre zeta function of an ideal | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 discuss the structure of the Weierstrass semigroup at a pair of points on an algebraic curve. It is known [see \textit{M. Homma}, Arch. Math. 67, 337-348 (1996; Zbl 0869.14015) and \textit{S. J. Kim} and \textit{J. Komeda}, Bol. Soc., Bras. Mat., Nova Sér. 32, No. 2, 149-157 (2001; Zbl 1077.14534)] that the Weierstrass semigroup at a pair \((P,Q)\) contains the unique generating subset \(\Gamma(P,Q)\). We find some characterizations of the elements of \(\Gamma(P,Q)\) and prove that, for any point \(P\) on a curve, \(\Gamma(P,Q)\) consists of only maximal elements for all except for finitely many points \(Q\neq P\) on the given curve. Also we obtain more results concerning special and non-special pairs. generalized Weierstrass point; Weierstrass semigroup of a pair; Weierstrass semigroup of a point Kang E., Kim S.J.: Special pairs in the generating subset of the Weierstrass semigroup at a pair. Geom. Dedicata 99, 167--177 (2003) Riemann surfaces; Weierstrass points; gap sequences, Compact Riemann surfaces and uniformization, Special divisors on curves (gonality, Brill-Noether theory), Applications to coding theory and cryptography of arithmetic geometry Special pairs in the generating subset of the Weierstrass semigroup at a pair | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let L be a very ample divisor on a smooth projective surface S, and let K denote the canonical divisor on S. Sommese and Van de Ven have proved that the linear system \(| L+K|\) is base-point free unless \((S,L)=({\mathbb{P}}^ 2,{\mathcal O}(i))\), \(i=1\) or 2, or \((S,L)\) is a scroll. The morphism \(\phi_{L+K}\) defined by \(| L+K|\) is called the adjunction mapping (associated to L). Let \(S\to^{\alpha}\hat S\to^{\beta}{\mathbb{P}}^ n\) be the Stein factorization of \(\phi_{L+K}\). The author concentrates on the structure of the morphism \(\beta\) and proves the following results:
Theorem 1: Suppose that \(\dim(\phi_{L+K}(S))=2\) and let K and S as above. Then \(\beta\) is an embedding except in the following cases: (i) S is obtained by blowing-up \({\mathbb{P}}^ 2\) at 7 points in general position, and either \(S=\hat S\) and \(L=-2K\), or the map \(\alpha: S\to \hat S\) is the blowing-up of \(\hat S\) at one point P and \(L=\alpha^*(-2\hat K)-E\), where \(E=\alpha^{-1}(P)\); (ii) S is obtained by blowing-up \({\mathbb{P}}^ 2\) at 8 points in general position, and \(L=-3K\); (iii) \(S={\mathbb{P}}({\mathcal E})\) is the projectivization of a rank 2 vector bundle \({\mathcal E}\) over an elliptic curve Y, where \({\mathcal E}\) is a non-split extension \(0\to {\mathcal O}_ Y\to {\mathcal E}\to {\mathcal O}_ Y(P)\to 0,\) \(P\in Y\). If \(\zeta ={\mathcal O}_{{\mathbb{P}}({\mathcal E})}(1)\) is the tautological invertible sheaf, then L is numerically equivalent to \(3\zeta\). - One observes that the finite map \(\beta\) is 2-to-1 in case (i), and 3-to-1 in cases (ii) and (iii).
Theorem 2: There exists an effective divisor \(C\in | L|\) which is a smooth hyperelliptic curve if and only if (S,L) belongs to one of the following cases: (a) \(({\mathbb{P}}^ 2,{\mathcal O}(i))\) with \(i=1, 2\) or 3; (b) S is a geometrically ruled surface over a hyperelliptic curve, and the restriction of L to a fibre has degree 2; (c) S is a rational ruled surface, and the restriction of L to a fibre has degree 2; (d) \(S={\mathbb{P}}({\mathcal E})\) is the projectivization of a rank 2 vector bundle \({\mathcal E}\) over an elliptic curve Y, where \({\mathcal E}\) is a non-split extension \(0\to {\mathcal O}_ Y\to {\mathcal E}\to {\mathcal O}_ Y(P)\to 0,\) with \(P\in Y\). - If \(\zeta ={\mathcal O}_{{\mathbb{P}}({\mathcal E})}(1)\) denotes the tautological invertible sheaf, and F is a fibre of \(S\to Y\), then L is numerically equivalent to \(2\zeta +F\); (e) \((S,L)\) is as described in cases (i) and (ii) of theorem 1. In cases (a),(b),(c) and (d), every smooth divisor \(D\in | L|\) is hyperelliptic, but in case (e) the general element of \(| L|\) is not hyperelliptic.
In this paper some bounds are also given for the degree of the fibres of a ruled surface in \({\mathbb{P}}^ n\). It is proved that a hyperelliptic curve C of genus \(g>0\) can be embedded in the rational surface \({\mathbb{P}}({\mathcal O}_{{\mathbb{P}}^ 1}\oplus {\mathcal O}_{{\mathbb{P}}^ 1}(-e))\) if and only if \(e\leq g+1\). If C is a general hyperelliptic curve of genus g and \(e\leq g+1\), then the curves in \({\mathbb{P}}({\mathcal O}_{P^ 1}\oplus {\mathcal O}_{P^ 1}(-e))\) isomorphic to C move in an algebraic family of dimension \(g+6\). hyperelliptic divisors; very ample divisor; canonical divisor; adjunction mapping; degree of the fibres of a ruled surface SERRANO F., ''The adjunction mapping and hyperelliptic divisors on a surface'', J. Reine Angew. Math. 381 (1987), 90--109. Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Families, moduli, classification: algebraic theory The adjunction mapping and hyperelliptic divisors on a surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(R\) be a homogeneous ring with irrelevant ideal \({\mathfrak m}\) and let \(S=\bigcap R_{\mathfrak p}\) \(({\mathfrak p}\in\text{Spec}(R)-\{{\mathfrak m}\})\), \(R^{(1)}=\bigcap R_{\mathfrak p}\) \(({\mathfrak p}\in\text{Spec}(R)\), \(ht({\mathfrak p})=1)\). In general we have (*) \(R\subseteq S\subseteq R^{(1)}\).
The author studies the case of equality in (*) from an algebraic point of view and he gives some geometric applications. \(S_ 2\); coordinates of a projective variety Relevant commutative algebra On the principal ideals of the ring of coordinates of a projective variety and the property \(S_ 2\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \((J, \Theta)\) be a \(g\)-dimensional principally polarized abelian variety and \(A\) the affine ring of \(J-\Theta\), \(J = {\mathbb{C}}^{g}/\Gamma\) - the quotient of the \(g\)-dimensional vector space by some lattice and \(\Theta\) as the zero locus of a theta function \(\theta(z)\) with \(z = (z_{1},\dots, z_{g})\) being linear coordinates of \({\mathbb{C}}^{g}\). In particular, that Jacobian varieties of hyperelliptic curves of genus \(g > 2\) and non-hyperelliptic curves of genus \(g > 4\) are excluded. Analytically \(A\) is isomorphic to the ring of meromorphic functions on \(J\) which have poles only on \(\Theta\). This means that \(A\) become a module over of ring of differential operators \(D = {\mathbb{C}}[\partial_{1},\dots, \partial_{g}]\), where \(\partial_{i} = \frac{\partial}{\partial_{i}}\). It is a very curious problem to determine generators and relations of the \(D\)-module \(A\). The aim of this paper is to study these problems for \((J, \Theta)\) with \(\Theta\) being non-singular. principally polarized abelian variety; theta function; Jacobian varieties of hyperelliptic curves and non-hyperelliptic curves; module over of ring of differential operators Cho K., Nakayashiki A., Differential structure of Abelian functions, Internat. J. Math., 2008, 19(2), 145--171 Theta functions and abelian varieties, Jacobians, Prym varieties, Relationships between algebraic curves and integrable systems Differential structure of abelian functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a non-singular connected projective curve over an algebraically closed field \(k\) and denote by \(U^s\) the set of isomorphism classes of stable vector bundles on \(X\) with given degree \(d\) and rank \(r\). We know, after \textit{C. S. Seshadri} and \textit{D. Mumford}, that \(U^s\) has a natural structure of quasi-projective variety, which can be compactified to a variety \(U\), by adding semistable vector bundles to the boundary. From a set-theoretical point of view \(U=R/G\), where \(R\) is a variety under the action of a reductive group \(G\), and the quotient is endowed with a scheme structure by techniques of geometric invariant theory (GIT for short). Without using GIT, \textit{G. Faltings} [J. Algebr. Geom. 2, 507-568 (1993; Zbl 0790.14019)] showed that \(R\) has enough theta functions to produce a \(G\)-invariant morphism \(\theta:R\to{\mathbb P}^N\), whose image is a closed subvariety \(U_\theta\), and \(\theta\) factors through a map \(\pi:U\to U_\theta\). In particular, there is a bijection between \(U\) and the normalization of \(U_\theta\), which induces a structure of projective variety on \(U\).
This paper contains a partial answer to the question, raised by Seshadri, about how close is the normalization map \(\pi:U\to U_\theta\) to being an isomorphism. The author proves that the map \(\pi\) is bijective and is an isomorphism over \(U^s\). moduli of vector bundles on a curve; geometric invariant theory; theta functions; normalization map Esteves, E.: Separation properties of theta functions. Duke Math. J. 98, 565--593 (1999) Theta functions and abelian varieties, Geometric invariant theory, Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic) Separation properties of theta functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians An abelian variety over a field which is complete with respect to a discrete valuation admits a uniformization in the rigid analytic sense. Using this uniformization one can show the existence of a Néron-model and the existence of a finite field extension for which the Néron-model has semi-abelian reduction. The present paper gives an exposition of this and can be read as an introduction to two papers of the author and \textit{W. Lütkebohmert} on the subject [see Math. Ann. 270, 349-379 (1985; Zbl 0554.14012) and Invent. Math. 78, 257-297 (1984; Zbl 0554.14015)]. abelian variety; uniformization; existence of a Néron-model Arithmetic ground fields for abelian varieties, Minimal model program (Mori theory, extremal rays), Local ground fields in algebraic geometry, Non-Archimedean analysis Neron models from the rigid analytic viewpoint | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We describe an algorithm for producing the smallest complex algebraic variety containing a given semi-algebraic set \(S\), and all the irreducible components of \(S\). Let \(S\) be defined by \(s\) polynomials of degrees less than \(d\) with integer coefficients of bit lengths less than \(M\). Then the complexity of the algorithm is bounded from above by a polynomial in \(M\), \(s^n\), \(d^{n^2}\). The degree of the complexification is less than \(s^nd^{O(n)}\), while the degrees of polynomials defining the complexification and irreducible components are less than \(d^{O(n)}\). smallest complex algebraic variety containing a given semialgebraic set; complexity of algorithm M.-F. ROY, N. VOROBJOV Computing the Complexification of a Semi-algebraic Set, Proc. of International Symposium on Symbolic and Algebraic Computations, 1996, 26-34 (complete version to appear in Math. Zeitschrift). Semialgebraic sets and related spaces, Analysis of algorithms and problem complexity Computing the complexification of a semi-algebraic set | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 gives various ways to obtain upper or lower bounds for the number of rational points on a curve over a finite field, better than the Weil-estimates. zeta-functions; number of rational points; curve over a finite field; Weil-estimates Serre, J.P.: Sur le nombre des points rationnels d'un courbe algébrique sur un corps fini. C. R. Acad. Sc. Paris 296, 397-402 (1983) Rational points, Finite ground fields in algebraic geometry, Arithmetic ground fields for curves, Enumerative problems (combinatorial problems) in algebraic geometry Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present article is a survey on various results of deformation theory of moduli spaces for vector bundles over smooth projective complex curves. The authors focus mostly on the main results obtained in this area over the recent years, their interdependence, and on the conceptual aspects of their proofs. The first part discusses the moduli spaces of semistable rank 2 vector bundles on a curve, whose determinant is a fixed line bundle. The deep results on their deformation spaces, basically obtained by \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math., II. Ser. 101, 391-417 (1975; Zbl 0314.14004)] and, more recently, by the authors themselves [Am. J. Math. 115, No. 2, 279-303 (1993; Zbl 0785.14015)] are lucidly outlined.
The second part concerns Torelli-type theorems for these moduli spaces, while the concluding third part is devoted to the deformations of the Picard bundle on them. Part 2 and 3 are essentially commented summaries of according results published in the authors very recent paper ``Deformations of Picard sheaves and moduli of pairs'' [Duke Math. J. 76, No. 3, 773-792 (1994; see the preceding review)]. The latter reference, by the way, provides all details of proof for the results surveyed in the present article. deformation theory of moduli spaces for vector bundles; semistable rank-2 vector bundles on a curve Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Formal methods and deformations in algebraic geometry, Picard schemes, higher Jacobians On the deformation theory of moduli spaces of 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 The author studies the topology of the quotient variety of a complex algebraic projective variety \(X\) with an action of a complex algebraic torus \((\mathbb{C}^*)^ n\). As a main result, he obtains an inductive formula for the intersection Betti numbers. The formula includes singular quotients. One can always find a rationally nonsingular quotient and a canonical map that is a small resolution in the sense of Goresky- MacPherson. The most important quotients under consideration are those that can be understood as symplectic reduced spaces.
The author starts with a nice introduction to this subject. The main part of the paper contains an adequate stratification, the proof that there exist small resolutions and the decomposition theorem for the intersection homology in the symplectic case. In a further step this is done analogously in the so-called semigeometric case which generalizes the symplectic one. But the results are smaller: Vanishing property of intersection homology in odd degree and isomorphism to rational intersection groups are transferable from the fixed-point set to the quotient. Finally, an application to flag varieties is given. See also \textit{F. C. Kirwan}, ``Cohomology of quotients in symplectic and algebraic geometry'', Math. Notes 31 (1984; Zbl 0553.14020).
[See also erratum to this paper in the following review.]. topology of the quotient variety; action of a complex algebraic torus; intersection Betti numbers; symplectic reduced spaces; intersection homology Y. Hu, The geometry and topology of quotient varieties of torus actions, Duke Math. J. 68 (1992) 151--184. Erratum: 68 (1992) 609. Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) The geometry and topology of quotient varieties of torus actions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(S=\{D\mid h^0(D)\geq 1\), \(h^1(D)\geq 1\}\) be the set of special divisors on a curve \(C\) of genus \(g>0\). Let \(S'\) be the subset of \(S\) consisting of divisors \(D\) with strictly negative Brill-Noether number \(\rho(D):= g-h^0(D)\cdot h^1(D)\). An index for \(C\) is a function \(\sigma: S\to\mathbb{Z}\) with \(\sigma(D)\) depending only on \(\deg(D)\), \(h^0(D)\), \(h^1(D)\) and satisfying
(1) \(\sigma(K-D)= \sigma(D)\), \(K=\) canonical divisor of \(C\),
(2) \(\sigma(D+P)> \sigma(D)\), \(P\in C\) a general point,
(3) \(\sigma(nD)\geq \sigma(D)\) for \(n\geq 1\) if \(h^0(nD)-1= n(h^0(D)-1)\).
A universal index of curves of genus \(g\) is an index which does not depend on the choice of the curve of genus \(g\). A well-known example is the Clifford index \(\text{cliff} (D):=\deg (D)- 2(h^0(D)-1)\).
Main result: If \(\sigma\) is a universal index which is either
(1) non-negative on \(S'\) or
(2) \(\sigma>\rho\) on \(S'\) and \(\sigma(D)= \deg(D)- m(D) (h^0(D)-1)\), \(m(D)\in \mathbb{Z}\) for all \(D\) with \(h^0(D)\geq 2\), \(h^1(D)\geq 2\), then \(\sigma=\text{cliff}\).
The author gives several interesting examples for different types of curves. divisors on a curve; Brill-Noether number; Clifford index Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Topological properties in algebraic geometry A remark on Clifford index and Brill-Noether number | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\subset\mathbb{C} \mathbb{P}^3\) be a closed integral curve, which is involutive with respect to the symplectic form \(dx_0\wedge dx_1+dx_2 \wedge dx_3\) and \(TC\) its tangent sheaf. Let \(s\) be the minimal degree of a surface of \(\mathbb{C}\mathbb{P}^3\) containing \(C\). Here we give a proof that
\[
h^0(C,TC(s-2)) \neq 0, \quad \deg(TC)\geq d(2-s)\quad\text{and}\quad\deg(TC)=d(2-s)\quad\text{iff }TC\cong {\mathcal O}_C(2-s).
\]
Assume that \(C\) is smooth. If \(\omega_C\not\approx {\mathcal O}_C(s-2)\) and
\[
(s-2)h^1(\mathbb{C} \mathbb{P}^3,{\mathcal I}_C(s-1))> (s-1)h^1 (\mathbb{C}\mathbb{P}^3, {\mathcal I}_C(s-2))
\]
(respectively, either
\[
h^1(\mathbb{C} \mathbb{P}^3,{\mathcal I}_C(s-1))=h^1 (\mathbb{C} \mathbb{P}^3, {\mathcal I}_C(s-1)) =0,
\]
or
\[
(s-2)h^1 (\mathbb{C}\mathbb{P}^3, {\mathcal I}_C(s-1))> s (h^1(\mathbb{C} \mathbb{P}^3,{\mathcal I}_C(s-2)))),
\]
then \(s\leq 4\) (respectively, \(s\leq 3)\). minimal degree of a surface containing a closed integral curve Plane and space curves, Special surfaces Involutive space curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper a numerical criterion for divisors on a smooth projective surface to be very ample is given. The idea is to restrict a given divisor to a sufficient number of (not necessarily irreducible nor reduced) curves on the surface and prove the very ampleness of the restriction. At the end we give an application to Bordiga surfaces. ampleness of divisors on a smooth projective surface; Bordiga surfaces DOI: 10.1007/BF02570459 Embeddings in algebraic geometry, Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus Embeddings of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A lot of work, both classical and modern, has been done in order to classify embedded projective varieties with small invariants, and more specifically with small degree and/or sectional genus. Another classical projective invariant that has not been much considered is the class. Recall that the class of a projective variety is the degree of its dual variety (if the latter is a hypersurface).
In the paper under review, the authors classify smooth projective surfaces with class \(\mu \leq 25\), as well as those with \(\mu - d \leq 16\), \(d\) being the degree. The key tool for this classification is to use Landman's formula \(\mu - d = 4(g-q) + b_ 2-2\) (here \(g\) is the sectional genus, \(q\) is the irregularity and \(b_ 2\) the second Betti number). This formula provides that \(g\) must be small enough to use known classifications of varieties with small sectional genus. Similarly, the authors classify varieties of dimension \(k \geq 3\) with \(\mu \leq 23\). codegree; small degree; class of a projective variety; degree of dual variety; small sectional genus C. Turrini and E. Verderio, Projective surfaces of small class, Geom. Dedicata 47 (1993), 1--14. Families, moduli, classification: algebraic theory, Projective techniques in algebraic geometry Projective surfaces of small class | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author considers the Picard vector bundles defined over Jacobi varieties. The rank \(g+1\) Picard bundle imbeds in the rank \(2^ g\) Clifford bundle, so the second order theta functions span the dual of the Picard bundle over each fiber. A result on the minimum number of such second order theta functions required to span the whole bundle at each point is proved. There is an application of using these functions to describe subvarieties of the Jacobian. Some comments are given on which functions one could use and finally the paper contains also generalizations to higher order theta functions. second order theta functions; dual of the Picard bundle; subvarieties of the Jacobian; higher order theta functions Jacobians, Prym varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Theta functions and abelian varieties Second order theta functions and vector bundles over Jacobi 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 \textit{L. Ein}, \textit{D. Eisenbud} and \textit{S. Katz} in Algebraic geometry, Proc. Conf., Sundance/Utah 1986, Lect. Notes Math. 1311, 51-70 (1988; Zbl 0699.14064) asked whether the general set of \(d\quad points\) in \({\mathbb{P}}^ r\) is the hyperplane section of a curve, or for that matter of a projectively normal curve. For the case \(r=3\) this problem was completely solved by \textit{L. Chiantini} and \textit{F. Orecchia} in algebraic curves and projective geometry, Proc. Conf. Trento/Italy 1988, Lect. Notes Math. 1389, 32-42 (1989; Zbl 0707.14027).
The first goal in the paper under review is to answer the first part of the above question, but in a more general context: any non-degenerate curvilinear zero-scheme in \({\mathbb{P}}^ r\) is the hyperplane section of a smooth rational curve.
In the remainder of the paper the authors consider higher genus. In section 2 they show that for a given reduced zero-scheme \(Z\subset {\mathbb{P}}^ 2\) there exists a smooth curve C in \({\mathbb{P}}^ 3\) with Z as hyperplane section and having genus g, for each g in the range \(0\leq 2g\leq d-2\). They also show that a set of points in linear general position in \({\mathbb{P}}^ n\) is the hyperplane section of a smooth elliptic curve. In section 3 they give examples of ``gaps'' in the possible genera of curves having a given set of points as hyperplane section. Throughout this work, k is an algebraically closed field of characteristic zero. curvilinear zero-scheme; hyperplane section of a smooth rational curve; higher genus E. Ballico, J. Migliore, Smooth curves whose hyperplane section is a given set of points, Comm. Algebra 18 (1990), 3015--3040. Projective techniques in algebraic geometry, Special algebraic curves and curves of low genus Smooth curves whose hyperplane section is a given set of points | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper contains a number of results involving Cartier points on curves, foremost among them a strengthening of Ekedahl's theorem that a superspecial curve in characteristic \(p>0\) has genus at most \(p(p-1)/2\) if it is non-hyperelliptic, and at most \((p-1)/2\) if it is hyperelliptic.
Suppose \(C\) is a curve of positive genus over an algebraically closed field \(k\) of characteristic \(p\). A point \(P\) on \(C\) is a ``Cartier point'' of \(C\) if the Cartier operator takes the vector space of meromorphic differentials on \(C\) vanishing at \(P\) to itself. The main theorem of this paper states that
(1) if \(C\) has Cartier points \(P_1,\ldots, P_p\) such that the divisor \(p(P_i - P_j)\) is non-principal for every \(i\neq j\), then the genus of \(C\) is at most \(p(p-1)/2\); and
(2) if \(p>2\) and \(C\) is hyperelliptic, and if a hyperelliptic branch point of \(C\) is a Cartier point, then \(C\) has genus at most \((p-1)/2\).
Ekedahl's result follows easily from this, because if a curve \(C\) is superspecial then the Cartier operator acts as \(0\) on the meromorphic differentials, so that every point is a Cartier point. The author notes that his proof of Ekedahl's theorem does not rely on Nygaard's characterization of the Jacobians of superspecial curves.
The author also provides an explicit bound (in terms of \(g\) and \(p\)) on the number of Cartier points on a non-superspecial genus-\(g\) curve over a field of characteristic \(p\), and gives several examples of curves for which the Cartier points can be computed explicitly. For example, he shows that in characteristic \(3\) the modular curves \(X_0(43)\) and \(X_0(61)\) have no Cartier points. Cartier operator; superspecial curve; characteristic \(p\); genus; meromorphic differentials; number of Cartier points Baker, M.: Cartier points on curves. Internat. math. Res. notices 7, 353-370 (2000) Special algebraic curves and curves of low genus, Finite ground fields in algebraic geometry, Commutative rings of differential operators and their modules, Arithmetic ground fields for curves Cartier points on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Abelian variety over a global field; Tate module; elliptic curve without complex multiplication Zarhin, Yu.G.: Abelian varieties, \ell-adic representations and SL2, Izv. akad. Nauk SSSR, ser. Mat. 43, 294-308 (1979) Arithmetic ground fields for abelian varieties, Elliptic curves, Arithmetic problems in algebraic geometry; Diophantine geometry, Global ground fields in algebraic geometry Abelian varieties, \(\ell\)-adic representations and \(\mathrm{SL}_2\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth irreducible projective curve. Denote by \(C_d\) the \(d\)-fold symmetric product of \(C\) that parameterizes effective divisors of degree \(d\) in \(C\). The effective cones of divisors and curves of \(C_d\) as well as the subcones generated by tautological classes have been studied intensively. In this paper the authors focus on cycles of intermediate dimension. Given a tuple of positive integers \(a_1\geq a_2 \geq \cdots \geq a_n \geq 1\) and \(\sum_{i=1}^n a_i = d\), consider \((p_1, \ldots, p_n) \mapsto \sum_{i=1}^n a_i p_i\) from \(C^n\) to \(C_d\), whose image cycle is called an \(n\)-dimensional diagonal. Let \(\mathcal D_n(C_d)\) be the cone generated by all the \(n\)-dimensional diagonals. The authors determine completely the extremal rays of \(\mathcal D_n(C_d)\). Moreover, they show that \(\mathcal D_n(C_d)\) is a rational polyhedral perfect face of the pseudoeffective cone of \(n\)-dimensional cycles Pseff\(_n(C_d)\) along which Pseff\(_n(C_d)\) is locally finitely generated (and the same conclusion holds for restricting to the tautological subcone). effective cone; symmetric product of a curve; diagonal Algebraic cycles, Special divisors on curves (gonality, Brill-Noether theory), Divisors, linear systems, invertible sheaves Effective cycles on the symmetric product of a curve. I: The diagonal cone | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(r,d\) be integers such that \(r\geq 3\), \(d\geq r+2\). Let \(C\subseteq {\mathbb P}^r\) be a nondegenerate, irreducible, projective curve of degree \(d\); \(C\) is called \(m\)-regular if \(H^i({\mathbb P}^r, {\mathcal I}_C(m-i))=0\) for \(i\geq 1\). The \textit{regularity} \(\text{reg}(C)\) of \(C\) is the least integer \(m\) such that \(C\) is \(m\)-regular. We have \(\text{reg}(C)\leq d-r+2\) [\textit{L. Gruson} et al., Invent. Math. 72, 491--506 (1983; Zbl 0565.14014)], and if equality holds, \(C\) is named of \textit{maximal regularity}. In this case, \(C\) is a nonsingular rational curve admiting a \((d-r+2)\)-secant line, the so-called \textit{extremal secant line} [loc. cit.].
In this paper the authors are interested in spaces that parametrize projective curves with a fixed regularity condition; in particular, they show that \(R=R^{d-r+2}_{r,d}\), the set of all maximal regularity curves in \({\mathbb P}^r\) of degree \(d\), is an irreducible variety of dimension \(3d+r^2-r-1\) (cf. Lemma 2.4 in [\textit{T. Johnsen} and \textit{S. L. Kleiman}, Commun. Algebra 24, No. 8, 2721--2753 (1996; Zbl 0860.14038)]. Moreover, as the automorphism group \(A=\mathrm{PGL}(r+1)\) of \({\mathbb P}^r\) acts in a natural way on \(R\), we can define in particular the stabilizer group \(A_C\) for each \(C\in R\); they show that \(A_C\) is finite and \(C\) cuts out the extremal secant line in at least 4 points. rational curve; \(m\)-regular curve; regularity of a curve; extremal secant line Special algebraic curves and curves of low genus, Stacks and moduli problems, Questions of classical algebraic geometry On the space of projective curves of maximal regularity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given a variety over a valued field, one can tropicalize it and construct a combinatorial object. Originally, these tropical varieties are polyhedral complexes which inherit topology from \(\mathbb{R}^n\). \textit{J. Giansiracusa} and \textit{N. Giansiracusa} [Duke Math. J. 165, No. 18, 3379--3433 (2016; Zbl 1409.14100)] combined \(\mathbb{F}_1\)-geometry with the notion of bend loci to equip tropical varieties with scheme structure, and therefore, obtained tropical schemes. This paper investigates Picard groups of tropical schemes as a first step towards building scheme-theoretic tropical divisor theory which can lead to finding a scheme-theoretic tropical Riemann-Roch theorem.
For monoid \(M\) one can pass from monoid scheme \(X = \text{Spec}\, M\) to scheme \(X_K = \text{Spec}\, K[M]\) by scalar extension to field \(K\). \textit{J. Flores} and \textit{C. Weibel} [J. Algebra 415, 247--263 (2014; Zbl 1314.14003)] show Picard groups \(\text{Pic} (X)\) and \(\text{Pic} (X_K)\) are isomorphic. The current paper proves this isomorphism in tropical setting: for irreducible monoid scheme \(X\) and idempotent semifield \(S\) Picard groups \(\text{Pic} (X)\) and \(\text{Pic} (X_S)\) are both isomorphic to certain sheaf cohomology groups, and hence, are isomorphic. They also construct the group \(\text{CaCl}\, (X_S)\) of Cartier divisors modulo principal Cartier divisors and show that \(\text{CaCl}\, (X_S)\) is isomorphic to \(\text{Pic} (X_S)\). tropical schemes; Picard groups; Cartier divisors; idempotent semiring Geometric aspects of tropical varieties, Picard groups, Semirings, Semifields, Ordered semigroups and monoids Picard groups for tropical toric schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 real algebraic curve of genus 2 with at least two real components \(B_1\) and \(B_2\). An embedding of \(C\) into the projective plane blown-up in a point allows an explicit description of the neutral real component \(\text{Pic}^0 (C)^0\) of the Jacobian of \(C\). The author uses an isomorphism \(\text{Pic}^0 (C)^0 \simeq B_1\times B_2\) which is a particular case of an isomorphism found by J. Huisman.
In particular the group law on \(B_1\times B_2\) is given by intersecting with conics when \(C\) is mapped as a quartic curve into \(\mathbb{P}^2\), and finally the author describes the 2- and 3-torsion points on \(B_1\times B_2\). real algebraic curve of genus 2; neutral real component; Jacobian; group law; quartic curve; 2- and 3-torsion points Fichou, G.: Loi de groupe sur la composante neutre de la jacobienne d'une courbe réelle de genre 2 ayant beaucoup de composantes réelles. Manuscripta math. 104, 459-466 (2001) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Special algebraic curves and curves of low genus, Real algebraic sets, Jacobians, Prym varieties Group law on the neutral component of the Jacobian of a real curve of genus 2 having many real components | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 completes previous work of \textit{G. D. Mostow} [Pac. J. Math. 86, 171-276 (1980; Zbl 0456.22012); Publ. Math., Inst. Hautes Etud. Sci. 63, 91-106 (1986; Zbl 0615.22009); Bull. Am. Math. Soc. 16, 225-246 (1987; Zbl 0639.22005); On discontinuous action of monodromy groups on the complex \(n\)-ball (to appear)] and \textit{P. Deligne} and \textit{G. D. Mostow} [Publ. Math., Inst. Hautes Etud. Sci. 63, 5-89 (1986; Zbl 0615.22008)]. In these previous works, certain monodromy subgroups of \(PU(1,n)\), related to hypergeometric differential systems, were defined and their possible discreteness was investigated. These groups arise as monodromy groups of the Picard-Fuchs equations obtained when the \(n + 3\) ramification points of a covering of \(\mathbb{P}^ 1\), with fixed local monodromy, are moved around, precisely as in the classical theory of ordinary hypergeometric functions. Mostow gave a necessary and sufficient condition for these groups to be discrete, holding when \(n>3\). This paper deals with the case \(n=2\). After a clear review of previous work, the author, inspired by some computer-work, completes the list of discrete groups arising for \(n=2\). He can then verify that the Mostow condition essentially holds also in the present case. He then computes the volumes of the fundamental domains for all of his discrete groups, and subsequently determines relative indexes, in case of mutual inclusions. He also proves certain isomorphisms among various groups. His conclusions are nicely summarized in several tables that include the results of Mostow and Deligne. The paper contains very explicit proofs of its statements, and should be of great help to both specialists and newcomers of the topic. isomorphisms among monodromy groups; lattices in \(PU(1,2)\); congruence subgroups; monodromy groups; Picard-Fuchs equations; hypergeometric functions; discrete groups; volumes of fundamental domains; relative indexes; tables Sauter., J., Isomorphisms among monodromy groups and applications to lattices in \({{\mathrm PU}(1,2)}\), Pacific J. Math.,, 146, 331-384, (1990) Discrete subgroups of Lie groups, Structure of modular groups and generalizations; arithmetic groups, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Structure of families (Picard-Lefschetz, monodromy, etc.), Connections of hypergeometric functions with groups and algebras, and related topics, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Fuchsian groups and their generalizations (group-theoretic aspects) Isomorphisms among monodromy groups and applications to lattices in \(PU(1,2)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a projective smooth variety over \(\mathbb{Q}\). Let \(S\) be a finite set of primes such that \(X\) has a projective smooth model \({\mathcal X}\) over \(U:=\text{spec} \mathbb{Z} [{1\over S}]\). We have the following exact localization sequence in algebraic \(K\)-theory:
\[
H^1({\mathcal X}, {\mathcal K}_2)\to H^1(X, {\mathcal K}_2) @>\partial>> \bigoplus_{p\in U}\text{Pic} (X_p)\to \text{CH}^2({\mathcal X})\to\text{CH}^2(X)\to 0
\]
where \(X_p\) is the fiber of \({\mathcal X}\) at the prime \(p\).
A conjecture of Beilinson on the special values of \(L\)-function and the Tate conjecture tell us that the cokernel of \(\partial\) is torsion. This means there should be enough elements in \(H^1(X,{\mathcal K}_2)\) so that any element of \(\text{Pic} (X_p)\otimes \mathbb{Q}\) is in the image of the map \(\partial\otimes \mathbb{Q}\).
The purpose of this paper is to give a potential method to construct elements in \(H^1(X,{\mathcal K}_2)\) in the case where \(X\) is a self-product of a curve: We present elements of \(H^1(C\times C,{\mathcal K}_2)\) for certain specific curves \(C\). The image of the element under the boundary map arising from the localization sequence above is the graph of Frobenius endomorphism of the reduction of the curve modulo 3. constructing elements in \(H^1\); self-product of a curve Applications of methods of algebraic \(K\)-theory in algebraic geometry, Whitehead groups and \(K_1\), Curves in algebraic geometry On \(K_1\) of a self-product of a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0553.00005.]
Let \(C_ m\) be a real irreducible curve of order m in the real projective plane. \textit{L. V. Golubina} and \textit{K. K. Tai} [in Diff. Integral'nye Uravn., Gor'kij 2, 130-136 (1978)] proved that there are exactly 41 types of real singular points of a curve \(C_ 5\). \textit{L. V. Golubina} and \textit{D. A. Gudkov} [ibid. 6, 126-132 (1982)] raised the question of classification of sets of singular points of a unicursal curve \(C_ 5\), and solved the problem of classifying two-point sets. In the present paper the authors give a complete classification of three- point sets of singular points of a unicursal curve \(C_ 5\). real singular points of a curve; three-point sets of singular points of a unicursal curve Singularities of curves, local rings, Special algebraic curves and curves of low genus Classification of three-point sets of singular points for 5-th order unicursal 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 \(\pi \colon \mathcal{X}\to B\) be a family whose general fibre \(X_b\) is a \((d_1,\,\ldots,\,d_a)\)-polarization on a general abelian variety, where \(1\leq d_i\leq 2, i=1,\,\ldots,\,a\) and \(a\geq 4\). We show that the fibres are in the same birational class if all the \((m,\,0)\)-forms on \(X_b\) are liftable to \((m,\,0)\)-forms on \(\mathcal{X} \), where \(m=1\) and \(m=a-1\). Actually, we show a general criteria to establish whether the fibres of certain families belong to the same birational class. extension class of a vector bundle; holomorphic forms; Albanese variety; families of varieties; infinitesimal invariant Families, moduli, classification: algebraic theory, Torelli problem, Variation of Hodge structures (algebro-geometric aspects), Birational geometry, \(n\)-folds (\(n>4\)) On birationally trivial families and adjoint quadrics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be the surface obtained by blowing-up the complex projective plane at \(r\quad points\) in general position (0\(\leq r\leq 8)\). Such a surface is called a Del Pezzo surface (the author looks at \({\mathbb{P}}^ 1\times {\mathbb{O}}^ 1\) as an exceptional member of the family). A real structure on X is defined by a complex conjugation acting on X; it induces an involution of the Picard group P of X preserving the structure given by the intersection form and the canonical \(class\quad k.\) Its conjugacy class \(\gamma\) in the group of isometries of P leaving k fixed is an important invariant of the real structure. Actually the author associates with \(\tau\) a Dynkin diagram \(\Delta\) (t) and shows that a rather complete description of \(X_{{\mathbb{R}}}\), the fixed point set of the complex conjugation, can be obtained directly from the knowledge of \(\Delta\) (\(\tau)\). In particular, the mod 2 Betti numbers of \(X_{{\mathbb{R}}}\) can be computed in terms of invariants of \(\Delta\) (\(\tau)\). A careful analysis of blowing-up in the real domain also allows the author to describe \(X_{{\mathbb{R}}}\) up to homeomorphism. Further geometrical properties of \(X_{{\mathbb{R}}}\) are discussed, e.g. for \(r=7\), \(X_{{\mathbb{R}}}\) is described according the various possible configurations of the real plane quartic curve which is the branch locus of the double cover \(X\to {\mathbb{P}}^ 2\) given by the anticanonical map. Finally, the above methods are used to classify real forms of rational elliptic surfaces without multiple fibres. blowing-up; Del Pezzo surface; real structure; Picard group; configurations of the real plane quartic curve; anticanonical map; real forms of rational elliptic surfaces without multiple fibres Wall, CTC, Real forms of smooth del Pezzo surfaces, J. Reine Angew. Math., 47, 47-66, (1987) Real algebraic and real-analytic geometry, Special surfaces, Topological properties in algebraic geometry Real forms of smooth de 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 In this article, even if it is known for general case by \textit{L. Sharifan} and \textit{R. Zaare-Nahandi} [J. Pure Appl. Algebra 213, No. 3, 360--369 (2009; Zbl 1167.13001)], we give the explicit minimal free resolution of the associated graded ring of certain affine monomial curves in affine \(4\)-space based on the standard basis theory. As a result, we give the minimal graded free resolution and the Hilbert function of the tangent cone of these families in \(A^4\) in the simple form according by Sharifan and Zaare-Nahandi [loc. cit.]. minimal free resolution; monomial curve; Cohen-Macaulayness; Hilbert function of a local ring; tangent cone Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities of curves, local rings, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) On minimal free resolution of the associated graded rings of certain monomial curves: new proofs in \(A^4\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors give a new proof of the existence of a non-singular model for a variety in characteristic zero. More precisely they prove the following theorem: Let \(X\) be a normal projective variety and let \(D\) be a proper subvariety of \(X\). Then there exist a smooth projective variety \(M\), a strict normal crossing divisor \(R\subset M\) and a birational morphism \(f:M \to X\) with \(f^{-1} D=R\).
The proof uses induction on \(n= \dim X\) and it is based on the construction of a finite morphism \(g: (X',D') \to(P,S)\) from a suitable blow up \((X',D')\) of \((X,D)\) to a pair \((P,S)\), where \(P= \mathbb{P} ({\mathcal O}\oplus {\mathcal L})\) for some line bundle \({\mathcal L}\) over \(\mathbb{P}^{n-1}\) and \(S\) is the image of a section of \({\mathcal L}\) and \(g\) is branched over a sum of sections. existence of a non-singular model for a variety; characteristic zero F.A. Bogomolov and T. Pantev: ''Weak Hironaka Theorem'', Math. Res. Let., Vol. 3, (1996), pp. 299--307. Minimal model program (Mori theory, extremal rays) Weak Hironaka 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 We extend the finiteness result on the \(p\)-primary torsion subgroup in the Chow group of zero cycles on the selfproduct of a semistable elliptic curve obtained in joint work with S. Saito [see \textit{A. Langer} and \textit{S. Saito}, ``Torsion zero-cycles on the self-product of a modular elliptic curve'', Duke Math. J. 85, No. 2, 315-357 (1996)] to primes \(p\) dividing the conductor. On the way we show the finiteness of the Selmer group associated to the symmetric square of the elliptic curve for those primes. The proof uses \(p\)-adic techniques, in particular the Fontaine-Jannsen conjecture proven by Kato and Tsuji. torsion zero cycles; Selmer group of the symmetric square; Hyodo-Kato cohomology; selfproduct of a semistable elliptic curve Andreas Langer, Selmer groups and torsion zero cycles on the selfproduct of a semistable elliptic curve, Doc. Math. 2 (1997), 47 -- 59. Elliptic curves, \(p\)-adic cohomology, crystalline cohomology, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Algebraic cycles Selmer groups and torsion zero cycles on the selfproduct of a semistable elliptic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0682.00009.]
The paper considers the problem of description of an irreducible, complete curve over the complex numbers of genus \(g\geq 5\), having a linear series \(g^ 2_ d\) such that the plane curve given by the rational map determined by the \(g^ 2_ d\) is smooth or it has only nodes as singularities; then the dual of the plane curve and linear series \(g^ 2_ d\) are recovered from the Gauss map on the subvariety \(W_{d-2}\) of the jacobian of the curve. linear series; Gauss map; jacobian of the curve Jacobians, Prym varieties, Divisors, linear systems, invertible sheaves The Gauss map on subvarieties of Jacobians of curves with \(g^ 2_ d\)'s | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Langlands has been asserting that there should exist an analytic theory of automorphic functions for geometric Langlands theory of complex algebraic curves. Motivated by his insistence, the paper under review outlines an approach, which is detailed in the work of the author et al. [``An analytic version of the Langlands correspondence for complex curves'' Preprint, \url{arXiv:1908.09677}].
The Langlands program was initiated by Langlands in the late 1960s. Let \(F\) be a number field, meaning a finite extension of \(\mathbb{Q}\), or a function field, meaning the field of rational functions on a smooth proper curve \(C\) over a field field. Let \(G\) be a reductive group over \(F\). A protagonist of the program is automorphic forms, certain functions on \(G(F)\backslash G(\mathbb{A}_F)/K\) where \(\mathbb{A}_F\) is the ring of adeles of \(F\) and \(K\) is a compact subgroup of \(G(\mathbb{A}_F)\). One can find a commutative algebra of Hecke operators acting on automorphic forms. Then a crucial insight of Langlands was to study their spectral decomposition and describe it in terms of, roughly speaking, homomorphisms from the Galois group \(\text{Gal}(\overline{F}/F)\) to the Langlands dual group \(\check{G}\).
Then from the early 80s, the geometric Langlands program was developed by Drinfeld and Laumon where one shifts interest from functions on the double coset to sheaves on a certain algebraic stack (for example, the moduli stack \(\text{Bun}_G\) of principal \(G\)-bundles on the curve \(C\) in the unramified case): in the case of function fields, one can recover the original formulation of Langlands if Grothendieck's function-sheaf correspondence is applied. An important point of departure from the initial formulation is that the sheaf-theoretic formulation makes sense even for a complex algebraic curve where there is no function-sheaf correspondence. In the complex setting, Beilinson and Drinfeld reformulated the crux of the program as a spectral decomposition of Hecke functors on the (derived) category of sheaves on \(\text{Bun}_G\). In particular, the entire framework was completely algebraic and there didn't seem to be a room to find an analytic theory as for the classical setting.
Recently, Langlands made a proposal for an analytic theory of spectral decomposition of Hecke operators for complex curves in his paper ``On the analytical form of the geometric theory of automorphic forms'' (written in Russian). In the paper under review, Frenkel identifies some issues in Langlands' proposal with providing explicit and concrete examples and suggests a different analytic theory of automorphic forms based on the joint work with Etingof and Kazhdan.
The main suggestion is to study spectral decomposition properties of a commutative algebra different from the one of Hecke operators. To identify the commutative algebra, let \(K\) be the canonical line bundle on \(\text{Bun}_G\). Consider the algebra of holomorphic differential operators on \(K^{1/2}\), its complex conjugate acting on \(\overline{K}^{1/2}\), and their tensor product. This is the algebra of interest which will play the role analogous to the one of Hecke operators. Namely, the author notes that the algebra naturally acts on a certain Hilbert space of sections of the line bundle \(K^{1/2}\otimes \overline{K}^{1/2}\) over \(\text{Bun}_G\) and studies their spectral decomposition property. It is worth mentioning that this spectral problem was considered earlier by Teschner in ``Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence''.
One may ask why it is reasonable to think of the given problem as the spectral problem relevant for the geometric Langlands program. The author provides an answer in Remark 4.1; he recalls what happens to automorphic forms of \(G\) over a number field \(F\) where one has to study representations of \(G(\mathbb{Q}_p)\) and \(G(\mathbb{C})\). That is, while the spherical Hecke algebra comes in for representation theory of \(G(\mathbb{Q}_p)\), one uses the center \(Z(\mathfrak{g})\) of the universal enveloping algebra \(U(\mathfrak{g})\) for representation theory of \(G(\mathbb{C})\). In the geometric setting where we would discuss representation theory of \(G(\mathbb{C}(\!(t)\!))\), this leads us to the study of the center of \(U(\mathfrak{g}(\!(t)\!))\) or its variant. In fact, the earlier work of the author with Feigin ``Affine Kac--Moody algebras at the critical level and Gelfand--Dikii algebras'' identifies such a center at the critical level, which is very closely related to the global differential operators on \(K^{1/2}\).
The paper ends with formulating a concrete conjecture for the spectral problem. To identify it, recall that Beilinson and Drinfeld (in the unpublished book ``Quantization of Hitchin's integrable system and Hecke eigensheaves'') identified the algebra of differential operators on \(K^{1/2}\) in terms of the algebra of functions on the space of \(\check{G}\)-opers, based on the work of Feigin and the author mentioned above. With this identification, the spectral problem conjecturally leads to the study of opers whose monodromy takes values in the split real form of \(\check{G}\). The paper ends with the example of \(G=\text{GL}_1\) and \(C=E_i=\mathbb{C}/(\mathbb{Z} + \mathbb{Z}i)\) to provide a simple illustration of the conjecture. Langlands program; automorphic function; complex algebraic curve; principal \(G\)-bundle; Jacobian variety; differential operator; oper Geometric Langlands program (algebro-geometric aspects), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Geometric Langlands program: representation-theoretic aspects Is there an analytic theory of automorphic functions for 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 [For the entire collection see Zbl 0675.00006.]
Let X be a complete intersection of three quadrics in \({\mathbb{P}}^{2n+4}\). Inside the projective plane of quadrics, which contain X, the set of singular quadrics forms a curve C of degree \(2n+5\). The smooth points of C correspond to quadrics of rank \(2n+4\); the last have two systems of \((n+1)\)-planes. That defines a two-sheeted covering \(\pi: \tilde C\to C\), which corresponds to X. - It is known (cf. theorem 2.2) that:
1. The intermediate Jacobian of X is isomorphic as a principally polarized abelian variety to the Prym variety of the covering \(\pi: \tilde C\to C;\)
2. The covering \(\pi: \tilde C\to C\) determines the variety X up to isomorphism.
In the present report is proved the Torelli theorem for the Prym varieties of coverings \(\pi: \tilde C\to C\), where C is a plane curve of degree \( \geq 9\) (i.e. \(\dim(X)\geq 5)\). - The result is an improvement of the one of R. Friedman and R. Smith, in which the Torelli theorem is proved for a generic point (for degree \(\geq 7).\) The methods used are close to the methods of Welters (cf. the bibliography). complete intersection of three quadrics; intermediate Jacobian; principally polarized abelian variety; Prym variety Picard schemes, higher Jacobians, Complete intersections, Jacobians, Prym varieties Une démonstration élémentaire du théorème de Torelli pour les intersections de trois quadriques génériques de dimension impaire. (Elementary demonstration of the Torelli theorem for the intersections of three generic quadrics of odd dimension) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The goal of this paper is to characterize one of the mappings found by \textit{J. Hietarinta} and \textit{C. Viallet} [Phys. Rev. Lett. 81, 325-328 (1998)] from the point of view of the theory of rational surfaces. As its space of initial values, the author obtains a rational surface associated with some root of indefinite type. Conversely the author recovers the mapping from the surface and consequently obtains an extension of mapping to its non-autonomous version. By considering the intersection numbers of divisors, the author presents a method to calculate the algebraic entropy of the mapping. Finally it is shown that the degree of the mapping is given by the \(n\)th power of a matrix that is given by the action of the mapping on the Picard group. non-autonomous cases; rational surfaces; intersection numbers of divisors; algebraic entropy; Picard group Takenawa, T.: A geometric approach to singularity confinement and algebraic entropy. J. Phys. A 34, L95--L102 (2001) Dynamical aspects of statistical mechanics, Discrete version of topics in analysis, Birational automorphisms, Cremona group and generalizations A geometric approach to singularity confinement and algebraic entropy | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0577.14021. level curve of a rational function on a smooth algebraic surface; points of indeterminacy; smooth prime curve; rational functions of; \({\mathbb{C}}^*\)-type Kizuka, T.: Rational functions of ?*-type on the two-dimensional complex projective space. Tôhoku Math. J. 38, 123-178 (1986) Families, moduli of curves (analytic), Special surfaces, Holomorphic functions of several complex variables Rational functions of \({\mathbb{C}}^*\)-type on the two-dimensional complex 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 The classical Hurwitz formula for finite, separable morphisms \(\pi:X\to Y\) of nonsingular curves, is extended to the case when X is singular, as follows: If \(p_ a(X)\) is the arithmetic genus of X, g(Y) the genus of Y and d the degree of \(\pi\), then \(2p_ a(X)-2=d(2g(Y)-2)+\deg {\mathcal R}\) where \({\mathcal R}\) is the discriminant divisor associated with \(\pi\). - Applications are given to the computation of the rank of X, under suitable assumptions on its singularities and on the characteristic of the ground field. singular curves; rank of a curve; Hurwitz formula Nadia Chiarli, A Hurwitz type formula for singular curves, C. R. Math. Rep. Acad. Sci. Canada 6 (1984), no. 2, 67 -- 72. Singularities of curves, local rings, Enumerative problems (combinatorial problems) in algebraic geometry, Ramification problems in algebraic geometry A Hurwitz type formula for singular curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Wenn \(a, b, c\) die Coordinaten eines Punktes auf der Curve
\[
a^3 + b^3 + c^3 = (b + c) \; (c + a) \; (a + b)
\]
sind, und
\[
(b^2 + c^2 - a^2)\; x = (c^2 + a^2 - b^2) \;y = (a^2 + b^2 - c^2)\;z,
\]
so sind \(xyz\) die Coordinaten eines Punktes auf derselben Curve. Coordinates of a curve Plane and space curves, Cubic and quartic Diophantine equations Solution of a question (4752). | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is shown that general projections of the orbit of a given space curve form an orbit in the space of plane nodal curves. projections of the orbit of a given space curve Group actions on varieties or schemes (quotients), Birational geometry A note on general projections 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 Hasse-Witt-matrix; projective hypersurface; perfect field of characteristic p; Cartier operator; linear variety; generic hypersurface; invertible Frobenius operator DOI: 10.2140/pjm.1972.43.443 Special surfaces, Classical real and complex (co)homology in algebraic geometry, Rational points The Hasse-Witt-matrix of special projective varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Take \(f\in \mathbb {C}[x,y]\) and a non-zero linear form \(l \in \mathbb {C}[x,y]\). The polar curve of the affine curve \(C:= \{f=0\}\) with respect to the line \(l=0\) is the singular locus of the map \((f,l): \mathbb {C}^2 \to \mathbb {C}^2\). Here the author considers two resolutions at infinity of \(C\), an intermediate one \(\Pi\) and a total one \(\Pi _{\text{tot}}\). For these resolution he studies the intersection of the strict transform of the polar curve with the counterimage of divisors at infinity. His main results state how the strict transforms intersect the ``zone de rupture''. polar curve; affine plane curve; resolution at infinity of a plane curve; zone de rupture D. Ivanovski, Résolution à l'infini et courbes polaires affines, thèse de l'Université Paul Sabatier à Toulouse sous la direction de Françoise Michel, 2006 Plane and space curves, Pencils, nets, webs in algebraic geometry, Classification of affine varieties, Rational and birational maps Resolution at infinity and affine polar curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians M-curve; hyperelliptic curve; realisation; gemms; real points of a curve S. M. Natanzon, Automorphisms of the Riemann surface of an \?-curve, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 82 -- 83 (Russian). Curves in algebraic geometry, Classification theory of Riemann surfaces, Real-analytic manifolds, real-analytic spaces Automorphisms of a Riemann surface of an M-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 I: In these notes there are studied certain deformations of the hyperquadric system that determines a hyperelliptic curve with genus g imbedded in \({\mathbb{P}}^{g+2}\). A deformation is obtained by quadratic liftings of the hyperquadrics.
II: In this paper it is constructed a deformation of a hyperelliptic curve with genus g and is proved that Kodaira-Spencer's mapping of this deformation in (0) has maximal rang. deformation of a hyperelliptic curve Families, moduli of curves (algebraic), Formal methods and deformations in algebraic geometry, Special algebraic curves and curves of low genus Deformation of hyperelliptic curves. I; II | 0 |
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