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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 [For the entire collection see Zbl 0588.00014.]
Let \(Y| X\) be a covering of non-singular, projective algebraic curves over the finite field \({\mathbb{F}}_ r\), and suppose the point \(\infty\) of X splits completely in Y. Using the theory of Drinfel'd modules, for each ''finite'' point \({\mathfrak p}\) of X, a certain abelian covering \(Y_ 1({\mathfrak p})| Y\) may be specified by ramification properties.
The principal result (3.5) of the article is a Kummer-type criterion for the non-vanishing of the p-component of the Picard group of \(Y_ 1({\mathfrak p})\) \((p=char {\mathbb{F}}_ r)\). The crux of the proof consists in comparing the values at negative integers of \((a)\quad classical\) complex-valued L-series of X resp. Y, and \((b)\quad certain\) char-p- valued L-series introduced in an earlier work of the author.
Details are too complicated to be presented here. Also, we advice the reader first to consult the author's paper in Duke Math. J. 49, 377-384 (1982; Zbl 0473.12013), where a special case of the above problem is treated. Iwasawa theory of totally real number fields; covering of algebraic curves over a finite field; Drinfel'd modules; Picard group; L-series David Goss, The theory of totally real function fields, Applications of algebraic \?-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 449 -- 477. Coverings of curves, fundamental group, Totally real fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arithmetic theory of algebraic function fields, Finite ground fields in algebraic geometry The theory of totally real function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0668.00005.]
Let F be a field and A an F-algebra. A monic polynomial of degree n in F[\(\lambda\) ], g(\(\lambda)\), is said to have a conjugate splitting over A if there are elements \(\zeta\), \(\theta\) in A such that \(\zeta^ n\) is a nonzero element of F, and \(g(\lambda)=\sum^{n}_{i=1}(\lambda - \zeta^{-(n-i)}\theta \zeta^{n-i}) \) in A[\(\lambda\) ].
From the author's abstract: ``In this paper we establish a connection between the Clifford algebra of a binary form f(u,v) of degree \( n\) and the relative Brauer group B(F(C)/F) where C is the curve given by \(f(u,v)-w^ n=0\). We then use this connection to produce examples of conjugate splittings for division algebras of prime degree over local number fields.'' Clifford algebra of a binary form; relative Brauer group; conjugate splittings Haile, D.: On Clifford algebras, conjugate splittings, and function fields of curves. Israel math. Conf. proc. 1, 356-361 (1989) Brauer groups of schemes, Algebraic functions and function fields in algebraic geometry, Algebras and orders, and their zeta functions, General binary quadratic forms On Clifford algebras, conjugate splittings, and function fields 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 It is well-known that the classical Weyl character formula for irreducible representations of a compact Lie group is a consequence of the classical Lefschetz fixed point formula applied to the corresponding generalized flag variety. In the context of Arakelov geometry, a fixed point formula of Lefschetz type has recently been formulated and proved by \textit{K. Köhler} and \textit{D. Roessler} [Invent. Math. 145, 333-396 (2001; Zbl 0999.14002)]. Again by applying that formula to generalized flag varieties (now over Spec(\(\mathbb Z\))), the authors present, in the paper under review, a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over \(\text{Spec}(\mathbb Z)\) [see \textit{J. C. Jantzen}, ``Representations of algebraic groups'' (1987; Zbl 0654.20039)], except for the three exceptional cases \(G_2\), \(F_4\) and \(E_8\). The proof involves the computation of the equivariant Ray Singer analytic torsion associated with certain vector bundles on the corresponding complex generalized flag variety.
In the special case the flag variety is Hermitean symmetric, this computation has been carried out by \textit{K. Köhler} in a previous paper [J. Reine Angew. Math. 460, 93-116 (1995; Zbl 0811.53050)]. In the general case, the authors decompose the flag variety into Hermitean symmetric flag varieties by various fibrations and inductively apply a special case of a formula of \textit{X. Ma} [Ann. Inst. Fourier 50, 1539-1588 (2000; Zbl 0964.58025)], which relates the equivariant analytic torsion of the total space of a fibration to the equivariant analytic torsion of its base and its fibre. The authors in fact give a proof of this special case based on the arithmetic Lefschetz formula again. In the final chapter of the paper under review, the authors use the Jantzen sum formula to derive explicit formulae for the global height of ample line bundles on an arbitrary generalized flag variety. This way they recover formulas for projective spaces proved by \textit{H. Gillet} and \textit{C. Soulé} [Ann. Math. (2) 131, 163-203 (1990; Zbl 0715.14018) and 205-238 (1990; Zbl 0715.14006)], and for quadrics proved by \textit{J. Cassaigne} and \textit{V. Maillot} [J. Number Theory 83, 226-255 (2000; Zbl 1001.11027)]. integral representations of Chevalley schemes; Jantzen sum formula; Arakelov geometry; generalized flag variety; equivariant Ray-Singer torsion; Hermitean symmetric space; arithmetic Lefschetz formula Kai Köhler and Damian Roessler, A fixed point formula of Lefschetz type in Arakelov geometry. I. Statement and proof, Invent. Math. 145 (2001), no. 2, 333 -- 396. , https://doi.org/10.1007/s002220100151 K. Köhler and D. Roessler, A fixed point formula of Lefschetz type in Arakelov geometry. II. A residue formula, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 1, 81 -- 103 (English, with English and French summaries). Christian Kaiser and Kai Köhler, A fixed point formula of Lefschetz type in Arakelov geometry. III. Representations of Chevalley schemes and heights of flag varieties, Invent. Math. 147 (2002), no. 3, 633 -- 669. Arithmetic varieties and schemes; Arakelov theory; heights, Grassmannians, Schubert varieties, flag manifolds, Determinants and determinant bundles, analytic torsion, Representation theory for linear algebraic groups, Heights A fixed point formula of Lefschetz type in Arakelov geometry. III: Representations of Chevalley schemes and heights of flag varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth curve of genus \(g \geq 4\). Let \(\text{Cliff}(X)\) be its Clifford index. Here the author gives conditions on a base-point-free line bundle \(L\) on \(X\) which assure that \(L\) is birationally very ample. All statements are sharp, e.g. if \(L\) is a \(g^r_d\), \(r \geq 3\), \(g-1-d+r \geq 1\), and \(\text{Cliff}(L) =\text{Cliff} (X) +4\), then \(L\) is birationally very ample, except \(5\) exceptional pairs \((X,L)\). Clifford index; Clifford index of a curve; linear series Special algebraic curves and curves of low genus, Special divisors on curves (gonality, Brill-Noether theory) Remarks on the Clifford index 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 Let \(\gamma\) be a non-negative integer. A pointed curve \((C,P)\) is called \textit{\(\gamma\)-hyperelliptic} if the Weierstrass semigroup \(H(P)\) at \(P\) has exactly \(\gamma\) even gaps; here by a \textit{curve} we mean a projective, non-singular, irreducible algebraic curve defined over an algebraically closed field of characteristic zero. Then the very semigroup property of \(H(P)\), see e.g. [\textit{F. Torres}, Semigroup Forum 55, No. 3, 364--379 (1997; Zbl 0931.14017)], implies \(H(P)=2\tilde H\cup\{u_\gamma<\ldots<u_1\}\cup\{2g+i: i\in{\mathbb N}_0\}\), where \(g=g(C)\) is the genus of \(C\), \(\tilde H\) is a numerical semigroup of genus \(\gamma\), and the \(u_i's\) are odd integers with \(u_1<2g\); in addition \(u_\gamma\geq 2g-4\gamma+1\, (*)\).
This paper deals with the question \(D(C,P,\gamma)\): If \((C,P)\) is \(\gamma\)-hyperelliptic, do exist a double covering of curves \(F:C\to \tilde C\) which is ramified at \(P\)? If the answer is positive, the Weierstrass semigroup at \(F(P)\) equals \(\tilde H\) above so that \(g(\tilde C)=\gamma\); in particular, \(g\geq 2\gamma\) by the Riemann-Hurwitz formula. If \(\gamma\leq 3\), \(D(C,P,g,\gamma)\) is indeed true; see [\textit{J. Komeda}, Semigroup Forum 83, No. 3, 479--488 (2011; Zbl 1244.14025)] and the references therein. From now we let \(\gamma\geq 4\).
If \(g(C)\geq 6\gamma+4\), \(D(C,P,\gamma)\) is true [\textit{F. Torres}, Manuscr. Math. 83, No. 1, 39--58 (1994; Zbl 0838.14025)]. To see this we consider the linear system \(D_{\gamma+1}:=|(6\gamma+2)P|\) which has dimension \(2\gamma+1\) by \((*)\) above (indeed, this follows provided that \(g(C)\geq 5\gamma+1\)). Then the degree \(t\) of the morphism \(F_1: C\to {\mathbb P}^{2\gamma+1}\) associated to \(D_{\gamma+1}\) is at most \(2\). If \(t=2\), the claimed answer follows. On the contrary, Castelnuovo's genus bound gives \(g(C)\leq \pi_0(6\gamma+2,2\gamma+1)=6\gamma+3\), a contradiction.
The present paper proves that \(D(C,P,\gamma)\) is even true whenever \(g(C)= 6\gamma+1, 6\gamma\). As a matter of fact, \(D(C,P,\gamma\) is also true for \(g(C)=6\gamma+3, 6\gamma+2\) which follow from the techniques used by the authors here.
Let \(g(C)=6\gamma+1\) and notation as above. We claim that \(t=2\). Let \(C_0:=F_1(C)\) and assume \(t=1\). Then \(g(C)=g(C_0)\leq g_a(C_0)\leq c_0(6\gamma+2, 2\gamma+1)=6\gamma+3\), where \(g_a\) is the arithmetic genus of \(C_0\). If \(g(C)=g_a(C_0)\), \(C\) is isomorphic to \(C_0\) and hence \((6\gamma+2)P\sim P+D\) with \(D\) a divisor on \(C\) such that \(P\not\in \text{supp}(D)\). Hence \(6\gamma+1\in H(P)\), a contradiction according to \((*)\) above. Now the number \(\pi_1(6\gamma+2,2\gamma+1)\) in Theorem 3.15 [\textit{J. Harris}, Curves in projective space. Montreal, Quebec, Canada: Les Presses de l'Universite de Montreal (1982; Zbl 0511.14014)] equals \(6\gamma+1\); hence \(C_0\subseteq S\subseteq {\mathbb P}^{2\gamma+1}\), being \(S\) a surface of degree \(2\gamma\) [loc. cit.]. Then by considering the minimal resolution of \(S\) and the adjunction formula, \(g_a(C_0)\) can be computed. Finally the proof that \(t=1\) is a contradiction proceeds via a carefully study of the condition \(1\leq g_a(C_0)-g(C_0)\leq 2\).
The case \(g(C)=6\gamma\) is treated in a similar way; however, here the linear system \(|(6\gamma-2)P|\), which is of dimension \(2\gamma-1\), is used. Weierstrass semigroup; double cover of a curve; rational ruled surface Plane curves of degree 4 Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group, Rational and ruled surfaces On \(\gamma \)-hyperelliptic Weierstrass semigroups of genus \(6\gamma +1\) and \(6\gamma \) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(E\) be an irreducible plane curve over the field \(\mathbb{C}\) of complex numbers, let \(\widetilde \nu : \widetilde E \to E \subset \mathbb{P}^ 2\) be the normalization morphism, and let \(\overline D\) be an arbitrary curve in \(\mathbb{P}^ 2\) such that \(\overline E \not \subset \overline D\). The main result of this paper says that if \(\overline E\) and \(\overline D\) intersect transversely, then
\[
\widetilde \nu_ * : \pi_ 1 (\widetilde E \backslash \widetilde \nu^{-1} (\overline E \cap \overline D)) \to \pi_ 1 (\mathbb{P}^ 2 \overline D)
\]
is an epimorphism. Lefschetz theorem; fundamental group of complement of a plane curve Homotopy theory and fundamental groups in algebraic geometry, Fundamental group, presentations, free differential calculus, Curves in algebraic geometry On the Lefschetz theorem for the complement of a curve 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 As an outcome of his study of Zariski pairs in the moduli of plane curves of degree six with three cusps of the type \(y^4-x^3=0,\) the author has observed several interesting features of the variety formed by these curves, which he describes in this paper. His first observation is that there are exactly two irreducible components, one consisting of torus curves and the other of non-torus curves. Here, a \textit{torus curve} (\textit{of type \((p,q)\)}) is a curve having an irreducible equation \(g^p+h^q=0\) of degree, say \(n,\) with \(g\) and \(h\) polynomials such that \(p\text{ deg}(g)=n=q\text{ deg}(h).\)
The author observes next that the duals of these curves are sextics with six cusps and three nodes. Denoting the moduli of sextics with six cusps and three nodes by \(\mathcal M,\) the author goes on to study the structure of \(\mathcal M\) in considerable detail.
Here are some of the results he obtains: The space \(\mathcal M\) has a natural compactification \(\widehat{\mathcal M}.\) This compactification is invariant under the operation of taking the dual of a curve, giving \(\widehat{\mathcal M}\) a natural symmetry. The duality preserves the torus or non-torus nature of a curve. The generic Alexander polynomial and the fundamental group for a curve in \(\widehat{\mathcal M}\) are determined explicitly. moduli of plane curves; Zariski pairs; cusp; torus curve; dual of a curve; non-torus curve M. Oka, Geometry of cuspidal sextics and their dual curves, to appear. Singularities of curves, local rings, Families, moduli of curves (algebraic), Plane and space curves Geometry of cuspidal sextics and their dual 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 After a brief review of maps, hypermaps and Belyi function, the author studies a linear representation of the automorphism group of a Riemann surface \(X\) generated by elements \(\alpha_1, \alpha_2, \alpha_3: =(\alpha_1 \alpha_2)^{-1}\). Then he applies the results obtained to the representation of the automorphism group of a Riemann surface on its abelian differentials \(\Omega(X)\) of the first kind, and gets relations among the multiplicities of the eigenvalues of the representation matrix, the degree of the irreducible representation, the number of its occurring times in the representation on \(\Omega(X)\) and the orders of \(\alpha_i\). As examples, the calculations of the representation for several groups are given.
The author also mentions the canonical curve \(C(X)\) of \(X\), and the informations at the fixed points on \(C(X)\) of the automorphisms \(\alpha_1, \alpha_2, \alpha_3\) is used to find the equations corresponding to a given map. hypermaps; Belyi function; automorphism group of a Riemann surface; canonical curve; fixed points Streit, Manfred, Homology, Belyĭ\ functions and canonical curves, Manuscripta Math., 90, 4, 489-509, (1996) Riemann surfaces; Weierstrass points; gap sequences, Birational automorphisms, Cremona group and generalizations, Differentials on Riemann surfaces, Coverings of curves, fundamental group Homology, Belyĭfunctions and canonical curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper rings of the type: \(R=k[u^{n_ d}, t^{n_ 1}u^{n_ d-n_ 1},\dots,t^{n_{d-1}}u^{n_ d-n_{d-1}} ,t^{n_ d}]\) are considered, where \(n_ 1,\dots,n_ d\) are positive integers, \(n_ d \geq 2\), \(GCD(n_ 1,\dots,n_ d)=1\) and \(k\) is a field with \(\text{char} k=0\). \(R\) is the coordinate ring of a monomial curve in \(\mathbb{P}^ d_ k\), i.e. the image of \(\mathbb{P}^ 1_ k\) via the map \((t:u)\to(u^{n_ d}:t^{n_ 1}u^{n_ d-n_ 1}:\dots:t^{n_{d-1}} u^{n_ d- n_{d-1}}:t^{n_ d})\) . The aim of the paper is to study the minimal number \(\mu\) of generators of \(\text{Der}_ k(R_{\mathfrak p})\), where \({\mathfrak p}\subseteq R\) is the maximal homogeneous ideal of \(R\). -- If \(S \subseteq \mathbb{N}^ 2\) is the semigroup \(S=\langle(0,n_ d),(n_ 1,n_ d-n_ 1),\dots,(n_{d-1}, n_ d-n_{d-1}), (n_ d,0)\rangle\) and \(S_ 1,S_ 2\) are its projection on \(\mathbb{N}\), we can denote \(R=k[S] \subseteq k[t,u]\), and in the same way we define the rings \(k[S_ 1]\), \(k[S_ 2]\) in \(k[t],k[u]\), respectively. Let \(\mu_ i=\text{card}\{x \in \mathbb{Z}] | x+(S_ i)-\{0\} \subseteq S_ i\}\), \(i=1,2\). Then the main result in the paper is the following:
\(R\) is Cohen-Macaulay \(\Rightarrow \mu=\mu_ 1'+ \mu_ 2'+2\), where \(\mu_ i' =\mu_ i\) if \(S_ i \neq \mathbb{N}\) and \(\mu_ i=1\) otherwise.
Bounds on \(\mu\) are given when \(R\) is Buchsbaum and also in the general case. minimal number of generators of derivation ring; Buchsbaum ring; coordinate ring of a monomial curve Molinelli S., Tamone G.,On the derivations of the homogeneous coordinate ring of a monomial curve in P k d . Comm. in Algebra20 (1992), 3279--3300. Morphisms of commutative rings, Plane and space curves On the derivations of the homogeneous coordinate ring of a monomial curve in \(\mathbb{P}_{k}^{d}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 provides the details to the results announced in [Dokl. Math. 59, No. 1, 96--98 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 364, No. 5, 596--598 (1999; Zbl 0976.14032)]. virtual Mordell-Weil rank; rationality of a curve; minimal bundle of non-hyperelliptic curves M.-H. SAITO - V. NGUYEN KHAC, On Mordell-Weil lattices for nonhyperelliptic fibrations of surfaces with zero geometric genus and irregularity, Izv. Ross. Akad. Nauk Ser. Mat., 66 (2002), pp. 137-154. Zbl1053.14043 MR1942097 Rational and ruled surfaces, Families, moduli of curves (algebraic), Rational and unirational varieties On Mordell-Weil lattices for non-hyperelliptic fibrations on surfaces with zero geometric genus and irregularity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result in this paper is to give some necessary conditions on \(C\subset \mathbb{P}^{g-1}\), the canonical embedding of a non-hyperelliptic curve, which imply that \(C\) is a hyperplane section of a surface distinct of a cone. One of these conditions is the vanishing of the cohomology groups \(H^1(\mathbb{P}^{g-1}, I^2_C(k))\), for \(k\geq 3\). The paper contains several results and a conjecture on this vanishing property. embedding of a non-hyperelliptic curve; hyperplane section; vanishing property Wahl, J, The cohomology of the square of an ideal sheaf, J. Algebraic Geom., 6, 481-511, (1997) Complete intersections, Vanishing theorems in algebraic geometry, Embeddings in algebraic geometry On cohomology of the square of an ideal sheaf | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is known, according to a result of Ellingsrud and Peskine, that there is an upper bound for the degree of smooth surfaces \(S\subset{\mathbb P}^4\) not of general type. The use of generic initial ideal theory to bound \(\chi{\mathcal O}_S\) and the sectional genus of \(S\) has allowed several authors to successively find better bounds for that degree. This paper presents one more improvement by finding a sharper bound on the number of sporadic zeros of a general hyperplane section of \(S\). By definition, a sporadic zero of a curve \(C\subset{\mathbb P}^3\), with coordinates \(x_0,x_1,x_2,x_3\) with the reverse lexicographical order, is a monomial \(x_0^ax_1^bx_2^c\) which does not belong to the generic initial ideal of \(C\), but such that its product with some positive power of \(x_2\) does. surface not of general type; degree of smooth surfaces; generic initial ideal; sporadic zero of a curve DOI: 10.1023/A:1000150519995 Families, moduli, classification: algebraic theory, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) A smooth surface in \(\mathbb{P}^4\) not of general type has degree at most 66 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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, by using the concept of resolvent of a prime ideal introduced by Ritt, we give methods for constructing a hypersurface which is birational to a given irreducible variety and birational transformations between the hypersurface and the variety. In the case of algebraic curves, this implies that for an irreducible algebraic curve \(C\), we can construct a plane curve which is birational to \(C\). We also present a method to find rational parametric equations for a plane curve if it exists. Hence we have a complete method of parametrization for rational algebraic curves. constructing a hypersurface which is birational to a given irreducible variety; construct a plane curve birational to a given curve; geometric modeling; resolvents; Gröbner bases; rational parametric equations Gao, X S; Chou, S C, On the parameterization of algebraic curves, Journal of Applicable Algebra in Engineering, Communication and Computing, 3, 27-3, (1992) Computational aspects of algebraic curves, Plane and space curves, Computational aspects of algebraic surfaces, Hypersurfaces and algebraic geometry, Special algebraic curves and curves of low genus On the parameterization 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 By a Picard curve the authors understand a projective plane curve with affine equation \(y^3=p(x)\), where \(p\) is a square-free polynomial of degree 4. Picard curves have genus 3 (being non-singular plane quartics) and a unique point \(\infty\) at infinity; hence any divisor \(D\) of degree 0 on a Picard curve is linearly equivalent to a divisor \(D^*-d^*\infty\), where \(d^*=\deg D^* \leq 3\), and \(D^*\) is an effective divisor supported on finite points, non-collinear if \(d^*=3\). The first part of the paper sketches an algorithm for effecting this reduction, i.e. for finding \(D^*\) given \(D\). The second part obtains an explicit description of the Jacobian of a Picard curve, using the divisors \(D^*-d^*\infty\) as representatives of classes of divisors of degree zero, analogous to the description of the Jacobian of a hyperelliptic curve given by \textit{D. Mumford} [``Tata Lectures on Theta. II: Jacobian theta functions and differential equations'', Prog. Math. 43 (1984; Zbl 0549.14014)].
The paper by \textit{E. R. Barreiro, J. Estrada Sarlabous} and \textit{J.-P. Cherdieu} [in: Coding theory, cryptography and related areas, Proc. Int. Conf., Guanajuato 1998, 13-28 (2000)] is a sequel to the paper under review, in which some improvements are made and some topics clarified. Picard curve; Jacobian Sarlabous, J. Estrada; Barreiro, E. Reinaldo; Barceló, J. A. Piñeiro: On the Jacobian varieties of Picard curves: explicit addition law and algebraic structure. Math. nachr. 208, 149-166 (1999) Computational aspects of algebraic curves, Jacobians, Prym varieties, Computational aspects of higher-dimensional varieties On the Jacobian varieties of Picard curves: Explicit addition law and algebraic structure | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given a smooth irreducible projective curve \(X\) defined over a \(p\)-adic field \(k\), the authors are interested in describing all central division algebras over \(k(X)\). The goal is to compute the \(m\)-torsion part of the Brauer group of \(X\) for any \(m\) and to find an explicit representation of the generators of \(_m\text{Br}(X)\) by central division algebras.
This program has been developed for elliptic curves \(X\) provided \(m=2\) and \(p\neq 2\) in a previous paper by \textit{V. I. Yanchevskij} and \textit{G. L. Margolin} [St. Petersb. Math. J. 7, No. 3, 473-505 (1996); translation from Algebra Anal 7, No. 3, 200-239 (1995)]. The paper under review treats the case of hyperelliptic curves with good reduction (also for \(m=2\)). It turns out that all the generators of \({}_2Br(X)\) can be represented by quaternion algebras.
See a recent paper by \textit{G. Margolin, U. Rehmann} and \textit{V. Yanchevskij} [``Quaternion generation of the 2-torsion part of the Brauer group'', Preprint (Bielefeld 1997)] for the case of a hyperelliptic quintic over a non-dyadic field without any restriction on its reduction. central division algebras over the function field of a curve; Brauer group; elliptic curves V. I. Yanchevskiĭ and G. L. Margolin, Brauer groups of local hyperelliptic curves with good reduction, Algebra i Analiz 7 (1995), no. 6, 227 -- 249 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 6, 1033 -- 1048. V. I. Yanchevskiĭ and G. L. Margolin, Erratum: ''Brauer groups of local hyperelliptic curves with good reduction'', Algebra i Analiz 8 (1996), no. 1, 237 (Russian). Brauer groups of schemes, Elliptic curves, Quaternion and other division algebras: arithmetic, zeta functions The Brauer groups of local hyperelliptic curves with good reduction | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be the field of fractions of a curve over \(R\) where \(R\) is the henselization of a regular local ring on an algebraic curve over an algebraically closed field of characteristic zero. In this paper, the classification of division algebras \(D\) of finite dimension over their center \(K\) is addressed, restricting to those fields \(K\) that have the \textit{exponent = degree} property for division algebras. One says that \(K\) has the \textit{exponent = degree} property if for any central \(K\)-division algebra \(D\) the exponent of the class \([D]\) in the Brauer group \(\text{Br}(K)\) is equal to the degree \(\sqrt {(D:K)}\) of the division algebra. -- The main result of this paper is formulated as follows:
Let \(k\) be an algebraically closed field of characteristic 0, and let \(X\) be an algebraic curve defined over \(k\). Let \({\mathfrak O}_{p,X}\) be the local ring at a regular point \(p\in X\) with the residue field \(k\), and let \(R= {\mathfrak O}^h_{p,X}\) be the henselization of \({\mathfrak O}_{p,X}\). Let \(C\) be an affine algebraic curve over \(R\) with a structure morphism \( \pi: C\to \text{Spec} (R)\) such that \(\pi\) is flat, of finite type and the fibers of \(\pi\) are algebraic curves, and \(C\) is connected. Then \(\pi\) has 2 fibres: the closed fiber \(\pi: C_0\to x_0\) over the closed point \(x_0\in \text{Spec} (R)\) is an algebraic curve over \(k\), and the open fiber \(\pi: C_\eta \to\eta\) over the open point \(\eta\in \text{Spec} (R)\) is an algebraic curve over the quotient field of \(R\). Assume that \(C_\eta\) is integral with \(K= K(C_\eta)\) the field of fractions.
Theorem. Assume that \(K\) has the \textit{exponent = degree} property. Let \(D\) be a central finite dimensional \(K\)-division algebra with exponent\((D)=n\). Then \(D\) is a cyclic algebra of degree \(n\).
The proof is along the line of the author's earlier articles [\textit{T. J. Fort}, Pac. J. Math. 147, No. 2, 301-310 (1991; Zbl 0668.16010) and \textit{T. J. Ford} and \textit{D. Saltman}, in: Ring theory 1989, Proc. Symp. Workshop, Jerusalem 1988/89, Isr. Math. Conf. Proc. 1, 320-336 (1989; Zbl 0696.16012)]. Brauer group of a curve; henselization of a regular local ring Ford, T.J.: The Brauer group of a curve over a strictly local discrete valuation ring. Israel J. Math. 96, 259--266 (1996) Brauer groups of schemes, Finite-dimensional division rings, Curves in algebraic geometry The Brauer group of a curve over a strictly local 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 The spin moduli space \(S_ g\) in genus \(g\) is the space parametrizing all couples (smooth genus \(g\) algebraic curve \(C\), theta characteristic on \(C)\). It has a natural structure of an algebraic variety and a well behaved compactification of \(S_ g\) was introduced by the author in ``Moduli of curves and theta characteristics'' (Lectures on Riemann surfaces, World Scientific, Singapore 1989). There were described also natural classes in the Picard group of this compactification. The answer to the problem whether these classes generate the Picard group is still not known. The author gives a complete answer to the question of what relations they satisfy. Finally it is shown that the Picard group of spin moduli space contains 4 torsion. generation of Picard group; spin moduli space; algebraic curve; theta characteristic M. Cornalba, A remark on the Picard group of spin moduli space,Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 2 (1991), 211--217. Families, moduli of curves (algebraic), Picard groups, Algebraic moduli problems, moduli of vector bundles, Theta functions and abelian varieties A remark on the Picard group of spin moduli 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 author determines a field \(E\) over \(\mathbb{Q}^{nr}_ p\), the maximal unramified extension of the \(p\)-adic number field \(\mathbb{Q}_ p\), over which the modular curve \(X_ 0(N)\) of level \(N\) has semi-stable reduction. \(E\) depends on the exponent of \(N\) at \(p\). The proof is based on Grothendieck's Galois criterion of semi-stable reduction [\textit{A. Grothendieck}, Sém. Géometrie algebrique, 1967-1969, SGA 7 I, Exposé 9, Lect. Notes Math. 288, 313-523 (1972; Zbl 0248.14006)], the decomposition theorem of the monodromy representation on the modular curve due to \textit{H. Carayol}, and the classification of Galois representation of degree 2 due to \textit{G. Henniart}. Jacobian; semistable reduction of modular curve; Galois representation Modular and Shimura varieties, Ramification and extension theory, Arithmetic ground fields for curves, Arithmetic ground fields (finite, local, global) and families or fibrations An extension of \(\mathbb{Q}_ p^{nr}\) on which \(J_ 0(N)\) is semi-stable | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X_0(N)/ \mathbb{Q}\) denote the modular curve associated with the subgroup \( \Gamma_0 (N)\) of \(Sl_2 (\mathbb{Z})\) and let \(J_0 (N)/ \mathbb{Q}\) be the Jacobian of \(X_0 (N)/ \mathbb{Q}\). The cusps of \(X_0 (N)\) generate the subgroup \(C_N\) of \(\mathbb{Q}\)-rational cuspidal points of \(J_0 (N)\). According to a result of Manin \(C_N\) is a subgroup of \(J_N\), the torsion subgroup of \(J_0 (N) (\mathbb{Q})\). For a prime \(p\) let \(\Phi_N (p)\) denote the group of components of the Néron model of \(J_0 (N)_{\mathbb{Q}_p}/ \mathbb{Q}_p\). If \(p = N\) the groups \(C_N\), \(J_N\) and \(\Phi_N (p)\) are pairwise isomorphic. In general this is not the case, however it is shown that for \(N = p^r\) with \(p \equiv 1 \pmod {12}\) the prime-to-\(2p\) parts of the groups \(C_N\) and \(J_N\) coincide. The paper also provides bounds for the orders of the groups \(C_N\), \(J_N\), and \(\Phi_N (p)\) and describes the reduction map \(\pi_{N,p} : J_N \to \Phi_N (p)\). Jacobian of modular curve; rational cuspidal points D. J. Lorenzini, Torsion points on the modular Jacobian \(J_{0}(N)\), Compositio Math., 96 (1995), 149-172. Jacobians, Prym varieties, Modular and Shimura varieties Torsion points on the modular Jacobian \(J_ 0 (N)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This very well written paper explains a way to compute the period matrix of a real algebraic curve when it is given via uniformization, and not via equations. It describes an algorithm which has as input Möbius transformations freely generating a Fuchsian group \(G\) of the second kind, and as output the period matrix of the real curve \(X = \Omega (G)/G\), where \(\Omega(G)\) is the domain of discontinuity of \(G\). This accounts for real curves of genus \(g\) which separate their complexification, the case where the real curve does not separate may be reduced to the former using a double covering. The main idea is to express the holomorphic abelian differentials on \(X\) via Poincaré series, where the summation is over \(G\). The needed facts (convergence of Poincaré series, basis for holomorphic differentials for \(X\) given by Poincaré series easily obtained from the generators of \(G)\) are explained following closely a paper by \textit{C. J. Earle} and \textit{A. Marden} [Ill. J. Math. 13, 202-219 (1969; Zbl 0169.102)]. -- The author gives an estimation of the error made when the Poincaré series is truncated, and indicates that there are in this estimation constants which may be difficult to calculate from the presentation of \(G\).
The paper closes with an example for genus 2, for which the computation time is given. compute the period matrix of a real algebraic curve; uniformization; Poincaré series Seppälä, M.: Computation of period matrices of real algebraic curves. Discrete comput. Geom. 11, No. 1, 65-81 (1994) Computational aspects of algebraic curves, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Special algebraic curves and curves of low genus, Symbolic computation and algebraic computation, Real algebraic sets Computation of period matrices of real algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study basic properties (e.g., algebraicity, reducibility, and dimension) of certain sets of matrices defined by means of the spark. spark of a matrix; generic determinantal variety; affine algebraic set; irreducible component; dimension; linear capacity SPARK (2014). http://www.spark-2014.org Determinantal varieties, Varieties and morphisms, Vector spaces, linear dependence, rank, lineability A note on spark 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 \(E\) be an elliptic curve over a finite field \(\mathbb{F}_p\) where \(p\) is a large prime number. Schoof's algorithm computes the number \(N\) of \(\mathbb{F}_p\)-rational points on \(E\) by calculating \(N\) modulo \(l\) for sufficiently many small prime numbers \(l\). The calculation of \(N\) modulo \(l\) uses the connection of \(N\) with the Frobenius endomorphism \(\pi_E\) acting on the \(l\)-adic Tate module of \(E\). The improvement of Atkin and Elkies makes use of those prime numbers \(l\) (called ``good primes''), where the eigenvalues of \(\pi_E\) are contained in \(\mathbb{Z}_l\). In this paper the authors propose a strategy which in addition makes use of higher powers \(l^n\) of ``good primes'' \(l\). Schoof algorithm; number of rational points; \(\ell\)-adic Tate module; elliptic curve over a finite field; Frobenius endomorphism Jean-Marc Couveignes and François Morain, Schoof's algorithm and isogeny cycles, Algorithmic number theory (Ithaca, NY, 1994) Lecture Notes in Comput. Sci., vol. 877, Springer, Berlin, 1994, pp. 43 -- 58. Computational aspects of algebraic curves, Rational points, Finite ground fields in algebraic geometry, Number-theoretic algorithms; complexity Schoof's algorithm and isogeny cycles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Usually, twistor spaces are certain complex 3--manifolds fibred over a Riemannian 4-manifold. It was shown by \textit{N. J. Hitchin} [Proc. Lond. Math. Soc., III. Ser. 43, 133-150 (1981; Zbl 0474.14024)] and by \textit{Th. Friedrich} and \textit{H. Kurke} [Math. Nachr. 106, 271-299 (1982; Zbl 0503.53035)] that only two of these spaces are projective algebraic. They are Fano varieties. The author of the paper under review relaxes the usual definition of twistor space as follows: he calls ``twistor space'' every complex projective algebraic 3-fold containing a twistor curve, that is a smooth rational curve with normal bundle \({\mathcal O}(1)\oplus{\mathcal O}(1)\). If there exists an irreducible family of such twistor curves sweeping out the whole twistor space, then he calls it ``minimal''. The two usual projective twistor spaces are, of course, minimal twistor spaces in the sense of the paper. These more general twistor spaces have many properties of usual twistor spaces. The author studies the following question:
Is a twistor space a Fano variety? Which Fano varieties arise in this way? Let \(X\) be a twistor space (in the sense of this paper) and \(X \subset \mathbb{P}_ n\) an embedding such that the twistor curves have degree \(d\). He shows that \(n < {d + 3 \choose 3}\). In particular, he obtains: the only twistor space admitting a projective embedding with \(d = 1\) is \(\mathbb{P}_ 3\). Then he focuses on the case \(d = 2\), the case of ``twistor conics''. He obtains the following theorem: if \(X \subset \mathbb{P}_ n\) is a twistor space in which twistor curves are conics (case \(d = 2)\), then one of the following holds:
(i) \(n = 9\), \(X \cong \mathbb{P}_ 3\) and \(X \subset \mathbb{P}_ 9\) is the Veronese embedding;
(ii) \(n \leq 8\), \(X\) is a Fano 3-fold of index 2;
(iii) \(n = 6\) and \(X\) is a quartic 3-fold whose generic hyperplane section is a rational scroll.
The author achieves this result by studying the generic hyperplane section of \(X\). An example of case (iii) is obtained by the birational image of \(\mathbb{P}_ 3\) under the rational map \(\mathbb{P}_ 3 \to \mathbb{P}_ 6\) given by the linear system of quadrics through a line in \(\mathbb{P}_ 3\). This defines a twistor space which is not minimal. The author conjectures that there is no other threefold in case (iii). In the rest of the paper he studies case (ii). The result is a complete list of index 2 Fano threefolds which are twistor spaces. rational curves on 3-folds; Fano variety of index two; twistor curve; minimal twistor spaces; Fano variety; Fano threefolds DOI: 10.1093/qmath/45.3.343 \(3\)-folds, Twistor theory, double fibrations (complex-analytic aspects), Fano varieties Twistor spaces and Fano 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 Prym varieties of curves with an involution without fixed points play an important role in the study of the Schottky problem for Jacobians and the moduli space of principally polarized abelian varieties (PPAV). Prym-Tjurin varieties were introduced by Tjurin and \textit{V. Kanev} [see e.g. Compos. Math. 64, 243-270 (1987; Zbl 0694.14009)] and are a generalization of Prym varieties. These varieties are important because every PPAV admits a ``Prym-Tjurin presentation'' and have been used to define stratifications of the moduli space of PPAV. Another possible generalization of Prym varieties are the abelian varieties \(P(\overline C, \sigma)\) associated with a smooth connected complete curve \(\overline C\) with an automorphism \(\sigma\) of prime order \(p\) or, in other words, a cyclic covering \(\pi: \overline C\to C=\overline C/ \langle\sigma \rangle\) of prime degree. These varieties are very useful in the study of curves with non-trivial automorphisms via their Jacobians. The abelian varieties \(P(\overline C, \sigma)\) are canonically Prym-Tjurin varieties of curves with asymmetric correspondence. If the order of \(\sigma\) is different from 2, this correspondence has fixed points and is non-effective (remark 2.2). On the other hand, the most important results obtained by Kanev (loc. cit.) are valid only for Prym-Tjurin varieties associated with symmetric effective correspondences without fixed points. The aim of this paper is to study Prym varieties \(P( \overline C,\sigma)\) and to obtain similar results to Kanev's for this case. In the first section, we study some properties of these varieties and give a criterion for the existence of principal polarizations on \(P(\overline C, \sigma)\) fulfilling certain conditions (theorem 2.5). When these polarizations do not exist, we prove that there are principally polarized abelian varieties \((P', \Theta')\) and isogenies \(P'\to P(C,\sigma)\) fulfilling those conditions (theorem 2.6). \textit{J. D. Fay} [in his book: ``Theta functions on Riemann surfaces'', Lect. Notes Math. 352 (1973; Zbl 0281.30013)] constructs, from the period matrices of the Jacobians, a PPAV and an isogeny when \(\sigma\) is free of fixed points.
In section 3 of this paper we prove the existence of a natural PPAV \((P' (\overline C,\sigma),\Theta')\) and an isogeny \(f:P'(\overline C,\sigma)\to P( \overline C,\sigma)\) without conditions on \(\sigma\), by fixing level structures on the Jacobians. Finally, in the last section we calculate the restriction of the polarization \(\Theta'\) on \(P'(\overline C,\sigma)\) to the curve \(\overline C\) when \(\sigma\) does not have fixed points. This result is interesting because, following the ideas developed earlier [\textit{E. Gómez González}, Math. Ann. 305, No. 1, 153-159 (1996; Zbl 0842.14033)], it can be useful in proving the existence of \(m\)-linear subvarieties of the projective space \(|2\Theta' |\) that meet the Kummer variety of \(P'(\overline C,\sigma)\) in \((m+2)\) points, for some \(m\in\mathbb{N}\). Prym-Tjurin varieties; Schottky problem; principally olarized abelian variety; automorphism of curve; cyclic covering Theta functions and curves; Schottky problem, Automorphisms of curves, Jacobians, Prym varieties, Coverings of curves, fundamental group, Picard schemes, higher Jacobians Prym varieties of curves with an automorphism of prime order | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the following a reduced irreducible complex space is called a complex variety. A compact complex variety X of algebraic dimension 0 is said to be simple, if there exists no covering family \(\{Z_ t\}_{t\in T}\) of analytic subvarieties \(Z_ t\) of X where \(0<\dim Z_ t<\dim X\) and T is a compact complex variety. (''Covering'' means that \(\cup_{t\in T}Z_ t=X.)\) X is called ''semisimple'', if it is isogenous to a product of simple compact complex varieties, where two compact complex varieties \(X_ 1\) and \(X_ 2\) of algebraic dimension 0 are said to be ''isogenous'', if there exists a finite to finite correspondence \(\Gamma \subseteq X_ 1\times X_ 2\) of \(X_ 1\) and \(X_ 2\). By a meromorphic fibre space \(f: X\to Y\) of compact complex varieties we mean that f is meromorphic and the natural projection \(\Gamma\) \(\to Y\) of the graph \(\Gamma \subseteq X\times Y\) of f has connected fibres. For a meromorphic map \(g: X\to W\) we can define the Stein factorization \(f: X\to Y,\) \(h: Y\to W,\) where f is a meromorphic fibre space and h is generically finite with \(g=h\circ f\). \(f: X\to Y\) is called the meromorphic fibre space associated with g. The main results in this paper are the following three theorems.
Theorem 1 (Semisimple reduction). Let X be a compact complex variety. Then there exists a meromorphic fibre space \(f: X\to Y\) such that Y is a semisimple compact complex variety and for any surjective proper meromorphic map \(g': X\to Y'\) with Y' semisimple there exists a unique meromorphic map \(h: Y\to Y'\) with \(g'=h\circ g\). Moreover, there exist Zariski open subsets \(U\subset X\) and \(V\subset Y\) such that \(g| U: U\to V\) is proper holomorphic. - The meromorphic map \(f: X\to Y\) in the above theorem is called a ''semisimple reduction''. For a compact complex variety X let \({\mathcal F}=(\rho: Z\to T,\quad \pi: Z\to X)\) be a covering family of X (that is \(\rho\) is a flat family of subvarieties of X with \(\cup_{t\in T}Z_ t=X,\) \(Z_ t=\rho^{-1}(t)),\) then \(^ t{\mathcal F}=(\pi: Z\to X,\quad \rho: Z\to T)\) can be considered as a covering family of T. Hence there is the associated meromorphic map \(\tau: X\to D_ T\) into the Douady space \(D_ T\) of T. Let Y be the image of X by \(\tau\). Then \(\tau: X\to Y\) is called the meromorphic map associated with the covering family \({\mathcal F}\). Let \(\{{\mathcal F}_{\alpha}\}_{\alpha \in A}\) be a set of covering families of X and \(f_{\alpha}: X\to Y_{\alpha}\) the associated meromorphic mappings. Then the normalized product \(f: X\to Y\) of \(f_{\alpha}'s\) is called the meromorphic map associated with \(\{\) \({\mathcal F}_{\alpha}\}_{\alpha \in A}\). (For the definition of the normalized product see lemma 1 of the paper.) A covering family \(\{Z_ t\}_{t\in T}\) is called a maximal family, if T is an irreducible component of the Douady space \(D_ X\) of X and the general member \(Z_ t\) is an irreducible maximal proper analytic subvariety of X. - Theorem 2. Let X be a compact complex variety in \({\mathcal C}\) (that is, X is a meromorphic image of a compact Kähler manifold). Let \(\{\) \({\mathcal F}_{\alpha}\}_{\alpha \in A}\) be the set of all maximal families of X whose members are of codimension \(>1\) in X. Then for a meromorphic fibre space \(f: X\to Y\) the following conditions are equivalent: (1) f is a semisimple reduction of X; (2) f is bimeromorphic to the meromorphic fibre space associated with \(\{\) \({\mathcal F}_{\alpha}\}_{\alpha \in A}\). - Theorem 3 (Relative semisimple reduction). Let \(f: X\to Y\) be a fibre space of compact complex varieties in \({\mathcal C}\). Then there exists a meromorphic fibre space \(g: X\to \bar X\) over Y with \(\bar X\) a compact complex variety over Y such that (1) for any surjective meromorphic Y-map \(g': X\to \bar X',\) where general fibres of \(\bar X'\to Y\) are semisimple, there exists a unique meromorphic Y-map \(u: \bar X\to \bar X'\) with \(g'=u\circ g,\) and (2) for general \(y\in Y\), g induces a meromorphic map \(g_ y: X_ y\to \bar X_ y\) which is the semisimple reduction of \(X_ y\). simple compact complex variety; meromorphic fibre space; semisimple reduction; Douady space; meromorphic image of a compact Kähler manifold FUJIKI (A.) . - Semi-simple reductions of compact complex varieties , Pub. Institut Élie Cartan, t. 8, 1983 , p. 79-133. MR 85m:32027 | Zbl 0562.32014 Compact analytic spaces, Moduli, classification: analytic theory; relations with modular forms, Holomorphic mappings and correspondences Semisimple reductions of compact complex 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 an overview over the entire collection see Zbl 0811.14019).]
This chapter discusses the algebro-geometric part of G. Faltings' proof of the Mordell conjecture. Following \S4 of \textit{G. Faltings'} original paper [cf. Ann. Math., II. Ser. 133, 549-576 (1991; Zbl 0734.14007)], and applying the product theorem presented in the previous chapter [cf. \textit{M. van der Put}, ``The product theorem'', in: these Proceedings, chapter VIII, 77-82 (1993; see this Zbl 0811.14006)], the author gives a detailed proof of the ampleness of certain rational line bundles on products of a subvariety of an abelian variety over an algebraically closed groundfield of characteristic zero. The author's carefully elaborated version, together with the preparatory material developed in the foregoing articles, is extremely helpful for the understanding of Faltings' approach, its complex structure, and its spectacular consequences. abelian variety; symmetric line bundles; ampleness of rational line bundles; Mordell conjecture; product theorem; products of a subvariety C. Faber , Geometric part of Faltings's proof , In: ''[EE]'', Chapitre IX, pp. 83 - 91 . MR 1289007 | Zbl 0811.14023 Rational points, Arithmetic ground fields for abelian varieties, Arithmetic varieties and schemes; Arakelov theory; heights, Algebraic theory of abelian varieties Geometric part of Faltings's proof | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 the third in a series of papers [J. Math. Sci., Tokyo 15, No. 1, 69--193 (2008; Zbl 1156.14038); Nagoya Math. J. 189, 63--138 (2008; Zbl 1169.14031)] whose aim is to re-work the theory of degenerations of abelian varieties by systematically using logarithmic geometry. In their framework, a family of abelian varieties degenerates to a ``log abelian variety'', which is in an appropriate sense both smooth, proper, and a group (which is of course impossible outside the log world!). As an example of the advantages of their set-up, one can consider the problem of obtaining modular interpretations of toroidal compactifications. After the work of Alexeev and Olsson (see [\textit{M. C. Olsson}, Compactifying moduli spaces for abelian varieties. Berlin: Springer (2008; Zbl 1165.14004)]), there is such an interpretation for the second Voronoi compactification of \(A_g\); however, the approach of Kajiwara-Kato-Nakayama provides in a sense a uniform modular interpretation of all toroidal compactifications at the same time. For all this we refer to the first two papers in the series (which treat the analytic and the algebraic theory, respectively).
In this paper the authors illustrate their theory by recasting the work of [\textit{P. Deligne} and \textit{M. Rapoport}, Lect. Notes Math. 349, 143--316 (1973; Zbl 0281.14010)] in their language -- that is, the case of degenerating elliptic curves. Deligne and Rapaport (working over a base where the level is invertible) provided a moduli-theoretic interpretation of compactified modular curves in terms of ``generalized elliptic curves''; a degenerate elliptic curve is for them a polygon of N rational curves, where \(N\) needs to depend on what kind of level structure the degenerate curve should support. In contrast, the log approach is far less ad hoc: Kajiwara-Kato-Nakayama can simply define \(X(N)\) as the moduli space parametrizing log elliptic curves equipped with an isomorphism of their N-torsion with \((\mathbb Z/N)^2\), and similarly for other kinds of level structure. (They repeatedly emphasize that the log approach allows one to treat smooth and degenerate elliptic curves on equal footing.) The main result is that the scheme \(X(N)\) defined by Deligne-Rapoport represents also the moduli functor defined using level structures on log elliptic curves.
In an appendix they explain precisely the relationship between the universal log elliptic curve over \(X(N)\) and the universal generalized elliptic curve. log abelian variety; log elliptic curve; generalized elliptic curve; modular curve Algebraic moduli of abelian varieties, classification, Families, moduli, classification: algebraic theory, Fibrations, degenerations in algebraic geometry Logarithmic abelian varieties. III: Logarithmic elliptic curves and modular curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope \(P\) in terms of the number of integral points in the interior of dilates of faces of dimension greater than or equal to \(\lceil\frac{\dim P}{2}\rceil\). lattice polytopes; lattice volumes of faces; degree of the discriminant of a smooth projective toric variety; number of integral points Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies A relation between number of integral points, volumes of faces and degree of the discriminant of smooth lattice polytopes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We present Bombieri's proof of the Riemann hypothesis for the zeta function of a curve over a finite field. We first briefly describe this zeta function and discuss the two-variable zeta function of Pellikaan. Then we give Naumann's proof that the numerator of this function is irreducible. Riemann hypothesis for a curve over a finite field; zeta function of a curve over a finite field; two-variable zeta function Zeta and \(L\)-functions in characteristic \(p\), Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses, Curves over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) The two-variable zeta function and the Riemann hypothesis for function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Take a smooth, connected and nondegenerate projective curve \(X\subset\mathbb{P}^r\), \(r\geq 2b+ 2\geq 6\), defined over an algebraically closed field with characteristic 0 and let \(\sigma_b(X)\) be the \(b\)-secant variety of \(X\). We prove that the \(X\)-rank of \(q\) is at least \(b+1\) for a non-empty codimension 1 locally closed subset of \(\sigma_b(X)\). secant variety; \(X\)-rank; tangential variety; join of two varieties; tangentially degenerate curve; strange curve E. Ballico, The b-secant variety of a smooth curve has a codimension 1 locally closed subset whose points have rank at least b + 1, Rivista Mat. Univ. Parma, 8 (2017), 345--351. arXiv: 1706.03633. Projective techniques in algebraic geometry, Plane and space curves The \(b\)-secant variety of a smooth curve has a codimension 1 locally closed subset whose points have rank at least \(b + 1\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathbb{F}_q\) be a finite field. The authors find nonsingular projective plane curves over \(\mathbb{F}_q\), of degree \(q+2\), whose \(\mathbb{F}_q\)-rational points fill the whole projective plane \(\mathbb{P}^2(\mathbb{F}_q)\). The degree \(q+2\) is minimal for a curve with this property. Let \(x,y,z\) be homogeneous coordinates of \(\mathbb{P}^2\), and \(U=y^qz-yz^q\), \(V=z^qx-zx^q\), \(W=x^qy-xy^q\). The curves have equations \(F_A=0\), where \(F_A=(x\;y\;z)A(U\;V\;W)^t\), for a \(3\times 3\) matrix \(A\) with entries in \(\mathbb{F}_q\), whose characteristic polynomial is irreducible. The automorphism groups of these curves are also determined.
The results of the paper are similar to those obtained by \textit{G. Tallini} in [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 30, 706--712 (1961; Zbl 0107.38104), Rend. Mat. Appl., V. Ser. 20, 431--479 (1961; Zbl 0106.35604)]. plane curve; finite field; rational point; automorphism group of a curve Keel, S., M\(^{\mathrm c}\)Kernan, J.: Rational Curves on Quasi-Projective Surfaces. Memoirs of the American Mathematical Society, vol. 140(669). American Mathematical Society, Providence (1999) Finite ground fields in algebraic geometry, Automorphisms of curves, Plane and space curves, Rational points, Curves over finite and local fields Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: Supplements to a work of Tallini | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 man durch eine Fläche zweiten Grades eine Ebene legt,so kann die dadurch entstehende Curve als die Focale einer neuen Fläche angesehen werden, welche um eine beliebige der ersteren homofocale Fläche zweiten Grades beschrieben ist. Solution; surface of \(2^nd\) degree; plane; resulting curve; focal of a surface; circumscribed; quadric homofocal to another one Surfaces in Euclidean and related spaces, Euclidean analytic geometry, Curves in Euclidean and related spaces, Families, moduli, classification: algebraic theory Solution of question 1034. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(V\to {\mathbb{P}}^ 1\) be a fibre space of Del Pezzo surfaces with suitable conditions. Each fibre contains a finite set of lines (e.g. 27 if it is a cubic surface), defining a finite branched covering \(C\to {\mathbb{P}}^ 1\). The incidence correspondence on C line meets line) defines an endomorphism i of Jac(C) satisfying a quadratic relation due to A. N. Tyurin; then the intermediate Jacobian JV of V is given by an isomorphism of principally polarized abelian varieties \(JV\cong \Pr ym-Tyurin(C,i).\) Here the right-hand side is studied by the author in the paper ''Principal polarizations of Prym-Tyurin varieties'' (preprint) and the theorem stated is proved by him in the paper ''Intermediate Jacobians and Chow groups of 3-folds with a pencil of Del Pezzo surfaces'' (preprint)]. intermediate Jacobians of threefolds; Del Pezzo surfaces; generalized Prym varieties; intermediate Jacobian Picard schemes, higher Jacobians, \(3\)-folds, Special surfaces Intermediate Jacobians of threefolds with a pencil of Del Pezzo surfaces and generalized Prym varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A curve X admits a deformation into a monomial curve \(X_ 0\) with the same semigroup of values S(X) and the same multiplicity E(X) as X if and only if E(X) and S(X) are compatible.
Here a numerical semigroup \(\Gamma \subset {\mathbb{Z}}_+\) is said to be compatible with a multiplicity sequence E of a curve if there exists a monomial curve X with \(S(X)=\Gamma\) and having E as its multiplicities sequence. The main tools are the Hamburger-Noether matrices associated to quadratic tranformations of algebroid curves, Arf closures of such curves and a construction of deformations of curves due to Teissier. deformation into a monomial curve; semigroup of values; multiplicity; algebroid curves Castellanos, J.: ''A relation between the sequence of multiplicities and the semigroup of values of an algebroid curve'', J.P.P.A. 43 (1986), 119--127 Formal methods and deformations in algebraic geometry, Singularities of curves, local rings, Special algebraic curves and curves of low genus A relation between the sequence of multiplicities and the semigroup of values of an algebroid 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 author studies the toroidal resolution of a plane curve \(C=\{f(z,w) = 0\}\), which is a resolution consisting of a finite composition of admissible toric blowing-ups. It is shown how to read Puiseux pairs and the intersection multiplicities among irreducible components using the data of the toroidal resolution. The results are applied to planar curves, which are obtained as an \(n\)-times iterated generic hyperplane section of a non-degenerated hypersurface, to get that:
(1) the resolution complexity of these curves are at most \((n+1)\);
(2) each irreducible component of the curve has at most one Puiseux pair; and
(3) no Puiseux pair implies smoothness. toroidal resolution of a plane curve; toric blowing ups; Puiseux pair M. Oka: Geometry of plane curves via toroidal resolution ; in Algebraic Geometry and Singularities (La Rábida, 1991), Progr. Math. 134 , Birkhäuser, Basel, 95-121, 1996. Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Geometry of plane curves via toroidal resolution | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 complex toric variety, we compare, on the one hand, the natural inclusion of the group of classes of invariant Cartier divisors into that to Weil divisors, and on the other, the Poincaré duality homomorphism between the second integral cohomology and the integral homology (with closed supports) in complementary degree. If the variety has finite fundamental group, we prove that the natural ``Chern class homomorphism'' from the group of classes of invariant Cartier divisors to cohomology and the ``homology class map'' from the group of classes of invariant Weil divisors to homology are both isomorphisms, thus identifying the inclusion of these divisor class groups with the Poincaré duality homomorphism. Using suitable Künneth formulae, that yields a result valid in the general case. -- The cohomology part of our main theorem generalizes a result stated implicitly in \textit{T. Oda}'s report ``Convex bodies and algebraic geometry. An introduction to the theory of toric varieties'' [Ergebnisse Math. Grenzgebiete, Bd. 15 (1988; Zbl 0628.52002)] for the compact case, and explicitly in \textit{W. Fulton}'s book ``Introduction to toric varieties'' [Ann. Math. Stud. 131 (1993; Zbl 0813.14039)] for the case where each orbit closure contains a fixed point.
These groups of classes of invariant divisors -- and hence the corresponding co-homology groups -- have explicit descriptions in terms of combinatorial-geometric data of the fan that defines the toric variety. They can be used for explicit computation of Betti numbers and torsion coefficients. As an application, we use these descriptions to discuss problems of invariance of Betti numbers for toric varieties; furthermore, we discuss the lifting of the ``anticanonical'' homology class to integral cohomology. complex toric variety; invariant Cartier divisors; Weil divisors; Poincaré duality; fan; Betti numbers; integral cohomology DOI: 10.2748/tmj/1178225338 Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, Classical real and complex (co)homology in algebraic geometry Invariant divisors and Poincaré homomorphism of complex toric varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0752.00052.]
Let \(A\) be an abelian variety, defined over \(\mathbb{Q}\) of dimension \(d\) with complex multiplication. \(N\) will denote the order of a torsion point of \(A(\mathbb{Q})\). The author gives explicit bounds for \(N\). If \(A\) is two- dimensional then: For simple \(A\) we have \(N\leq 840\), with only one exception. If \(A\) is not simple then \(N\leq 28560\).
If \(A\) is three dimensional and simple then \(N\leq 31122\).
The results are refinements of previous results of \textit{A. Silverberg} [Compos. Math. 68, No. 3, 241-249 (1988; Zbl 0683.14002)]. abelian variety; complex multiplication; order of a torsion point Van Mulbregt, P., \textit{torsion-points on low dimensional abelian varieties with complex multiplication}, \textit{p}-adic methods in number theory and algebraic geometry, 205-210, (1992), American Mathematical Society, Providence, RI Complex multiplication and abelian varieties, Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\), Complex multiplication and moduli of abelian varieties Torsion-points on low dimensional abelian varieties with complex multiplication | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Relative invariants are invariants of rings parametrized by certain localizations of some ground ring. Picard groups, Brauer groups and class groups of rings obtain in this way several relative versions. (One should mention that the term relative Brauer group used in this theory is different from the one used ``classically'' in the theory of Brauer groups.) - The theory of relative invariants of rings can be found in the books by \textit{F. Van Oystaeyen} and the author with the title ``Relative invariants of rings. The commutative theory'' (1983; Zbl 0568.13001) and ``Relative invariants of rings. The non-commutative theory'' (1984; Zbl 0555.16001). One can find there also that relative invariants of a commutative ring R have a natural geometric interpretation if one views them as invariants of the affine scheme Spec(R). The question then rises to extend the theory in a general scheme theoretic frame work. This is what this book is about.
One classical problem which is dealt with is the behavior of quasicoherent and coherent sheaves on closed subsets of a scheme X with respect to extensions to the whole scheme. This problem turns out to be naturally related to the relative point of view since in geometrical terminology the relative invariants may be viewed as sections over subspaces of the classical invariants. - The last two chapters leave the general setting to study relative invariants of Krull schemes and to introduce the technique of Hecke actions on relative Picard groups.
The book pretends to be a self contained introduction to the theory. For this reason the first four chapters bring together definitions and facts from sheaf theory.
The subject of relative invariants is (at least to my taste) a very specialized one. I therefore think it might be interesting but is not essential to algebraists, algebraic geometers etc. as is claimed in the commercial on the cover of the book. But of course such opinions of taste are (as everything) relative. relative invariants of sheaves; Picard groups; Brauer groups; class groups Verschoren A., Relative Invariants of Sheaves (1986) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Brauer groups of schemes, Applications of methods of algebraic \(K\)-theory in algebraic geometry Relative invariants of sheaves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given an elliptic plane curve \(Q\), the authors consider the problem of constructing holomorphic self-maps \(f\) of \(\mathbb P^2\) that leave \(Q\) invariant. The criterion for a self-map of \(Q\) to extend to \(\mathbb P^2\) is stated. The authors look at the singular points of \(Q\).
In contrast with the smooth case, most singular elliptic curves do not admit nontrivial self-maps. The obstructions given by the singular points of \(Q\) are discussed. Two invariants are defined in terms of Weierstrass' \(\sigma\) and \(\zeta\) functions, and an invariance criterion for the elliptic plane curves with ordinary singularities is stated.
The authors prove that there do not exist self-maps of \(\mathbb P^2\), for which \(Q\) is critical and invariant, and the backward orbit of any point of \(Q\) is dense in the Julia set of \(f\). The case of a smooth cubic \(Q\) is discussed. The classic tangent process on \(Q\) provides examples of self-maps that leave \(Q\) invariant. If it is required \(f\) to leave invariant a line of lines, \(Q\) must be isomorphic to the Fermat cubic. The case when \(f\) has minimal degree 2 is also discussed.
When an elliptic plane curve has enough symmetries, the invariants associated to its singular points can be calculated easily. The simplest case is the dual of a smooth cubic. Special families of elliptic quartics with two singular points are considered. Computer-generated pictures illustrate tangent processes on such curves. elliptic plane curve; holomorphic self-maps; obstructions; invariants at singular points; ordinary singularities; backward orbit; invariant critical components; Julia set; invariant smooth cubics; elementary maps; dual of smooth cubic; Fermat cubic; tangent process; symmetries; Weierstrass' \(\sigma\) and \(\zeta\) functions; elliptic quartics with two singular points; Cassini quartic; quartics with a cusp and a node; mixed quartic; invariant cuspidal quartic [BD02]A. Bonifant and M. Dabija, \textit{Self-maps of }P 2\textit{with invariant elliptic curves}, in: Complex Manifolds and Hyperbolic Geometry (Guanajuato, 2001), Contemp. Math. 311, Amer. Math. Soc., Providence, RI, 2002, 1--25. Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Projective techniques in algebraic geometry, Elliptic curves Self-maps of \(\mathbb{P}^2\) with invariant 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 \textit{M. Martin-Deschamps} and \textit{D. Perrin} obtained sharp bounds for the Rao function \(h^1(I_C(n))\) of a locally Cohen-Macaulay purely 1-dimensional curve \(C\subset\mathbb{P}^3\) [C. R. Acad. Sci., Paris Sér. I, 317, No. 12, 1159-1162 (1993; Zbl 0796.14029)] and characterized the extremal curves (i.e. the non-arithmetically Cohen-Macaulay curves achieving equality for their bound) in terms of certain Koszul modules in Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 6, 757-785 (1996; Zbl 0892.14005). In Boll. Unione Mat. Ital. (7), A 9, No. 3, 593-607 (1995; Zbl 0866.14020), \textit{P. Ellia} recovered the bounds of Martin-Deschamps and Perrin and characterized extremal curves as the schematic union of a line and of a plane curve. Moreover he proved that a curve \(C\) of degree \(d\leq 5\) and genus \(g\), which is neither arithmetically Cohen-Macaulay, nor extremal, verifies the stronger bound: \(h^1(I_C(n))\leq 1+(d-3)(d-4)/2-g\). All previous papers assumed the characteristic 0 hypothesis.
In this article, without any particular assumption on the characteristic of the ground field, Scott Nollet improves Ellia's results, by proving, for neither arithmetically Cohen-Macaulay, nor extremal curves \(C\), the bounds:
\[
h^1\bigl( I_C(n)\bigr)\leq\begin{cases} 0\quad & \text{if }n<1+g-(d-3)(d-4)/2\\ (d-3)(d-4)/2-g+n \quad & \text{if }1+g- (d-3)(d-4)/2\leq n<1\\ 1+(d-3)(d-4)-g\quad & \text{if }1\leq n\leq d-3\\ 1+(d-2)(d-3)/2-g-n\quad & \text{if }d-3<n\leq(d-2)(d-3)/2-g\\ 0\quad & \text{if }n>(d-2)(d-3)/2-g\end{cases}
\]
(theorem 2.11), proves that the above bounds are sharp and classifies subextremal curves (i.e. neither arithmetically Cohen-Macaulay, nor extremal curves achieving their bound) as the curves obtained from extremal ones by elementary biliaison of height 1 on a quadric surface (theorem 2.14). -- A key-tool to get the previous results is the notion of spectrum of a curve (see section 1), studied by \textit{Enrico Schlesinger} in his Ph. D. thesis [see also J. Pure Appl. Algebra 136, 267-283 (1999)]. In particular Nollet proves that a subextremal curve has the spectrum: \(\{1+g-(d-3)(d-4)/2\}\cup\{0,1^2,2,\dots,d-3\}\) (theorem 2.14).
Some examples conclude the paper in order to observe that, unlike the extremal case, subextremal curves are not determined by their spectrum (see 2.15) and to suggest that there are no natural stronger bounds on the Rao function for curves, which are neither arithmetically Cohen-Macaulay, nor extremal, nor subextremal (see 2.16, 2.17, 2.18). subextremal curves; biliaison; spectrum of a curve; Rao function for curves Nollet S.: Subextremal curves. Manuscr. Math. 94(3), 303--317 (1997) Plane and space curves, Classical real and complex (co)homology in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Linkage Subextremal curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This work shows that for a smooth complex curve C of genus \(g\geq 5\) the singular locus of the Theta divisor of its Jacobian is reduced and it is reducible exactly when C is either trigonal, superelliptic or an unramified double cover of a non-hyperelliptic genus 3 curve. theta divisor of Jacobian of smooth complex curve M. Teixidor i Bigas, For which Jacobi varieties is {\(\operatorname{Sing} \Theta\)} reducible? \textit{J. Reine Angew. Math.}\textbf{354} (1984), 141-149. Jacobians, Prym varieties, Theta functions and abelian varieties For which Jacobi varieties is Sing \(\Theta\) reducible? | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this paper is to study the theta-dual and the cohomological support loci for the twisted ideal sheaves of an Abel-Prym curve contained in the Prym variety associated to an etale double cover of smooth projective non-hyperelliptic curves. Given a coherent sheaf \({F}\) on a smooth projective variety \(X,\) the \(i-\)th cohomological support locus of \({F}\) is
\[
V^{i}({F}) : = \{\alpha \in \mathrm{Pic}^{0}(X) : h^{i}(X, {F}\otimes \alpha) > 0\} \subset \mathrm{Pic}^{0}(X).
\]
Let \(P\) be the Prym variety of dimension \(g-1\) associated to the etale double cover \(\widetilde{C}\rightarrow C\) of irreducible smooth projective non-hyperelliptic curves of genus \(\widetilde{g} = 2g - 1\) and \(g > 2,\) respectively. Let \(\Sigma\) be the canonical principal polarization. Denote by \(\tau_{\widetilde{C}}\) the ideal sheaf of \(\widetilde{C}\) inside \(P.\)
Theorem A. Let \(\widetilde{C}\) be a non-hyperelliptic Abel-Prym curves embedded in its Prym variety \((P, \Sigma).\) Then have :
1. The theta-dual of \(\widetilde{C}\) has dimension \(\dim(V(\widetilde{C})) = \dim P - 3 = g - 4.\)
2. The cohomological support loci of \(\tau_{\widetilde{C}}(\Sigma)\) can be (non-canonically) identified with \(V^{0}(\tau_{\widetilde{C}}(\Sigma)) = V^{1}(\tau_{\widetilde{C}}(\Sigma)) = V(\widetilde{C})),\) \(V^{2}(\tau_{\widetilde{C}}(\Sigma)) = P,\) \(V^{>2}(\tau_{\widetilde{C}}(\Sigma)) = \emptyset.\)
3. The cohomological support loci of \(\tau_{\widetilde{C}}(2\Sigma)\) can be (non-canonically) identified with \(V^{0}(\tau_{\widetilde{C}}(2\Sigma)) = P,\) if \(g > 3\) and \(V^{0}(\tau_{\widetilde{C}}(2\Sigma)) = \emptyset,\) if \(g = 3\) ; \(V^{1}(\tau_{\widetilde{C}}(2\Sigma)) = V^{2}(\tau_{\widetilde{C}}(2\Sigma)) = \{0\},\) \(V^{>2}(\tau_{\widetilde{C}}(2\Sigma)) = \emptyset.\) theta-dual; cohomological support loci for the twisted ideal sheaves of an Abel-Prym curve; Prym variety; non-hyperelliptic curve; non-hyperelliptic curve S. Casalaina Martin, M. Lahoz, and F. Viviani, Cohomological support loci for Abel-Prym curves, Matematiche (Catania) 63 (2008), 205-222. Jacobians, Prym varieties, Subvarieties of abelian varieties, Vanishing theorems in algebraic geometry Cohomological support loci for Abel-Prym curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author presents a new approach to the theory of logarithmic Kodaire dimension. The main advantage of his method is its independence of desingularization theory, so that it works over any field of any characteristic. Given a variety V, he does not choose a good model (e.g., a smooth one with boundary being a divisor having only normal crossing singularities), as Iitaka did, but uses all the models of V. There are two candidates for the log-Kodaira dimension, both of which coincide with the classical one if a good model exists. Many classical results are generalized for one or both of them: Iitaka's fibration theorem, the so- called easy addition formula, the finiteness of proper birational automorphisms of a variety of log-general type, etc. The proofs are mostly omitted or just sketched. logarithmic Kodaire dimension; Iitaka's fibration theorem; easy addition formula; birational automorphisms of a variety of log-general type Z.-H. Luo, An invariant approach to the theory of logarithmic Kodaira dimension of algebraic varieties . Bull. A.M.S. (N.S.)vol. 19, 1 (1988) 319-323. Families, moduli, classification: algebraic theory, Birational geometry An invariant approach to the theory of logarithmic Kodaira dimension of algebraic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians To a locally Cohen-Macaulay equidimensional curve C in \({\mathbb{P}}_ k^ 3\) (k an algebraically closed field) one can associate its Hartshorne-Rao module \(M(C)=\oplus_{n\in {\mathbb{Z}}}H^ 1({\mathbb{P}}^ 3,{\mathcal I}_ C(n)) \) (where \({\mathcal I}_ C\) is the ideal sheaf of C), which is a graded module of finite length over the polynomial ring \(S=k[X_ 0,X_ 1,X_ 2,X_ 3]\). This paper is a continuation of the authors' program to understand the connection between the Hilbert scheme \(Hilb_{d,g}{\mathbb{P}}^ 3\) of C and the so-called variety of module structures of the module M(C).
The main theorem of the present paper is the following: Let \({\mathcal G}=\{G_ i,\phi_ i\}\) be a finite connected oriented commutative graph without loops. The vertices of \({\mathcal G}\) correspond to finite length graded S-modules and the arrows to surjective morphisms of graded S- modules. Then for infinitely many (d,g) there exists an irreducible component U of a suitable Hilbert scheme \(Hilb_{d,g}{\mathbb{P}}^ 3\) such that the general element of U is arithmetically Cohen-Macaulay, and to each \(G_ i\) there is an irreducible flat family \(V_ i\) of curves having Hartshorne-Rao module isomorphic to \(G_ i\) (suitably shifted). An arrow \(\phi_ i: G_ i\to G_ j\) in \({\mathcal G}\) implies that \(V_ i\) is contained in the closure of \(V_ j\). Finally, all curves of all families have the same speciality.
The main tool used is the so-called Rao construction, and the authors also apply it to produce curves with specified cohomology, over any infinite field (not necessarily algebraically closed).
See also the authors' paper in Arch. Math. 54, No.4, 397-408 (1990; Zbl 0715.14019). deficiency module; liaison; locally Cohen-Macaulay equidimensional curve; Hartshorne-Rao module; Hilbert scheme; variety of module structures Linkage, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Families, moduli of curves (algebraic), Linkage, complete intersections and determinantal ideals Construction of families of curves from finite length graded modules | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 real closed field, V an affine (non-necessarily irreducible) R-variety and \(f_ 1,...,f_ r\in \Gamma (V,{\mathcal O}_ V)\) polynomial functions on V. If \(S(f_ 1,...,f_ r)\) is the semi-algebraic set of points of V in which \(f_ 1,...,f_ r\) are strictly positive and \(\bar S(f_ 1,...,f_ r)\) is the set of points in which \(f_ 1,...,f_ r\) are non-negative then the sets \(S(f_ 1,...,f_ r)=S\) are called basic open sets and the sets of the form \(F=\bar S(f_ 1,...,f_ r)\) are called basic closed sets on V. Let s(S) be (resp. \(s(F)\) the minimal number of inequalities necessary to describe S (resp. F). Then the supremum s(V) of all the numbers s(S) when S are non empty basic open sets on V is finite and it is called the geometric stability index of the variety V. Also, \(\bar s(S)\) is defined as the supremum of s(F). \textit{L. Bröcker} proved that there are upper bounds for the numbers s(V), \(\bar s(V)\), which depend on the dimension of V. More precisely, in a preprint L. Bröcker showed that if \(\dim(V)=n>0\) then \(n+2\leq \bar s(V)\leq n(n+1)\) for \(n\geq 3\) and \(\bar s(V)=n(n+1)\) for \(n=1\) or \(n=2.\)
In the present paper this result is reproved but it is also presented the following complete result: if \(n>0\) then \(\bar s(V)=n(n+1)\). - The method to prove this is to obtain by an inductive argument that one has also \(\bar s(V)\geq n(n+1).\)
On the geometric stability index s(V), L. Bröcker showed that \(s(V)=\dim (V)=n\) if \(1\leq n\leq 3\) and that in general \(n\leq s(V)\leq 3\cdot 2^{m-1}\) if \(n=2m\), \(n\leq s(V)\leq 2^ m\) if \(n=2m-1\). In the present paper it is established the following precise result: if V is a real n-dimensional variety, \(n>0\), then \(s(V)=n\) (theorem 2, corollary 4). To prove this result, the author utilises: 1. the real spectrum of a ring developed by M. Coste and M.-F. Roy in 1982 and other authors in more recent works. 2.\textit{M. Marshall}'s theory of spaces of orderings [Trans. Am. Math. Soc. 258, 505-521 (1980; Zbl 0427.10015)] in which theory the author had obtained new useful results. real variety; semi-algebraic set; geometric stability index; real spectrum of a ring; spaces of orderings Scheiderer, C.,Stability index of real varieties. Invent. Math.97 (1989), 467--483. Semialgebraic sets and related spaces, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Real polynomials: analytic properties, etc. Stability index of real varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this paper is to define an \(''L^ 2\)-réseau'' of a regular holonomic \({\mathcal D}_ X\)-module on a smooth complex variety and give some applications of it. Let X be a smooth complex variety, Y a closed analytic subset of pure codimension in X and S a hypersurface in Y such that Y-S is non-singular. For a regular holonomic \({\mathcal D}_ X\)- module \({\mathcal M}\) whose support is contained in Y, we may define a canonical sub \({\mathcal O}_ X\)-module of \({\mathcal M}\) associated to the \(L^ 2\)-extension. We call it the \(L^ 2\)-réseau and denote it by \(L^ 2(Y,{\mathcal M})\). In particular, when \(Y=X\) and \({\mathcal M}\) has no non-trivial section supported in S, the \(L^ 2\)-réseau contains the réseau of Deligne. By using \(L^ 2(X,{\mathcal M})\), the author discusses a condition in order that \({\mathcal M}\) is generated by a ''standard'' distribution on X. singularity of a variety; regular holonomic \({\mathcal D}_ X\)-module; smooth complex variety; \(L^ 2\)-réseau; distribution D. Barlet and M. Kashiwara, Le réseau L2 d'un syst`eme holonome régulier, Invent. Math. 86 (1986), no. 1, 35-62. Complex spaces, Analytic subsets and submanifolds, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Local cohomology of analytic spaces Le réseau \(L^ 2\) d'un système holonome régulier. (The \(L^ 2\)- réseau of a regular holonomic system) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 moduli spaces of vector bundles over a singular curve can be defined in several different ways and are still largely unknown [cf. \textit{R. Pandharipande}, ``A compactification over \(\overline M_g\) of the universal moduli space of slope-semistable vector bundles'', J. Am. Math. Soc. 9, No. 2, 425-471 (1996)]. The author uses the notion of Hilbert semistability (a numerical condition), developed by \textit{D. Gieseker} and \textit{I. Morrison} [J. Differ. Geom. 19, 1-29 (1984; Zbl 0573.14005)]. He extends the work of Gieseker on rank-2 bundles over an irreducible curve with one node to the case where the curve has two components meeting in a simple node. The resulting moduli spaces have two irreducible components meeting transversally along a divisor; they are described in greater detail for the case of bundles with fixed determinant. The crux of this work is to relate bundles over the curve with bundles over its two components, including the semistability condition; for certain numerical data, a Hecke transformation is necessary, in order to have a universal bundle. blow-up; blow-down; moduli spaces of vector bundles over a singular curve; Hilbert semistability; bundles with fixed determinant; Hecke transformation Huashi, Xia, Degenerations of moduli of stable bundles over algebraic curves, Compos. Math., 98, 305-330, (1995) Families, moduli of curves (algebraic), Vector bundles on curves and their moduli, Singularities in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Degenerations of moduli of stable bundles over algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) and \(F\) be subfields of \({\mathbb{C}}\) and \(U\) be a smooth algebraic variety. We define the category \(M_ k(U, F)\) by the triple \({\mathcal M}= (({\mathcal E}, \nabla), V, \rho)\) consisting of
(1) A locally free \({\mathcal O}_ U\)-module \(\mathcal E\) of finite rank with an integrable connection \(\nabla: {\mathcal E}\to {\mathcal E}\otimes \Omega^ 1_ U\) which is regular singular along the boundary.
(2) A local system \(V\) of \(F\)-vector spaces on the complex manifold \(U^{an}\).
(3) An isomorphism \(\rho: V\otimes_ F {\mathbb{C}}\simeq \text{Ker }\nabla^{an}\) of local systems on the complex manifold \(U^{an}\).
For an object \(\mathcal M\) of \(M_ k(U, F)\), by the comparison theorem of Deligne, one can define the object \(H^*(U, {\mathcal M})\). We describe the determinant object \(\text{det }H^*(U, {\mathcal M})\) of \(H^*(U, {\mathcal M})\) in terms of the pairing of the relative Chow group and the motivic Picard group \(\text{MPic}_ k(U, F)\). We give an outline of the construction of the pairing by using \(K\)-theory. This is an announcement of the full paper. determinant of period integrals; motive of rank 1; tame symbol; determinant object; relative Chow group; motivic Picard group Takeshi Saito and Tomohide Terasoma, A determinant formula for period integrals, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 5, 131 -- 135. \(K\)-theory of schemes, de Rham cohomology and algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Determinantal varieties, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Symbols, presentations and stability of \(K_2\) A determinant formula for period integrals | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(F_ 4:X^ 4+Y^ 4+Z^ 4=0\) be the Fermat curve of degree 4 over \(\mathbb{Q}\). Each automorphism of the curve \(F_ 4\) induces an endomorphism of the corresponding jacobian variety \(J_ 4\). This defines a map \(\mathbb{Q} [\text{Aut} (F_ 4)] \to \text{End} (J_ 4) \bigotimes \mathbb{Q}\). The author shows by explicit calculations that \(\text{End} (J_ 4)\) is precisely the image of the group ring \(\mathbb{Z} [\text{Aut} (J_ 4)]\). It is then a consequence that \(J_ 4\) is not isomorphic to a product of elliptic curves. automorphism of Fermat curve; endomorphism of jacobian Jacobians, Prym varieties, Global ground fields in algebraic geometry, Special algebraic curves and curves of low genus The Fermat curve of fourth degree | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The problem of a diameter of a plane complex algebraic curve was initially proposed by S. Frenkel. He conjectured that the diameter of a complex algebraic curve in the projective plane, \(P^2\), is bounded in the induced metric independently of the degree of the curve. We assume that the metric on the projective plane is a standard Fubini-Study metric with the diameter of each projective line equal to 1. For example, the diameter, i.e., maximal distance in the induced metric, on any union of projective lines never exceeds 2. On the other hand, M. Gromov conjectured that in fact diameters of complex algebraic curves in \(P^2\) are unbounded unless there are some restrictions on the curves considered.
This article contains a proof of Gromov's conjecture. I describe further a procedure which enables one to construct, starting with any smooth curve, a new plane smooth curve with a diameter which approximately doubles the diameter of the initial curve. Hence by repeating this procedure we can produce smooth plane curves with a diameter exceeding any positive number. It gives a negative answer to the question of S. Frenkel and thus proves the Gromov conjecture. It is interesting to remark that the diameter is nevertheless bounded independently of the degree on a rather big class of plane curves. diameter of a plane complex algebraic curve; projective plane; degree Bogomolov, E.A. On the diameter of plane algebraic curves.Math. Res. Lett.,1, 95--98, (1994). Other complex differential geometry, Special algebraic curves and curves of low genus, Vector bundles on curves and their moduli, Projective techniques in algebraic geometry, Analytic subsets and submanifolds On the diameter of plane algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present book is the second part of a trilogy (for the first and third part see [Zbl 1298.51001] and [Zbl 1290.53001], respectively).
``During the seventeenth century, Fermat and Descartes introduced the basic concepts of analytic geometry''. In the beginning, this approach was considered the inelegant sister of synthetic geometry. The book describes the first steps of the triumphal march of analytic geometry through affine, projective, Euclidean, and Hermitian geometry as far as figures of degree one or two are concerned. Of course, in the Euclidean and Hermitian case the underlying field is real or complex, respectively, for the discussion of affine and projective geometry the author confines himself to the commutative case; in view of cryptographical applications also finite fields are admitted.
The first chapter of the book offers a very interesting historical description of the beginnings of analytical geometry, whereas from Chapter 2 up till 6 the book follows the progress of fundamental linear algebra, the reader is assumed to be familiar with. The final Chapter 7 provides together with the appendices a fine elementary introduction to algebraic plane curves in the complex projective plane and concludes with rational curves.
This beautiful, comprehensive, well-written and clearly structured book addresses all those who have to teach or learn geometry at an undergraduate level.
More than 110 very aesthetical figures accompany the text and appeal to the geometric intuition of the reader; 41 references are recommended for further reading.
The text is divided into 7 chapters, each begins with an overview and ends with problems and exercises. Eight appendices support the reader's understanding of the main text. In the following, the contents of the chapters are given in detail.
Chapter 1 (The birth of analytic geometry) tells that plane analytic geometry was introduced independently by the French mathematicians Pierre de Fermat (1601--1655) and René Descartes (1596--1650) around 1630. The solution of a geometric problem by the methods inherited from the Greek geometers -- that is, via constructions with ruler and compass -- often required incredible imagination and could by no means be systematised. Fermat and Descartes showed that many of these geometric problems could be solved instead by routine algebraic computation. ``Fermat considers only positive values for his coordinates, which was a serious handicap for the further development, hence \(ax=by\) and \(ax+by=c^2\) are equations of half-line and segment, respectively''. Fermat's approach to equations of degree two and Descartes' approach via the Pappus problem are presented. ``It took another century before mathematicians started to recognize and systematically use the new algebraic approach to geometry in terms of coordinates. Leonhard Euler (1707--1783) was particularly influential in this respect (parameter equations, theory of coordinate changes).'' [\dots] ``While Greek geometry [\dots] confined to the study of lines and conics, the new analytic geometry opened the way to the study of an incredibly wider class of curves. This is the reason behind its success.'' John Wallis (1616--1703) was the first to use negative numbers. Further, polar, spherical, and cylindrical coordinates, the computation of distances and angles, and the canonical scalar product on \({\mathbb R}^n\) are considered. Jakob Hermann (1678--1733) studies properties of planes and surfaces of degree two in solid geometry, equation and parametric description of a plane in solid geometry. The cross product of two vectors of \({\mathbb R}^3\) and its properties as well as generalization to the \(n\)-dimensional case are studied. Forgetting the origin, inspired by the theory of forces, equipollent vectors (vectors of the plane that constitute a vector space over the real numbers) and vector subspaces are studied. Considering tangents to a curve, Fermat's and Descartes' approaches are presented. A list of conics (the ellipse, the hyperbola, the parabola) and a list of quadrics in solid space are given. For ruled quadrics, Christopher Wren 1669 stated: ``Through every point of the hyperboloid of one sheet pass two lines entirely contained in the surface.'' The case of the hyperbolic paraboloid was known to Gaspard Monge (1746--1818).
The first part of Chapter 2 (Affine geometry) is free of (affine) coordinates. The definition of an affine space via an arbitrary vector space over a commutative field, examples of affine spaces, affine subspaces, parallel subspaces (with Euclid's fifth postulate), generated subspaces, supplementary subspaces, lines and planes, affinely independent points, barycenters, barycentric coordinates, triangles (Ceva, Menelaus), parallelograms, affine transformations, affine isomorphisms, translations, parallel projection, symmetries, homotheties, affinities, and the intercept Thales theorem are studied. The second part deals with affine coordinates, affine basis, the change of coordinates, equations of a subspace, the matrix of an affine transformation, the quadrics (the underlying field is not of characteristic \(2\)), the reduced equation of a quadric, the symmetries of a quadric, and the equation of a non-degenerate quadric.
Chapter 3 (More on real affine spaces) considers segments, order, orientation (clockwise and counter-clockwise), screws with a right(left)-hand thread, orientation of a real affine space, direct and inverse affine isomorphisms, parallelepipeds, half spaces, Pasch's theorem (In the real affine plane, consider a triangle \(ABC\) and a line \(d\) not containing any vertex of the triangle. If the line \(d\) intersects the perimeter of the triangle, it intersects it at exactly two points.), and the affine classification of real quadrics.
In Chapter 4 (Euclidean geometry), the definition of Euclidean space as affine space endowed with a scalar product is given. Distance, the Schwarz inequality, angles, orthogonal vectors, norm on a real vector space, metric spaces, the Minkowski inequality, Pythagoras' theorem, rectangles, diamonds, squares, examples of Euclidean spaces, orthonormal basis, Gram-Schmidt process, polar coordinates in a Euclidean plane, orthogonal projection, approximation by the law of least squares, Fourier approximation, the classification of isometries, rotations, similarities, and Euclidean quadrics are studied.
Chapter 5 (Hermitian spaces) deals with the (sesquilinear) Hermitian product, norm, distance, right angle, Pythagoras' theorem, orthonormal basis, the Gram-Schmidt process, the metric structure of Hermitian spaces, and complex quadrics (there exist quadrics, in finite dimensional Hermitian spaces, which do not admit a reduced equation with respect to any orthonormal basis.).
In Chapter 6 (Projective geometry), the origin of projective ideas is discussed in the first part of the trilogy. It provides a good intuitive base for the following algebraic representation such as the definition of the \(n\)-dimensional projective space \( P_n(\mathbb K)\) (\(n<\infty\)) over a commutative field, projective subspaces, projective hyperplane, and the dimension formula. The duality principle, homogeneous coordinates, and the projective basis are also studied. ``The anharmonic ratio of four points on a line can be regarded as the projective substitute of the notion `proportionality' in the affine space. We clarify this point and show how powerful this notion is.'' Furthermore, projective transformations (projective function, central projection, decomposition of a projective transformation between two distinct lines into central projections), computational proofs of Desargues' and Pappus' theorem in \(P_2(\mathbb K)\), Fano's theorem (In \(P_2(\mathbb K)\) with \(\text{char}\, \mathbb K \not=2\) the diagonal points of a quadrilateral are not on the same line.), and harmonic quadruples in \(P_2(\mathbb K)\) with \(\mathrm{char}\, \mathbb K \not=2\) are considered. The axioms of an abstract projective plane (taken from the first part of the trilogy) are valid in \(P_2(\mathbb K)\) with \(\mathrm{char}\, \mathbb K \not=2\). Projective quadrics in \(P_n(\mathbb K)\) (\(n<\infty\)) with \(\mathrm{char}\, \mathbb K \not=2\), symmetric bilinear forms, non-degenerate quadrics, regular quadrics, the duality with respect to a quadric, conjugate subspaces, pole and polar hyperplane, the tangent space to a quadric, the duality principle with respect to a quadric, projective conics, the existence question for conics containing five (or six) given points, the bijective correspondence between any non-degenerate regular conic and every projective line, and the anharmonic ratio along a conic, an elegant proof of Pascal's theorem, Brianchon's theorem, the construction of a \(2\)-dimensional affine space from \(P_2(\mathbb K)\), real quadrics, the statement that all non-empty regular projective real conics are projectively equivalent (this does not generalize to quadrics in higher dimensions), the topology of projective real spaces, and the statements that the real projective plane is compact Hausdorff and that the real projective line is homeomorphic to a circle are also treated.
Chapter 7 (Algebraic curves) provides an elementary study of algebraic plane curves. ``We restrict our attention to the projective plane \(P_2({\mathbb C})\) over the field \({\mathbb C}\) of complex numbers'' and ``limit our study to a Bezout theorem which involves only the intersection points and their multiplicities''. The definition of an algebraic curve in \(P_2({\mathbb C})\) including circular points is motivated. Algebraic plane curves (irreducible, component, degree, double point, point of multiplicity \(k\), tangents), examples of singularities (Descartes folium, clover leaf, quadrifolium, bifolium, semi-cubic parabola), inflexion points (Hessian curve), and Bezout's theorem (Consider two algebraic plane curves with respective degrees \(n\) and \(m\), without any common components. Let these curves admit the intersection points \(B_1,\ldots,B_k\), the point \(B_i\) being of multiplicity \(r_i\) on the first curve and \(s_i\) on the second curve. Then \( \sum_{i=1}^{k}r_is_i\leq\,nm \) .) are studied. In general one does not have equality in the Bezout theorem which is shown by an example of two cubics with \( \sum_{i=1}^{k}r_is_i=5 \). ``One can reach equality in the Bezout theorem via a theory of branches and multiplicity of the contact between branches. However this is far beyond the scope of this introductory text.'' Curves through given points (a generalization of Pascal's theorem), the limitation of the possible number of multiple points on a algebraic plane curve, Conics, and cubics are considered. Cramer's paradox is formulated especially for cubics and quartics. The following theorem is proven: If two cubics intersect in exactly nine points, every cubic passing through eight of these points passes through the ninth point.
A cubic of \(P_2({\mathbb C})\), without multiple point, possesses nine inflexion points. Every line through two of these inflexion points contains a third one. The group of a cubic, the motivation of the definition of a rational algebraic plane curve in \(P_2(\mathbb{C})\), and a criterion for rationality are treated. Note the problems 7.16.16 and 7.16.17: ``The inversion transforms a rational curve to a rational curve. All irreducible conics are rational curves.''
Appendix A (Polynomials over a field) considers polynomials versus polynomial functions, integral domain, Euclidean division, Bezout's theorem for two non-zero polynomials over a field, irreducible polynomials, the greatest common divisor of two non-zero polynomials over a field, roots of a polynomial, the addition of roots to a polynomial, the extension of a field, the derivative of a polynomial over a field, the connection between derivative and multiplicity, and Taylor's formula (for polynomials in one variable) in case of \(\mathrm{char}\, \mathbb K=0\).
Appendix B (Polynomials in several variables) deals with roots, the difference between polynomials in one and those in several variables, an example of a polynomial over \({\mathbb R}\) in two variables which does not factor in a product of polynomials of degree \(1\) in any extension of \({\mathbb R}\), polynomials domain over a field, quotient fields, irreducible polynomials (The author restricts to polynomial domains over a field \({\mathbb K}\) in which every non-constant element over \({\mathbb K}\) admits a unique factorization as a product of irreducible elements, up to non-zero multiplicative constants over \({\mathbb K}\).), Eisenstein's criterion, partial derivatives of a polynomial, and Taylor's formula (for polynomials in two or more variable) in case of \(\mathrm{char}\, \mathbb K=0\).
Topics of Appendix C (Homogeneous polynomials) are a criterion of homogeneity, Euler's formula and its iteration, and non-homogeneous polynomials and their associated homogeneous polynomial.
Appendix D (Resultants) considers the resultant of two non-homogeneous or homogeneous polynomials \(p(x)\), \(q(x)\) or \(P(X,Y)\), \(Q(X,Y)\), respectively, over a polynomial domain and a common irreducible factor and roots versus divisibility.
Appendix E (Symmetric polynomials) treats elementary symmetric polynomials and the fundamental theorem of the theory of symmetric polynomials.
In Appendix F, (Complex numbers) moduli, arguments, exponentials, and the fundamental theorem of algebra are considered. It is shown that every irreducible polynomial \(p(x)\) over the reals has degree one or two.
Appendix G (Quadratic forms) contains regular symmetric bilinear forms, conjugate vectors, isotropic vectors, definite symmetric bilinear forms, conditions that a basis of conjugate vectors exists, real quadratic forms (Sylvester's theorem), quadratic forms on Euclidean spaces, and complex quadratic forms (adjoint matrix).
Appendix H (Dual spaces) shows that in finite dimension every vector space is isomorphic to its dual. Further the Kronecker symbol and mixed orthogonality (between vectors and linear forms) are treated. Pierre de Fermat; René Descartes; Leonhard Euler; affine space; barycenter; real affine space; Pasch's theorem; Euclidean space; metric space; Gram-Schmidt process; approximation by the law of least squares; Fourier approximation; Hermitian space; projective space; duality principle; Fano's theorem; projective quadric; Pascal's theorem; Brianchon's theorem; topology of projective real spaces; algebraic plane curves; Bezout's theorem; Hessian curve; Cramer's paradox; group of a cubic; rational algebraic plane curve; Taylor's formula for polynomials in one or more variables; Eisenstein's criterion; Euler's formula; fundamental theorem of algebra; Sylvester's theorem Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry, General histories, source books, Linear incidence geometric structures with parallelism, Affine analytic geometry, Projective analytic geometry, Euclidean analytic geometry, Questions of classical algebraic geometry, Algebraic functions and function fields in algebraic geometry, Projective techniques in algebraic geometry An algebraic approach to geometry. Geometric trilogy II | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Torsors of finite commutative group schemes defined over the ring of integers of a local field are studied. An estimate for ramification of the splitting field of such a torsor is obtained by the Fontaine method. It is shown, how to apply this estimate to the study of the arithmetic of abelian varieties; namely, it is obtained an estimate for the rank of the Jacobian of the Fermat curve in cyclic extensions. In the case of the field \(\mathbb{Q}\) itself, this estimate is close to the known D. K. Faddeev estimate. torsors of finite commutative group schemes; local field; Fontaine method; rank of the Jacobian; Fermat curve; cyclic extensions Group schemes, Homogeneous spaces and generalizations, Ramification and extension theory, Local ground fields in algebraic geometry, Algebraic theory of abelian varieties, Cyclotomic extensions The Fontaine inequality for torsors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 generalized Remmert-Van de Ven problem can be formulated as follows: Suppose \(\phi:X\to Y\) is a morphism from a complete variety in a given class onto a smooth projective variety, possibly with some additional assumptions on \(\phi\) and \(Y\) (e.g. the Picard number \(\rho(Y)\) equals 1); then is it true that \(Y\simeq\mathbb{P}^n\) unless \(\phi\) is an isomorphism? The affirmative answer is known for \(X=\mathbb{P}^n\) [\textit{R. Lazarsfeld} in: Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale 1983, Lect. Notes Math. 1092, 29-61 (1984; Zbl 0547.14009)], \(X\) a smooth quadric [\textit{K. H. Paranjape} and \textit{V. Srinivas}, Invent. Math. 98, 425-444 (1989; Zbl 0697.14037) and \textit{K. Cho} and \textit{E. Sato}, Math. Z. 217, 553-565 (1994; Zbl 0815.14035)], \(X\) an irreducible Hermitian symmetric space [\textit{I. Tsai}, J. Differ. Geom. 33, 717-729 (1991; Zbl 0718.53042)], \(X\) rational homogeneous with \(\rho(X)=1\) [\textit{J.-M. Hwang} and \textit{N. Mok}, Invent. Math. 136, 209-231 (1999; Zbl 0963.32007)], \(X\) a Fano 3-fold [\textit{E. Amerik}, Doc. Math., J. DMV 2, 195-211 (1997; Zbl 0922.14007) and \textit{C. Schuhmann}, J. Algebr. Geom. 8, 221-244 (1999; Zbl 0970.14022)].
In the paper under review, the affirmative answer is given for \(X\) a complete toric variety and \(\rho(Y)=1\). The idea of the proof is similar to that of Lazarsfeld. The authors use Mori's result [\textit{S. Mori}, Ann. Math. (2) 110, 593-606 (1979; Zbl 0423.14006)]: Suppose that there exists a dominating family \(\mathcal{V}\) of rational curves \(f:\mathbb{P}^1\to Y\) of minimal degree such that for general \(y\in Y\) and any curve \(f\in\mathcal{V}\), \(f(0)=y\), the bundle \(f^{*}TY\) is ample; then \(Y\simeq\mathbb{P}^n\). Another ingredient is the Euler-Jaczewski sequence
\[
0\to\Omega_X\to\mathcal{P}_X^{\vee}\to H\otimes\mathcal{O}_X\to 0
\]
corresponding to \(\text{Id}\in\text{End}H= \text{Ext}^1(H\otimes\mathcal{O}_X,\Omega_X)\), where \(H=H^1(X,\Omega_X)\) [see \textit{K. Jaczewski} in: Classification of algebraic varieties, Algebr. Geom. Conf., L'Aquila 1992, Contemp. Math. 162, 227-247 (1994; Zbl 0837.14042)]. The setup is easily reduced to the case where \(X\) is smooth projective and \(\phi\) generically finite. Assuming the contrary and using the Euler-Jaczewski sequence, one gets a generically surjective map \(\bigoplus_{D\in J}\mathcal{O}(D)\to\phi^{*}TY\), where \(J\) is the set of invariant prime divisors on \(X\) not contracted by \(\phi\). This leads to a contradiction using some technical property of minimal dominating families of curves.
As an application, the authors prove a structure result for smooth projective varieties \(X\) admitting two projective bundle structures \(X\to Y\) and \(X\to Z\). If \(\dim X=\dim Y+\dim Z\), then it is easy to deduce from Lazarsfeld's theorem that \(X\simeq\mathbb{P}^r\times\mathbb{P}^s\). If \(\dim X=\dim Y+\dim Z-1\), then one deduces from the main theorem that either \(Y=Z=\mathbb{P}^n\), \(X=\mathbb{P}(T\mathbb{P}^n)\), or \(Y,Z\) are projective bundles over a smooth curve \(C\) and \(X=Y\times_CZ\). toric variety; Picard number; dominating family of curves; Euler-Jaczewski sequence; classification of projective bundle; morphism Occhetta, G; Wiśniewski, JA, On Euler-jaczewski sequence and remmert-Van de ven problem for toric varieties, Math. Z., 241, 35-44, (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Projective techniques in algebraic geometry, Families, moduli of curves (algebraic), Rational and birational maps On Euler-Jaczewski sequence and Remmert-Van de Ven problem for toric varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth complex projective curve of genus \(g\geq 2\). Let \({\mathcal M}_{\tau}(r, L)\) be the moduli space of \(\tau\)-stable pairs \((E, \Phi)\) consisting of a rank \(r\) vector bundle \(E\) on \(X\) of determinant \(L\) and a section \(\Phi \in H^0(E)\). The paper under review proves rationality of these moduli spaces in some cases. General results concerning rationality of such moduli spaces are contained in \textit{N. Hoffmann's} paper [Int. Math. Res. Not. 2007, No. 3, Article ID rnm010, 30 p. (2007; Zbl 1127.14031)]. a complex projective curve; moduli spaces of pairs; rationality; stable rationality Indranil, Biswas; Marina, Logares; Vicente, Munoz, Rationality of the moduli space of stable pairs over a complex curve, (Nonlinear analysis, Springer Optim. Appl., Vol. 68, (2012), Springer New York), 65-77 Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Rational and unirational varieties Rationality of the moduli space of stable pairs over a complex 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 0672.00003.]
We will interpret the ``rank 4 quadrics problem'' in terms of the deformation theory of singularities. This problem was solved by \textit{M. Green} [Invent. Math. 75, 85-104 (1984; Zbl 0542.14018)], over the complex numbers. We will use the deformation theory of Kodaira-Spencer, Grothendieck, and Schlessinger to solve the problem, first over the complex numbers and then also in characteristic p. The precise statement is given as follows:
Theorem. Let C be a nonhyperelliptic curve of genus \(g\geq 5\) over an algebraically closed field of characteristic \(\neq 2.\) Let \(\Theta\) be a theta divisor in the jacobian of C, and let \(I_ 2(C)\) denote the vector space of quadratic polynomials vanishing on the canonical model of C in \({\mathbb{P}}^{g-1}\). Then the tangent cones to \(\Theta\) at rank 4 double points span \(I_ 2(C)\). deformation of singularities; rank 4 quadrics problem; nonhyperelliptic curve; theta divisor in the jacobian R. Smith and R. Varley , Deformations of singular points on theta divisors , Theta Functions- Bowdoin 1987, Proceedings Symp. Pure Math. vol. 49, Part I, A.M.S., 1989, 571-579. Local deformation theory, Artin approximation, etc., Theta functions and abelian varieties, Jacobians, Prym varieties Deformations of singular points on theta divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0734.00015.]
In this entertaining paper, the author describes the difficulties he encountered when searching the literature for a text discussing the foundational theory of algebraic curves over an arbitrary basefield. He points out that for coding theoretical purposes the base field is usually a finite field, and that therefore, restricting to a perfect basefield seems useful and sufficient. He compares various texts and spells out how to translate the different points of view into one another. He concludes that there is not really a text that satisfies his purposes.
In the second part of the paper, the author discusses in some detail an algorithm to desingularize a curve over a finite field. This algorithm does not involve any extensions of the basefield. curve over a finite field; foundational theory of algebraic curves over an arbitrary basefield Vasquez, A. T.: Rational desingularization of a curve defined over a finite field. In: Chudnovsky, D. V. et al. (eds.). Number Theory: New York Seminar 1989--1990 Berlin, Heidelberg, New York: Springer 1991 Other types of codes, Finite ground fields in algebraic geometry, Arithmetic codes, Arithmetic theory of polynomial rings over finite fields, Singularities of curves, local rings Rational desingularization of a curve defined over a finite 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 Recently, Miyaoka and Sakamaki have achieved interesting results on Calabi-Yau threefolds in positive characteristics, which seem to suggest the possibility of generalizing the theory of \(K3\) surfaces to higher dimensions.
The main questions we are interested in here are: What pathological phenomena can occur if we consider Calabi-Yau threefolds in positive characteristics? Can we relate such phenomena to some numerical invariants such as the height of the Artin-Mazur formal groups? Does the Hodge spectral sequence of these Calabi-Yan threefolds degenerate?
In this paper, we construct some Calabi-Yau threefolds which have fibrations whose generic fiber is nonsmooth. We obtain Calabi-Yau threefolds with fibrations over \(\mathbb{P}^1\) whose general fiber is a direct product of an elliptic curve and a rational curve with an ordinary cusp in \(p=2,3\). In section 4, we observe that these Calabi-Yau threefolds have other fibrations whose general fiber is a \(K3\) surface, a normal surface with rational double points whose resolution is a \(K3\) surface. It also turns out that all these Calabi-Yau threefolds are unirational and the height of their Artin-Mazur formal groups is infinity. So they are in a class which should be called supersingular Calabi-Yau.
In the final section, applying arguments of Rudakov and Shafarevich, we show that the Hodge duality holds for these Calabi-Yau threefolds. non smooth generic fiber; Calabi-Yau threefolds; positive characteristics; Hodge spectral sequence; fibrations; product of an elliptic curve and a rational curve; Hodge duality Hirokado M. (2001). Calabi-Yau threefolds obtained as fiber products of elliptic and quasi-elliptic rational surfaces. J. Pure Appl. Algebra 162: 251--271 Calabi-Yau manifolds (algebro-geometric aspects), \(3\)-folds, Rigid analytic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects) Calabi-Yau threefolds obtained as fiber products of elliptic and quasi-elliptic 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 The author defines a class of nodal algebraic curves in \({\mathbb{C}}^ 2\), the so called ``negative to infinity'' curves. This class includes the curves of the form \(C\setminus L\subset {\mathbb{P}}^ 2\setminus L\), where C is a nodal curve in \({\mathbb{P}}^ 2\) and L is a general line. This includes also the general curve parametrized by two polynomials of one variable of fixed degrees or the general curve having a prescribed Newton polygon. Let \(K\subset {\mathbb{C}}^ 2\) be a curve as above. The main result asserts that one can associate to each irreductible component of K an element in \(\pi_ 1({\mathbb{C}}^ 2\setminus K)\) in such a way that these elements generate \(\pi_ 1({\mathbb{C}}^ 2\setminus K)\) and commute with each other when the corresponding components intersect. As the inclusion \(({\mathbb{P}}^ 2\setminus L)\setminus (C\setminus L)\hookrightarrow {\mathbb{P}}^ 2\setminus C\) induces a surjection of fundamental groups, one refinds the result of Fulton and Deligne on the commutativity of \(\pi_ 1({\mathbb{P}}^ 2\setminus C)\). A computer algorithm to calculate \(\pi_ 1({\mathbb{C}}^ 2\setminus K)\) in terms of coefficients of the equations defining the curve is also included. fundamental group of the complement of a plane algebraic curve; nodal algebraic curves; computer algorithm S. Yu. Orevkov, ''The fundamental group of the complement of a plane algebraic curve,''Mat. Sb. [Math. USSR-Sb.],137 (179), No. 2, 260--270 (1988). Coverings in algebraic geometry, Singularities of curves, local rings, Software, source code, etc. for problems pertaining to algebraic geometry, Surfaces and higher-dimensional varieties The fundamental group of the complement of a plane algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(p\) be a prime not equal to 2 or 3. We determine the group of all modular units on the modular curve \(X_1(2p)\), and its full cuspidal class number. We mention a fact concerning the non-existence of torsion points of order 5 or 7 of elliptic curves over \(\mathbb Q\) of square-free conductor \(n\) as an application of a result by \textit{A. Agashe} [Rational torsion in elliptic curves and the cuspidal subgroup, preprint, \url{http://arxiv-web3.library.cornell.edu/abs/0810.5181}] and the cuspidal class number formula for \(X_0(n)\). We also state the formula for the order of the subgroup of the \(\mathbb Q\)-rational torsion subgroup of \(J_1(2p)\) generated by the \(\mathbb Q\)-rational cuspidal divisors of degree 0.
In the erratum we correct a theorem on the conductor of elliptic curves over \(\mathbb Q\) given in the introduction. modular curve; modular unit; cuspidal class number; elliptic curve; Jacobian variety; torsion subgroup T. Takagi, The cuspidal class number formula for the modular curves \(X_{1}(2p)\), J. Math. Soc. Japan, 64 (2012), 23-85. Arithmetic aspects of modular and Shimura varieties, Elliptic curves over global fields, Elliptic and modular units, Rational points, Modular and Shimura varieties The cuspidal class number formula for the modular curves \(X_1(2p)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author defines the notion of a ``Swan conductor'', which is the measure of wildness of ramification for an element of the Brauer group of a curve over a local field. Then there is a relationship established between these Swan conductors and those defined by \textit{K. Kato} in 1989 [e.g. in: Algebraic analysis, geometry and number theory, Proc. JAMI Inaugur. Conf., Baltimore 1988, 191-224 (1989; Zbl 0776.14004)] for Brauer groups of Henselian discrete valuation fields. Swan conductor; wildness of ramification; Brauer group of a curve over a local field; Henselian discrete valuation fields Yamazaki T.: On Swan conductors for Brauer groups of curves over local fields. Proc. Amer. Math. Soc. 127, 1269-1274 (1999). Ramification problems in algebraic geometry, Brauer groups of schemes, Arithmetic ground fields for curves, Curves over finite and local fields, Local ground fields in algebraic geometry, Ramification and extension theory On Swan conductors for Brauer groups of curves over local fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is a lovely, user-friendly introduction to curves defined over a finite field and various connections to coding theory and finite geometry. The Hasse-Weil bound (and Serre's improvement) for the number of rational points on a nonsingular model of a plane curve is discussed. A plane nonsingular curve of genus \(g\) over \(GF(q)\) is called maximal if the number of rational points on it is \(q+ 1+2g \sqrt q \). In particular, the Hermitian curve in \(PG (2,q)\) is maximal, since this curve has \(q \sqrt q +1\) points in \(PG (2,q)\) and its genus is \({1\over 2} ( q - \sqrt q)\). It turns out that the genus of any maximal curve is bounded above by the genus of the Hermitian curve. Moreover, a result of \textit{H. G. Rück} and \textit{H. Stichtenoth} [J. Reine Angew. Math. 457 185-188 (1994; Zbl 0802.11053)] shows that any maximal curve over \(GF(q)\) of genus \({1\over 2} ( q - \sqrt q)\) is isomorphic to the Hermitian curve.
Beginning with the Hermitian curve \(X^{\sqrt q +1} + Y^{\sqrt q +1} + Z^{\sqrt q +1}\), one can write this polynomial in different affine forms, and then find curves that it covers. In particular, the Fermat curve \({\mathcal F}_t\) is given by \(x^{(\sqrt q +1)/t} + y^{(\sqrt q+1)/t} + 1\). The paper concludes by mentioning the following recent result of A. Cossidente, G. Korchmáros, F. Torres, and the author: any (plane, nonsingular) maximal curve of degree \({1\over 2} ( \sqrt q +1)\) is isomorphic to the Fermat curve \({\mathcal F}_2\). maximal curves; genus; Hasse-Weil bound; Hermitian curve; Fermat curve; curves over a finite field; configurations; number of rational points Curves over finite and local fields, Arithmetic ground fields for curves, Combinatorial aspects of finite geometries Curves and configurations in finite spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a curve in the projective \(n\)-space over an algebraically closed field of positive characteristic, having the property that every secant line to \(C\) is at least a trisecant line (this cannot happen in characteristic zero). The authors prove results on the postulation and the index of regularity of a generic hyperplane section \(S\) of \(C\). They also prove that certain natural groups cannot be isomorphic to the monodromy groups for the family of hyperplane sections of \(C\). hyperplane sections of a curve; positive characteristic; trisecant line; postulation; index of regularity; monodromy groups Ballico, E.; Cossidente, A.: On the generic hyperplane section of curves in positive characteristic. J. pure appl. Algebra 102, 243-250 (1995) Special algebraic curves and curves of low genus, Finite ground fields in algebraic geometry On the generic hyperplane section of curves in positive characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies the existence of smooth curves passing through the vertex of the cone over a (possibly singular) plane curve \(D\) of degree \(d\). Given a positive integer \(m\), he gives conditions for there to exist such curves that meet each ruling of the cone \(m\) times. In particular, they exist if \(D\) is rational -- in this case, the proof consists in looking at curves on the desingularization of the cone. If \(D\) is not rational, such curves exist under certain conditions on \(m\) and \(D\) as well as on the characteristic of the base field. This proof involves a generalization of Cayley's classical monoid construction.
The results can be applied to the problem of finding smooth space curves which are set-theoretic complete intersections of a cone with some other surface. cone over a plane curve; ruling of a cone; space curves; complete intersections David B. Jaffe, Smooth curves on a cone which pass through its vertex, Manuscripta Math. 73 (1991), no. 2, 187 -- 205. Projective techniques in algebraic geometry, Plane and space curves, Rational and ruled surfaces, Complete intersections Smooth curves on a cone which pass through its vertex | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Picard sheaves have been studied from differents aspects. \textit{Y. Li} [Int. J. Math. 2, 525-550 (1991; Zbl 0751.14019)] proves the stability of the Picard bundle \({\mathcal W}\) over the moduli space \({\mathcal M}(n,d)\) of stable bundles of rank \(n\) and degree \(d\). In general the restrictions of stable bundles need not be stable. In this paper we study the restriction \({\mathcal W}_\xi\) of the Picard bundle \({\mathcal W}\) to the subvariety \({\mathcal M}(n,\xi)\) of stable bundles with fixed determinant \(\xi\). We give a condition to get polystability. If such a condition is satisfied for rank 2 then \({\mathcal W}_\xi\) is stable and the connected component of the moduli space of stable bundles over \({\mathcal M}(2,\xi)\) with the same Hilbert polynomial as \({\mathcal W}_\xi\) containing \({\mathcal W}_\xi\) is isomorphic to the Jacobian \(J\) of the curve. Picard sheaves; polystability; moduli space of stable bundles; Jacobian Brambila-Paz L, Hidalgo-Solís L and Muciño-Raymondo J, On restrictions of the Picard bundle. Complex geometry of groups (Olmué, 1998) 49--56;Contemp. Math. 240;Am. Math. Soc. (Providence, RI) (1999) Vector bundles on curves and their moduli, Picard groups, Fine and coarse moduli spaces, Families, moduli of curves (algebraic) On restrictions of the Picard bundle | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The article gives a description of the author's results on the fundamental group of the complement of a plane curve \(C\) obtained by studying the global and local Alexander invariants attached to it. Globally this is the product of the invariants of the module \(H\), first homology group (with coefficients in a field) of the infinite cyclic covering of \(\mathbb{P}^2\) branched along \(C\) and a line \(L\) in general position. Locally they are the characteristic polynomials of the local monodromies around the singularities of \(C\). The author establishes a divisibility relation between the two. Applying it to cuspidal curves he obtains some informations on a torsion of \(H\) (which can be deduced also from the paper of \textit{R. Randell} [in; Topology, Proc. Symp., Siegen 1979, Lect. Notes Math. 788, 145--164 (1980; Zbl 0433.14020)]) and on the irregularity of a finite cyclic cover of \(\mathbb{P}^2\) branched along \(C\) and \(L\) (see also {\S} 3, lemma 10 of the reviewer's paper in [Invent. Math. 68, 477--496 (1982; Zbl 0489.14009)]). Finally he gives examples where the Alexander invariants are computable in terms of superabundance of suitable linear systems.
[For the entire collection see Zbl 0509.00008.] singularities; fundamental group of the complement of a plane curve; Alexander invariants A. Libgober, ''Alexander invariants of plane algebraic curves,'' In:Proc. Symp. Pure. Math., Vol. 40 (1983), pp. 29--45. Coverings in algebraic geometry, Singularities in algebraic geometry, Covering spaces and low-dimensional topology, Topological properties in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Special algebraic curves and curves of low genus Alexander invariants of plane algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mu\) (X) be (up to sign) the defect of the topological Euler characteristic of a hypersurface X of a compact complex manifold. In the paper the elementary properties of the behaviour of \(\mu\) (X), when X varies in a linear system of divisors, are established. They are used to give a new proof of the Dimca-Némethi formula for the multiplicity of the dual variety and to generalize it to the case of any codimension. hypersurface of compact complex manifold; defect of the topological Euler characteristic; linear system of divisors; multiplicity of the dual variety Parusiński A., Bull. London Math. Soc 23 pp 428-- (1991) Hypersurfaces and algebraic geometry, Duality theorems for analytic spaces, Divisors, linear systems, invertible sheaves Multiplicity of the dual 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 paper is a detailed study of the ring of global differential operators on a rational projective curve \(X\). The authors prove that this ring \(D(X)\) is a Noetherian domain of Krull dimension one and global dimension two. It is a finitely generated \(k\)-algebra with a unique minimal nonzero ideal \(J(X)\). The quotient \(D(X)/J(X)\) is of finite dimension over \(k\) \((k\) is assumed to be algebraically closed and of characteristic 0 throughout.) A complete description of \(D(X)\) in terms of matrix algebra is also obtained in case the normalisation map is injective. It is proved that \(D(X)\) is not Morita equivalent to \(D(\mathbb{P}^ 1)\) in contrast with the affine case [see \textit{S. P. Smith} and \textit{J. T. Stafford}, Proc. Lond. Math. Soc., III. Ser. 56, No. 2, 229-259 (1988; Zbl 0672.14017)]. For any singular projective rational curve \(Y\) with injective normalisation map, \(D(X)\) is Morita equivalent to \(D(Y)\). The simple example of cuspidal plane cubic is worked out in detail in \(\S5\). The final section deals with differential operators with coefficients in an invertible sheaf.
Let \(D_ X\) denote the sheaf of differential operators on \(X\), then \(D(X)=\Gamma(X,D_ X)\). The authors show that tensoring with \(D_ X\) defines an exact functor \(F:(D(X)\)-\(\bmod)\to(D_ X\)-\(\bmod)\) making the latter the quotient category of the former. However, this is not an equivalence of categories unlike the case \(X=\mathbb{P}^ 1\). The reason is that \(\Gamma(X,-):(D_ X\)-\(\bmod)\to(D(X)\)-\(\bmod)\) is not exact. The only \(D(X)\)-modules killed by \(F\) are copies of \(H^ 1(X,{\mathcal O}_ X)\). ring of global differential operators on a rational projective curve Holland, M. P.; Stafford, J. T.: Differential operators on rational projective curves. J. algebra 147, 176-244 (1992) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Special algebraic curves and curves of low genus, Commutative rings of differential operators and their modules Differential operators on rational projective curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians As is well-known, the concept of divisors lies in the center of algebraic geometry, and its theory is well established for normal integral schemes. In recent times, the needs are felt to have a more general theory of ``generalized divisors'' over nonnormal schemes, e.g., in the classification theory of space curves. The author developed previously such a theory for Gorenstein curves. In this paper, he describes a satisfying general theory of generalized divisors. The schemes \(X\) under considerations are not necessarily integral but are only assumed to satisfy the following condition: the local rings \({\mathfrak O}_{X,x}\) are Gorenstein whenever \(\text{depth} ({\mathfrak O}_{X,x}) \leq 1\), i.e., \({\mathfrak O}_{X,x}\) is ``quasi-normal'' in the sense of Vasconcelos. After necessary preliminaries concerning reflexive modules, Gorenstein rings and duality, a generalized divisor is defined to be a reflexive coherent \({\mathfrak O}_X\)-submodule \(I\) of \({\mathcal K}_X\), the sheaf of total quotient rings of \({\mathfrak O}_X\). A generalized divisor \(I\) is said to be Cartier (resp. almost Cartier) if it is invertible (resp. invertible out of a closed subset of codimension \(\geq 2)\). The sum and the inverse of generalized divisors are defined, but in general the set of generalized divisors does not form a group. The set \(\text{GPic} (X)\) and the groups \(\text{APic} (X)\), \(\text{Pic} (X)\) of linear equivalence classes of generalized divisors, almost Cartier divisors and Cartier divisors are defined, and the general properties of these set and groups are examined in detail. As an application (or a dessert?) of the general theory, the author brushes up the foundations of the theory of liaison (or linkage) of subschemes of projective space, which have recently attracted some attention both in algebraic geometry and commutative algebra. The general theory is illustrated in the case of curves, some singular surfaces in \(\mathbb{P}^3\), and ruled cubic surfaces in detail, and the divergences from the classical theory of divisors on normal schemes are carefully examined.
This clear exposition of the fundamental notions will be expected to provide useful tools in the study of algebraic geometry in future. generalized divisor; Cartier divisors; liaison; linkage Hartshorne, R.: Generalized divisors on Gorenstein schemes. In: Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), \textbf{8}, pp.~287-339 (1994) Divisors, linear systems, invertible sheaves, Linkage, Vector bundles on curves and their moduli, Linkage, complete intersections and determinantal ideals, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Generalized divisors on Gorenstein 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 This is a short announcement of a recent result of the author. The result asserts that for a modulus \({\mathfrak m}\) of a smooth projective surface X over a field k, the category of rational maps from X to any commutative algebraic groups with modulus \({\mathfrak m}\) has an initial object; that is to say, there exists the generalized Albanese variety of (X,\({\mathfrak m})\). Concerning the concept of the modulus for a surface one can refer to \textit{K. Kato} and \textit{S. Saito} [see Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 103-152 (1983; Zbl 0544.12011)]. generalized Albanese variety; modulus for a surface H. Önsiper, Generalized albanese varieties for surfaces , Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 2, 331-332. Picard schemes, higher Jacobians Generalized Albanese varieties for surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review concerns special subvarieties of moduli spaces. Let \(\mathcal A_g\) denote the moduli space of principally polarized abelian varieties of dimension \(g\). A classical problem asks whether \(\mathcal A_g\) contains any Shimura curves which are generically contained in the Torelli locus \(\mathcal J_g\), i.e.\ the locus of Jacobians of smooth curves of genus \(g\). Previously a series of examples have been constructed by de Jong, Moonen, Mumford, Noot, Oort and others, mostly arising as Galois covers of \(\mathbb P^1\) with varying branch points. Recently there have been several new developments, starting from the non-abelian Galois covers studied in [\textit{P. Frediani} et al., Int. Math. Res. Not. 2015, No. 20, 10595--10623 (2015; Zbl 1333.14023)] and the purely analytic constructions in genus 3 by the authors in [\textit{S. Grushevsky} and \textit{M. Möller}, Int. Math. Res. Not. 2016, No. 6, 1603--1639 (2016; Zbl 1338.14046)] (which produce, in fact, infinitely many Shimura curves contained in the locus of hyperelliptic Jacobians).
The present paper introduces a geometric construction for infinitely many Shimura curves generically contained in \(\mathcal J_4\), using \(\mathbb Z/3\mathbb Z\) Galois covers of elliptic curves following \textit{G. P. Pirola} [J. Reine Angew. Math. 431, 75--89 (1992; Zbl 0753.14040)]. The proof builds on a method going back to \textit{G. Shimura} [Ann. Math. (2) 78, 149--192 (1963; Zbl 0142.05402)] to work out moduli spaces of abelian varieties of given endomorphism ring and polarization (the so-called PEL-Shimura varieties). In particular, the authors compute explicitly the period matrices involved.
Independently, the same curves (and other covers of elliptic curves) were discovered by \textit{P. Frediani} et al. [Geom. Dedicata 181, 177--192 (2016; Zbl 1349.14104)].
Note that for sufficiently large \(g\), it has recently been proved that the Torelli locus does not contain special subvarieties of certain types, for instance Shimura curves of Mumford type, of maximal variation or of hyperelliptic Jacobians [\textit{X. Lu} and \textit{K. Zuo}, J. Math. Pures Appl. (9) 108, No. 4, 532--552 (2017; Zbl 1429.14016)], [\textit{P. Frediani} et al., Geom. Dedicata 181, 177--192 (2016; Zbl 1349.14104)]. Meanwhile the Shimura curves constructed presently contain some curves which are of neither type. abelian variety; Jacobian; Shimura curve; Torelli locus; Galois cover; period matrix Modular and Shimura varieties, Jacobians, Prym varieties Explicit formulas for infinitely many Shimura curves in genus \(4\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a prime number \(p\) and a positive integer \(r\), let \(C_0(p^r)\) be the \({\mathbb Q}\)-rational cuspidal subgroup of the Jacobian \(J_0(p^r)\) of the classical modular curve \(X_0(p^r)\) over \({\mathbb Q}\). In Math. USSR, Izv. 6(1972), 19-64 (1973); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 36, 19-66 (1972; Zbl 0243.14008), \textit{Yu. I. Manin} has shown that \(C(p^r)\) is a finite abelian group, and one can state the question to find this group for any such \(p,r\). For \(r = 1\) and any prime \(p\) it is computed by \textit{A. P. Ogg} that the group \(C(p)\) is cyclic of order \((p-1)/(p-1,12)\) [see: Analytic Number Theory, St. Louis Univ. 1972, Proc. Symp. Pure Math. 24, 221-231 (1973; Zbl 0273.14008)]. For \(p \geq 5\) let \(a = a(p) = (p-1)/(p-1,12)\) and \(b = b(p) = (p+1)/(p+1,12)\). For such \(p\), \(a = a(p)\), \(b= b(p)\) it is shown by Lorenzini that the prime-to-2p part \(C(p^r)^{(2p)}\) of \(C(p^r)\) is isomorphic to the prime-to-2 part of \(({\mathbb Z}/a{\mathbb Z})^r \times ({\mathbb Z}/b{\mathbb Z})^{r-1}\), provided \(p \not\cong 11 \bmod 12\) [\textit{D. I. Lorenzini}, Compos. Math. 96, No. 2, 149-172 (1995; Zbl 0846.14017)].
In the paper under review the group \(C(p^r)\) is computed entirely for all primes \(p \geq 3\) and any positive integer \(r\). This result is applied to determine the component group \({\Phi}_{p^r}\) of the Néron model of \(J_0(p^r)\) over \({\mathbb Z}_p\) for \(p \geq 5\), this way extending a result of Lorenzini (loc. cit.). It is studied also the action of the Atkin-Lehner involution on the \(p\)-primary part of \(C(p^r)\), as well the effect of degeneracy maps on the component groups. modular curve; cuspidal subgroup of the Jacobian; Atkin-Lehner involution; degeneracy maps Ling, S., On the Q-rational cuspidal subgroup and the component group of \(J_0(p^r)\), Israel J. Math., 99, 1, 29-54, (1997) Jacobians, Prym varieties, Modular and Shimura varieties, Arithmetic ground fields for curves On the \(\mathbb{Q}\)-rational cuspidal subgroup and the component group of \(J_0(p^r)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a normal affine variety over an algebraically closed field \(k\). The author of this paper studies the group \(\mathcal{O}^*(X)\) of units in the ring \(\mathcal{O}(X)\) of regular functions on \(X\). (If \(X\) is projective, \(\mathcal{O}(X)=k\), so the question is not interesting.)
The author considers two main cases: cyclic covers of the affine space (Section 2) and restriction of a cyclic cover of the projective space to an open set (Section 3). In the first case, let \(A = k[x_1, \ldots, x_m]\) and \(T = A[z]/(z^n-f)\) for some irreducible square-free \(f \in A\) and an \(n\) invertible in \(k\). The author conjectures that \(T^*=k^*\) and proves it when \(n\) is a prime number or \(n=4\). The principal tool is Galois cohomology, of the Galois group of \(T/A\) (more precisely, of the Galois extension \(T[z^{-1}] / A[f^{-1}]\)). In the second case, let \(\pi: Y \to \mathbb{P}^m\) be a cyclic cover, and \(\pi: X \to U\) be a restriction to an open set. This case is split into two sub-cases, depending on whether \(\pi\) is ramified over \(\mathbb{P}^m \setminus U\). The results obtained in the two sub-cases are applied to affine curves and certain affine hypersurfaces in \(\mathbb{A}^m\).
Throughout the paper, there are numerous examples, making the paper very friendly to the reader. units in a ring; affine algebraic variety; group of units; class group; Galois cohomology; étale cohomology DOI: 10.1142/S0219498814500650 Divisibility and factorizations in commutative rings, Divisors, linear systems, invertible sheaves, Affine geometry The group of units on an affine 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 note analyzes the properties of \(| \omega_D| \) where \(D\) is a 1-connected curve on a smooth algebraic projective surface \(S\). In particular, the authors study the fixed part \(F\) of \(| \omega_D| \) showing that every irreducible component of \(F\) is a smooth rational curve. This can be done by considering \(0\)-maximal sections of the sheaf \(\omega_Z\) where \(Z\) is a curve in \(F\).
For any reducible curve \(C\) on a smooth surface \(S\) and for any invertible sheaf \(\mathcal L\) on \(C\) with \(H^0(C, \mathcal L)\neq 0\), let us consider a section \(s\neq 0\) such that \(s\) vanishes identically on some component of \(C\). Let \(C_s\) the biggest curve in \(C\) with \(s_{| C_s}=0\). Then \(s\) is a \(0\)-maximal section if there is no global section \(t\neq 0\) such that \(C_s\leq C_t\).
Using this tool, one can compute the numerical data of the irreducible components of \(F\) and obtain information about the restriction maps \(H^0(D,\omega_D)\longrightarrow H^0(C, \omega_D)\) for these components. dualizing sheaf; base component of the canonical system; 1-connected curve; effective divisor on a surface K. Konno and M. Mendes Lopes, The base components of the dualizing sheaf of a curve on a surface, Arch. Math. (Basel) 90 (2008), 395--400. Divisors, linear systems, invertible sheaves, Surfaces and higher-dimensional varieties, Special algebraic curves and curves of low genus The base components of the dualizing sheaf of a curve 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 See the preview in Zbl 0512.14032. smooth complete curve; principal bundle; stable rational variety; moduli variety of stable vector bundles Ballico E.: Stable rationality for the variety of vector bundles over an algebraic curve. J. Lond. Math. Soc. 30(1), 21--26 (1984) Rational and unirational varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special algebraic curves and curves of low genus, Algebraic moduli problems, moduli of vector bundles Stable rationality for the variety of vector bundles over an algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) denote an irreducible curve of degree \(d\) in the projective space \(\mathbb{P}^ n\) over the algebraic closed field \(K\). For any smooth point \(P\in C\) one can find a linear space \(V_ i\) of dimension \(i-1\) that is osculating \(C\) at \(P\), i.e. such that it intersects \(C\) at \(P\) more than any other linear space of the same dimension. Let \(b_ i(P)\) denote the order of contact of \(V_ i\) and \(C\) at \(P\) and let \(b_ i\) denote the value for general \(P\). If \(\text{char} (K)=0\) one has \(b_ i=i\) but if \(\text{char} (K)=p>0\) it may happen that \(b_ i>i\). -- Two extremal cases are considered: (a) \(b_ n=d\) and (b) \(b_{n-1} \geq d-2\). In particular the authors prove the following theorem:
Assume \(d \geq 6\), \(d>n+1\) and \(b_{n-1} \geq d-2\), then \(p_ a(C) \leq 1\). If either \(p \neq 2\) or \(d\) is odd, then the normalization of \(C\) is rational. osculating flag on a projective curve; characteristic \(p\); order of contact [BR] Ballico E., Russo B.,On the general osculating flag to a projective curve in characteristic p, Comm. in Algebra (to appear). Projective techniques in algebraic geometry, Special algebraic curves and curves of low genus, Finite ground fields in algebraic geometry On the general osculating flag to a projective curve in characteristic \(p\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0694.00007.]
The author gives a survey on arithmetical properties of Dirichlet series and their implications with respect to the arithmetic of algebraic number fields and elliptic curves. The main topic treated concerns the latest development in the theory of L-series of elliptic curves. In particular, the famous conjecture of Birch and Swinnerton-Dyer and the proof of partial cases by Coates and Wiles, Rubin, Gross and Zagier and by Kolyvagin are mentioned.
Further topics discussed include the Hasse-Weil conjecture regarding the analytic continuation and functional equation of the L-series of an elliptic curve, the Shimura-Taniyama conjecture according to which every elliptic curve over \({\mathbb{Q}}\) is modular and Serre's conjecture concerning modular representations arising from Galois action on p- division points of elliptic curves over \({\mathbb{Q}}\). The report culminates in the author's own contributions to the surprising discoveries that Serre's conjecture implies the Shimura-Taniyama conjecture and that Fermat's last theorem follows from the Shimura-Taniyama conjecture.
The article makes fascinating reading if one disregards some misleading notations and a considerable number of (minor) misprints. Euler product; arithmetic surface; Jacobian zeta function; modular curve; survey; Dirichlet series; L-series of elliptic curves; conjecture of Birch and Swinnerton-Dyer; Hasse-Weil conjecture; analytic continuation; functional equation; Shimura-Taniyama conjecture; Serre's conjecture; modular representations Elliptic curves over global fields, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Elliptic curves, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Other Dirichlet series and zeta functions L-series of elliptic curves: Results, conjectures and consequences | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(X@>f>>B\) be a family of abelian varieties of dimension \(g\) over an algebraically closed field \(k\), and let \({\mathcal L}\) be a symmetric line bundle on \(X\), trivialized along the zero section \(B@>s>>X\), which defines a polarization of type \(D=(d_1, \dots, d_g)\) on the fibres. A general theorem of \textit{G. Faltings} and \textit{Ch.-L. Chai} [``Degenerations of abelian varieties'' (1990; Zbl 0744.14031)], which even holds for abelian schemes, states that \(8d^3\cdot \det(f_*{\mathcal L})= -4d^4\cdot s^*\omega_{X/B}\) in \(\text{Pic}(B)\), where \(\omega_{X/B}\) is the relative dualizing sheaf on \(X\) and \(d:=d_1\cdot\dots\cdot d_g\).
The author proves a refinement of this theorem for the special case of complex-analytic abelian varieties. More precisely, he shows that the torsion factor in the Faltings-Chai identity can be improved in the complex case. The main results are the following two theorems.
Theorem A: Let \((f:X\to B,{\mathcal L})\) be a family of complex abelian varieties of dimension \(g\) and polarization type \(D=(d_1, \dots, d_g)\), where \(d_1|\cdots |d_g\) and \(d:=d_1\cdot\dots\cdot d_g\). Then
\[
8\cdot\det (f_*{\mathfrak L})= -4d\cdot s^*\omega_{X/B}
\]
in \(\text{Pic}(B)\), except when \(3 |d_g\) and \(\text{gcd} (3,d_{g-1})=1\), in which case the identity becomes \(24\cdot \det(F_*{\mathcal L})= -12d\cdot s^*\omega_{X/B}\).
Theorem B: If, under the assumptions of theorem A, the line bundle \({\mathfrak L}\) is totally symmetric and \(g\geq 3\), then \(\det(f_*{\mathfrak L})= -12d\cdot s^*\omega_{X/B}\), except when \(3|d_g\) and \(\text{gcd} (3,d_{g-1})=1\), in which case the identity becomes \(3\cdot\det (f_*{\mathfrak L})=-{3d\over 2}\cdot s^*\omega_{X/B}\).
At the end of the paper, theorem B is applied to the case of the universal Jacobian fibration \(f:J^{g-1}\to M_g\) over the moduli space \(M_g\) of compact Riemann surfaces of genus \(g\). This fibration is an abelian torsor which parametrizes line bundles of degree \(g-1\) on the fibres of the universal curve \(\psi: {\mathcal C}\to M^0_g\) over the moduli space \(M_g^0\subset M_g\) of Riemann surfaces without automorphisms, i.e., over the smooth locus in \(M_g\). This application transpires the following: Let \(\Theta\) be the canonical theta bundle on the universal Jacobian \(J^{g-1}\) and let \(\lambda :=\det (\psi_* \omega_{C/M^0})\) be the Hodge line bundle on \(M_g^0\). Then
\[
\det \bigl(f_* (\Theta^{\otimes n})\bigr)= \textstyle {1\over 2} n^g(n-1)\cdot \lambda
\]
in \(\text{Pic} (M^0_g)\) for any natural number \(n\).
Theorems A and B are proved by carefully refining some theta transformation formulae, in order to obtain suitable transition functions for the line bundle \(\det(f_*{\mathcal L})\) in \(\text{Pic}(B)\). As for theorem C, the author gives two different proofs, one of which utilizes special calculations in the Grothendieck-Riemann-Roch algebra for universal Jacobians à la \textit{L. Moret-Bailly} [``Princeaux de variétés abeliennes'', Astérisque 129 (1985; Zbl 0595.14032)].
The paper is written in a very thorough, comprehensive and clear style, and the results presented here are certainly important and beautiful. Picard groups; moduli spaces of curves; Abel-Jacobi mapping; family of abelian varieties; symmetric line bundle; polarization; Jacobian fibration; theta bundle Algebraic moduli of abelian varieties, classification, Theta functions and abelian varieties, Structure of families (Picard-Lefschetz, monodromy, etc.), Fibrations, degenerations in algebraic geometry Theta line bundles and the determinant of the Hodge bundle | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It has been known for a long time that the context of the (finite) Morley rank theory involves the intuition of algebraic geometry and, more precisely, the methodology of finite-dimensional algebra over algebraically closed fields. From this point of view this context is essentially poorer than the actual Algebraic Geometry since the latter involves, in the complex case, the powerful techniques of the theory of complex manifolds and analytic methods, which are virtually applicable in the positive characteristic case too. One of the aims of the paper is to show that some classic ``transcendental'' structures are in fact very interesting structures of finite Morley rank. On the other hand, as is shown in last sections, model-theoretic analysis under certain assumptions leads to a theory of `infinitesimals' or to a `non-standard analysis' which introduces in a purely algebraic context analytic methods.
The main theorem of the paper states that any compact complex manifold considered as a structure in a natural language is \(\omega\)-stable of finite Morley rank. This theorem had been obtained first directly, but immediately after the proof had been found it was realized that the properties of the compact complex manifolds used in the proof are very close to those E. Hrushovski postulated for a strongly minimal structure to get a field definable in the structure (Talk given at European Summer Meeting of ASL, Berlin, 1989). Combining his idea with the non-trivial examples of arbitrary dimensions we jointly came to the axioms defining a Zariski-type structure. Since then the theory has been essentially developed. The main result of the theory states that a one-dimensional non-locally modular pre-smooth structure can be basically identified with an algebraic curve. The techniques developed in the theory also show that to a great extent the Model Theory of Zariski-type structures is adequate to Algebraic Geometry. survey; analytic cover of a torus; ultrametric analysis; infinitesimals; finite-dimensional algebra over algebraically closed fields; finite Morley rank; analytic methods; compact complex manifold; strongly minimal structure; Zariski-type structure; algebraic curve B. Zilber, On model theory, non-commutative geometry and physics, manuscript. Model-theoretic algebra, Foundations of algebraic geometry, Nonstandard models, Classification theory, stability, and related concepts in model theory Model theory and algebraic geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0747.00024.]
The following generalization of the well-known result of Fulton-Deligne on the abelianness of the fundamental group of the complement of a nodal curve in \(\mathbb{P}^ 2\) is proved.
Theorem. Let \(\Gamma_ 1\), \(\Gamma_ 2\) be (not necessarily irreducible) curves in \(\mathbb{P}^ 2\), \(E\subset\mathbb{P}^ 2\) a line and \(\mathbb{C}^ 2=\mathbb{P}^ 2-E\). Suppose \(\Gamma_ 1\), \(\Gamma_ 2\) intersect transversally and \(\Gamma_ 1\cap\Gamma_ 2\cap E=\varphi\). Then \(\pi_ 1(\mathbb{C}^ 2-\Gamma_ 1\cup\Gamma_ 2)=\pi_ 1(\mathbb{C}^ 2- \Gamma_ 1)\times\pi_ 1(\mathbb{C}^ 2 -\Gamma_ 2)\).
The important point is that \(\Gamma_ i\) is not assumed to intersect \(E\) transversally. Since \(\Gamma_ 1\), \(\Gamma_ 2\) are allowed to be reducible, the above theorem extends easily to finitely many curves \(\Gamma_ 1\), \(\Gamma_ 2,\ldots,\Gamma_ n\) with appropriate assumption. -- The proof uses linear automorphisms of \(\mathbb{C}^ 2\) to deform one of the curves and Zariski's method of pencils. plane curve; complement of a nodal curve in \(\mathbb{P}^ 2\); Zariski problem; abelianness of the fundamental group Homotopy theory and fundamental groups in algebraic geometry, Curves in algebraic geometry, Coverings in algebraic geometry Fundamental group of the complement of affine plane curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be a non-singular complete intersection in a projective space over the complex number field. It is known that the cohomology \(H^{2i}(X,{\mathbb{Q}})\) is generated by the powers of the class of the hyperplane section except for the case \(2i=\dim X\). In this case, the dimension of the space of algebraic cycles of codimension i viewed as a \({\mathbb{Q}}\)-subvectorspace of \(H^{2i}(X,{\mathbb{Q}})\) is called the middle Picard number of X. \textit{P. Deligne} [SGA 7 II, Lect. Notes Math. 340 (1973] showed that X has the middle Picard number 1 if X has a type different from (2;3), (2n;2), (2n;2,2). Here a complete intersection is said to be of type (N;\b{a}), where \b{a}\(=a_ 1,...,a_ d\) is a sequence of positive integers, if it is a complete intersection of hypersurfaces of degree \(a_ 1,...,a_ d\) in \(P^{N+d}\). The paper under review proves that for a type as above, there exists a complete intersection of middle Picard number 1. algebraic cycle; type of a complete intersection; middle Picard number T.~Terasoma. Complete intersections with middle {P}icard number 1 defined over {\({\mathbf Q}\)}. {\em Math. Z.}, 189(2):289--296, 1985. https://doi.org/10.1007/BF01175050; zbl 0579.14006; MR0779223 Picard groups, Complete intersections Complete intersections with middle Picard number 1 defined over \({\mathbb{Q}}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0747.00028.]
After reviewing a modern treatment of the classical approach to Fano 3- folds via the double projection, the author discusses another one via vector bundles. Instead of proving the existence of a line on a Fano 3- fold \(X\), a vector bundle is built on a general surface section and extended to \(X\). This gives rise to an embedding into a Grassmann variety and the image of \(X\) is described as a linear section of a homogeneous space. A new proof of the genus bound is given as well as some results in higher dimension. canonical curve section; genus; Fano 3-folds; vector bundle; section of a homogeneous space S.~Mukai. Fano 3-folds. In {\em Complex projective geometry ({T}rieste, 1989/{B}ergen, 1989)}, volume 179 of {\em London Math. Soc. Lecture Note Ser.}, pages 255--263. Cambridge Univ. Press, Cambridge, 1992. Also \url{http://www.kurims.kyoto-u.ac.jp/~mukai/paper/Trieste.pdf}. zbl 0774.14037; MR1201387 \(3\)-folds, Fano varieties Fano 3-folds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 general K 3-surface X in the 3-dimensional flag variety \({\mathbb{F}}\) of \({\mathbb{P}}^ 2\) comes with two 2-fold coverings \(X\to {\mathbb{P}}^ 2\) branched along two sextic curves. Starting from the M. Noether theorem which says \(Pic(X)={\mathbb{Z}}^ 2\) we determine by elementary methods the group of automorphism as \(Aut(X)={\mathbb{Z}}_ 2*{\mathbb{Z}}_ 2\), the free product of two cyclic groups of \(rank^ 2.\) Each factor is generated by the Galois transformation belonging to one of the two coverings.
The proof makes use of the faithful representation \(Aut(X)\hookrightarrow O^ +(Pic(X))\) in the orthochronous Lorentz group of the hyperbolic lattice Pic(X), and the latter group can be computed with the help of Dirichlet's unit theorem for the number field \({\mathbb{Q}}[\sqrt{3}]\). general K3-surfaces in the 3-dimensional flag variety projective 2-space; group of automorphisms; orthochronous Lorentz group; Picard group J. Wehler, \(K\)3-surfaces with Picard number 2. Arch. Math. (Basel) 50(1), 73-82 (1988) Special surfaces, Quadratic extensions, Group actions on varieties or schemes (quotients), \(K3\) surfaces and Enriques surfaces, Automorphisms of surfaces and higher-dimensional varieties K3-surfaces with Picard number 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 smooth projective complex curve of genus \(g\geq 2\), and denote by \(U_X(r,0)\) the moduli space of semi-stable vector bundles of rank \(r\) and degree 0 on \(X\). Let \(SU_X(r)\) be the moduli space of rank-\(r\) vector bundles on \(X\) with trivial determinant bundle. The purpose of the paper under review is to introduce and study the push-forward bundles of the pluri-theta line bundles on \(U_X(r,0)\) to the Jacobian \(J(X)\) by the determinant map \(\det: U_X(r,0)\to J(X)\). These bundles \(E_{r,k}\) on \(J(X)\) are called the Verlinde bundles (on the Jacobian), and they are precisely defined by \(E_{r,k} :=\det_* {\mathcal O}(k\cdot \Theta_N)\), where \(\Theta_N\) is the generalized theta divisor on \(U_X(r,0)\) associated with a line bundle \(N\in\text{Pic}^{g-1} (X)\). Then the fibers of those Verlinde bundles are precisely the well-knovm Verlinde spaces of level-\(k\) theta functions on \(SU_X(r)\), the dimension of which is determined by the famous Verlinde formula.
The author's strategy is to study the geometry of the Verlinde bundles \(E_{r,k}\) on the Jacobian \(J(X)\), which provides a more geometric approach to a complete understanding of the Verlinde spaces. In particular, the behavior of the Verlinde bundles under natural operations associated to vector bundles over abelian varieties is thoroughly analyzed, where special emphasis is put on their Fourier-Mukai transforms. This leads to some new duality theorems for certain bundles and their spaces of sections, on the one hand, and to explicit results in the study of effective global generation and normal generation for pluri-theta line bundles on \(U_X(r,0)\), on the other hand.
As to the duality results for Verlinde bundles obtained here, the author's theorems generalize some earlier formulae due to \textit{R. Donagi} and \textit{L. Tu} [Math. Res. Lett. 1, 345-357 (1994; Zbl 0847.14027)] and \textit{A. Beauville}, \textit{M. S. Narasimhan} and \textit{S. Ramanan} [J. Reine Angew. Math. 398, 169-179 (1989; Zbl 0666.14015)], the latter one being known as ``the strange duality at level one'', whereas the explicit bounds for global generatedness, ampleness, or very-ampleness of pluri-theta bundles on \(U_X(r,0)\) are seemingly new and highly interesting. Also, extending a construction method due to \textit{M. Raynaud} (1982), the author shows that the Fourier-Mukai transforms of the dual Verlinde bundles, restricted to certain embeddings of the curve \(X\) into its Jacobian \(J(X)\), provide new examples of base points in the linear system of the determinant bundle \(L\) on \(U_X(r,0)\). In the last section of the paper, it is explained how the foregoing methods and results could be extended to the moduli spaces \(U_X(r,d)\) of rank-\(r\) vector bundles of arbitrary degree \(d\) over a curve \(X\).
Altogether, this is an important contribution towards the understanding of the geometry of moduli spaces of vector bundles over a curve, with a strong potential for generalization to vector bundles on smooth algebraic surfaces. The paper is very well written, lucid, detailed, rigorous and enlightening. moduli of vector bundles on curves; theta divisors; generalized theta functions; Fourier-Mukai transform; non-abelian theta functions; pluri-theta line bundles; Verlinde bundles; Verlinde formula --. --. --. --., Verlinde bundles and generalized theta linear series , Trans. Amer. Math. Soc. 354 (2002), 1869--1898. JSTOR: Vector bundles on curves and their moduli, Theta functions and abelian varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Verlinde bundles and generalized theta linear series | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 algebraic variety over an algebraically closed field \(k\) with singular locus \(\text{Sing}(X)\) and \(S\subseteq X\) a closed subvariety. Let \(T=\text{Spec} k[[t]]\) and \(T_ N=\text{Spec} k[[t]]/t^{N+1}\). An analytic arc is \(\gamma:T\to X\) (resp. \(T_ N\to X)\); \(\gamma\) is called an \(S\text{-arc}\) if \(\gamma(0)\in S\). Using the strong approximation theorem one can prove that an \(N\)-truncated arc can be lifted to an arc if \(N\) is big enough. The coefficients of an \(N\)- truncated \(S\)-arc define a point in some \(k^ M\). The closure of these points (coming from all \(N\)-truncated \(S\)-arcs) is called the Nash variety \(V(X,S,N)\).
The paper gives a foundation of the theory of Nash varieties. They may carry a non-reduced structure which turns out to be useful to characterize the smoothness of \(X\) or \(S\) in terms of \(V(X,S,N)\). Each irreducible component of \(V(X,S,N)\) contains a dense open set the points of which correspond to a family of \(N\)-truncated \(S\)-arcs. It is proved that this family considered as family of irreducible curve singularities is equisingular. Using this fact one may use the corresponding invariants (which are constant in an equisingular family) as invariants of the singularities of \(X\). equisingular family of irreducible curve singularities; truncated \(S\)- arcs; Nash variety A. Nobile, On Nash theory of arc structure of singularities, Ann. Mat. Pura Appl. (4) 160 (1991), 129--146. Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Real-analytic and Nash manifolds, Singularities in algebraic geometry On Nash theory of arc structure of singularities | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The well-known Bernstein-Kushnirenko theorem from the theory of Newton polyhedra relates algebraic geometry and the theory of mixed volumes. Recently, the authors have found a far-reaching generalization of this theorem to generic systems of algebraic equations on any algebraic variety. In the present note we review these results and their applications to algebraic geometry and convex geometry. Bernstein-Kushnirenko theorem; semigroup of integral points; convex body; mixed volume; Alexandrov-Fenchel inequality; Brunn-Minkowski inequality; Hodge index theorem; intersection theory of Cartier divisors; Hilbert function Kaveh, K., Khovanskii, A.G.: Algebraic equations and convex bodies. In: Itenberg, I., Jöricke, B., Passare, M. (eds.) Perspectives in Analysis, Geometry, and Topology, on the Occasion of the 60th Birthday of Oleg Viro, Progress in Mathematics, vol. 296, pp. 263-282. Birkhäuser Verlag Ag (2012) Mixed volumes and related topics in convex geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), \(n\)-dimensional polytopes, Special polytopes (linear programming, centrally symmetric, etc.) Algebraic equations and convex bodies | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 constructing unramified coverings of the affine line in characteristic \(p\), a general theorem about good reductions modulo \(p\) of coverings of characteristic zero curves is proved. This is applied to modular curves to realize \(\text{SL} (2,\mathbb{Z}/n \mathbb{Z})/ \{\pm 1\}\), with \(\text{GCD} (n,6)=1\), as Galois groups of unramified coverings of the affine line in characteristic \(p\), for \(p=2\) or 3. It is applied to the Klein curve to realize PSL(2,7) for \(p=2\) or 3, and to the Macbeath curve to realize PSL(2,8) for \(p=3\). By looking at curves with big automorphism groups, the projective special unitary groups \(\text{PSU}(3,p^ \nu)\) and the projective special linear groups \(\text{PSL} (2,p^ \nu)\) are realized for all \(p\), and the Suzuki groups \(Sz(2^{2\nu+1})\) are realized for \(p=2\). Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels. characteristic \(p\); good reduction; constructing unramified coverings of the affine line; modular curves; Galois groups of unramified coverings of the affine line; Klein curve; Macbeath curve; big automorphism groups; Jacobian varieties Coverings of curves, fundamental group, Inverse Galois theory, Finite ground fields in algebraic geometry, Jacobians, Prym varieties, Special algebraic curves and curves of low genus Construction techniques for Galois coverings of the affine 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 Let \(({\mathbf C},\otimes,1)\) be an abelian, closed symmetric monoidal category containing all small limits and colimits. Denote by \(\text{Comm}({\mathbf C})\) the category of commutative monoid objects in \({\mathbf C}\). Since \({\mathbf C}\) is a closed monoidal category, any commutative monoid \(A\in\text{Comm}({\mathbf C})\) defines another closed symmetric monoidal category \((A\text{-Mod},\otimes_A, A)\), namely the category of \(A\)-modules. Then \(\text{Aff}_{{\mathbf C}}:= \text{Comm}({\mathbf C})^{op}\) is called the category of affine schemes over \({\mathbf C}\). If \(A\) is an object of \(\text{Comm}({\mathbf C})\), then the corresponding object in \(\text{Aff}_{{\mathbf C}}\) will be denoted by \(\text{Spec}(A)\). This kind of generalized algebraic geometry (over an abelian, closed symmetric monoidal category) has been studied widely in the past, mainly in the context of ringed toposes, Grothendieck sites, and Tannaka categories.
Very recently, \textit{B. Toën} and \textit{M. Vaquié} [J. K-Theory 3, No. 3, 437--500 (2009; Zbl 1177.14022)] have introduced the notion of Zariski coverings in the category \(\text{Aff}_{{\mathbf C}}\), thereby determining a Grothendieck site such that the representable presheaves on \(\text{Aff}_{{\mathbf C}}\) are in fact sheaves. Furthermore, these authors defined the concept of a scheme over \({\mathbf C}\) essentially by using sheaves of sets on \(\text{Aff}_{{\mathbf C}}\) and suitable Zariski coverings of them. A scheme \(X\) over \({\mathbf C}\) comes equipped with a functorially defined structure sheaf \({\mathcal O}_X\), with values taken in \(\text{Comm}({\mathbf C})\).
In the paper under review, the author extends this approach by Toën and Vaquié by establishing the notion of relative Picard functor on the category of schemes over \({\mathbf C}\). To this end, quasi-coherent \({\mathcal O}_X\)-modules on a scheme \((X,{\mathcal O}_X)\) over \({\mathbf C}\) are suitably defined, and their basic functorial properties for a particular class of schemes, the so-called bicomplete schemes, are described in full detail. Then, for certain morphisms \(h: (X,{\mathcal O}_X)\to (S,{\mathcal O}_S)\) of bicomplete schemes over \({\mathbf C}\), a relative Picard functor \(\text{Pic}_{X/S}: (\text{Sch}_{{\mathbf C}}/S)^{op}\to Ab\) together with its associated sheaf \(\text{Pic}_{X/S}\) on \(\text{Sch}_{{\mathbf C}}/S\) is defined and analyzed, where \(\text{Sch}_{{\mathbf C}}/S\) stands for the subcategory of bicomplete schemes over a bicomplete base scheme \((S,{\mathcal O}_S)\). The author's main result, in this context, is a theorem (Theorem 1.1.) stating the following:
(a) The relative Picard functor \(\text{Pic}_{X/S}\) defines a separated presheaf on \(\text{Sch}_{{\mathbf C}}/S\).
(b) Under the additional assumption that the structure morphism \(h: (X,{\mathcal O}_X)\to (S,{\mathcal O}_S)\) has a section \(g: (S,{\mathcal O}_S)\to(X,{\mathcal O}_X)\), the relative Picard functor \(\text{Pic}_{X/S}\) defines a sheaf on \(\text{Sch}_{{\mathbf C}}/S\).
This result generalizes a classical theorem on the relative Picard functor in the usual algebraic geometry of schemes over a fixed commutative ring. A lucid discussion of that classical theorem can be found in \textit{S. Kleiman's} masterly essay ``The Picard Scheme'' [in: Fundamental algebraic geometry: Grothendieck's FGA explained. Mathematical Surveys and Monographs 123. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1085.14001)], where it appears as Theorem 9.2.5. in Section 2.
As the author points out, there is some recent similar work on the relative Picard functor on algebraic stacks by \textit{S. Brochard} [Math. Ann. 343, No. 3, 541--602 (2009; Zbl 1165.14023)]. generalization of schemes; symmetric monoidal categories; Grothendieck sites; sheaves; quasi-coherent sheaves; Picard groups; relative Picard functor Banerjee, A., The relative Picard functor on schemes over a symmetric monoidal category, Bull. Sci. Math., 135, 4, 400-419, (2011) Picard schemes, higher Jacobians, Monoidal categories (= multiplicative categories) [See also 19D23], Generalizations (algebraic spaces, stacks), Picard groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Closed categories (closed monoidal and Cartesian closed categories, etc.) The relative Picard functor on schemes over a symmetric monoidal category | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The theory of DAHA-Jones polynomials is extended from torus knots to iterated torus knot, for any reduced root systems and weights. This is inspired by \textit{P. Samuelson}'s construction for the \(\mathfrak{sl}_2\) case [``Iterated torus knots and double affine Hecke algebras'', Preprint, \url{arXiv:1408.0483}]. The paper proves polynomiality, duality and other properties, and computes several examples. They conjecture that these polynomials specialize to Khovanov-Rozansky polynomials, which was since proven by \textit{H. Morton} and \textit{P. Samuelson} [Duke Math. J. 166, No. 5, 801--854 (2017; Zbl 1369.16034)].
The same authors have since extended the DAHA-Jones polynomials to iterated torus links [\textit{A. Beliakova} (ed.) and \textit{A. D. Lauda} (ed.), Categorification in geometry, topology, and physics. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1362.81007)]. double affine Hecke algebra; Jones polynomials; HOMFLY-PT polynomial; Khovanov-Rozansky homology; iterated torus knot; cabling; MacDonald polynomial; plane curve singularity; generalized Jacobian; Betti numbers; Puiseux expansion Cherednik, I.; Danilenko, I., DAHA and iterated torus knots, Algebr. Geom. Topol., 16, 843-898, (2016) Singularities of curves, local rings, Knots and links in the 3-sphere, Hecke algebras and their representations, Braid groups; Artin groups, Compact Riemann surfaces and uniformization, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Lie algebras of linear algebraic groups, Singular homology and cohomology theory DAHA and iterated torus knots | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is an expository presentation of a completely integrable Hamiltonian system of Clebsch top under a special condition introduced by Weber. After a brief account of the geometric setting of the system, the structure of the Poisson commuting first integrals is discussed following the methods by \textit{F. Magri} and \textit{T. Skrypnyk} [``The Clebsch System'', Preprint, \url{arXiv:1512.04872}]. Introducing supplementary coordinates, a geometric connection to Kummer surfaces, a typical class of K3 surfaces, is mentioned and also the system is linearized on the Jacobian of a hyperelliptic curve of genus two determined by the system. Further some special solutions contained in some vector subspace are discussed. Finally, an explicit computation of the action variables is introduced. integrable Hamiltonian system; Clebsch top; Kummer surfaces; K3 surfaces; Jacobian; hyperelliptic curve of genus Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.), Relations of finite-dimensional Hamiltonian and Lagrangian systems with algebraic geometry, complex analysis, special functions, Integrable cases of motion in rigid body dynamics, \(K3\) surfaces and Enriques surfaces, Relationships between algebraic curves and integrable systems The rigid body dynamics in an ideal fluid: Clebsch top and Kummer 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 a prime number \(p\) we denote by \(\mathbb{Q}_p^{\text{nr}}\) the maximal unramified extension of \(\mathbb{Q}_p\). Suppose \(p^v\) exactly divides \(N\) for a given integer \(N\geq 1\). Then we use Carayol's works and the local class field theory to find an extension \(E_v\) of \(\mathbb{Q}_p^{\text{nr}}\) on which the Jacobian \(J_0\) of the modular curve \(X_0(N)\) has a semi-stable reduction and we estimate its degree. unipotent representation; Jacobian; local class field theory; Tate module; exponent of Artin character; maximal unramified extension; semi-stable reduction; modular curve Krir, M.: Degré d'une extension de \({\mathbb{Q}}_p^{\mathrm nr}\) sur laquelle \(J_0(N)\) est semi-stable. Ann. Inst. Fourier (Grenoble) \textbf{46}(2), 279-291 (1996) Representation-theoretic methods; automorphic representations over local and global fields, Abelian varieties of dimension \(> 1\), Arithmetic aspects of modular and Shimura varieties, Varieties over finite and local fields, Ramification and extension theory, Class field theory; \(p\)-adic formal groups, Arithmetic ground fields for curves, Modular and Shimura varieties Degree of an extension of \(\mathbb{Q}_ p^{\text{nr}}\) on which \(J_ 0 (N)\) is semi-stable | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author tries to find new examples of Cohen-Macaulay rings among the coordinate rings of the unions of regular varieties. The method used in this article is to represent a ring as the section ring of a sheaf on a poset. In particular, section \(4\) is devoted to a class of new examples: the generalized face rings. The main result says that the generalized face ring of a simplicial complex K is Cohen-Macaulay if and only if K itself is Cohen-Macaulay. seminormality; poset; Cohen-Macaulay rings; section ring of a sheaf; generalized face ring [Y2] Yuzvinsky, S.: Flasque sheaves on posets and Cohen-Macaulay unions of regular varieties. Adv. Math.73, 24--42 (1989) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Flasque sheaves on posets and Cohen-Macaulay unions of regular varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a convenient note for those who want to know the details of constructing the jacobian varieties of complete non-singular curves. The general construction of Picard varieties is given by \textit{A. Grothendieck} [``Fondements de la géométrie algébrique'', Extrait du Séminaire Bourbaki 1957-1962 (Paris 1962; Zbl 0239.14002)]. jacobian varieties of complete non-singular curves; Picard varieties Jacobians, Prym varieties Sur la construction de Weil de la jacobienne d'une courbe algébrique. (On the Weil construction of the jacobian of an algebraic curve) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a readable survey on the main trends of research on algebraic curves over finite fields that have arisen in the two last decades from the remarkable and unexpected applications of this theory to fields like Information Transmission, Physics or Cryptography.
The survey starts with a review of the zeta function of a curve \(X\) defined over a finite field \(\mathbb F_q\) of characteristic \(p\) and the Hasse-Weil bound on the number \(|X(\mathbb F_q)|\) of \(\mathbb F_q\)-rational points. Then, linear error-correcting codes are introduced, with the attention focused on the problem of determining the weight distribution of some classical codes like Reed-Muller codes or the dual Melas codes, for which this problem comes down to determining the number of rational points of certain families of curves: supersingular curves for Reed-Muller codes and curves with equation \(y^p-y=ax+b/x\) for dual Melas codes.
The utmost striking role of Goppa codes is duly stressed. A section is devoted to the construction of geometric Goppa codes from data in a curve \(X\) and to see how the genus of \(X\) measures the obstruction to attain the Singleton bound. Whereas classical Goppa codes attain the Gilbert-Varshamov bound, it is reminded how Tsfasman, Vladuts and Zink used the reduction of modular curves to construct a sequence of Goppa codes whose relative distance and transmission rate converge to values surpassing the Gilbert-Varshamov bound for \(q\) a square, \(q\geq 49\). Beyond this success, Goppa codes changed the insights and trends in Coding Theory and renewed the interest on curves over finite fields.
In general, finding good Goppa codes translates into the problem of finding curves with the quotient \(|X(\mathbb F_q)|/g\) as large as possible. This leads to the problem of determining the value of \(A(q):=\limsup_{g\to\infty}N_q(g)/g\), where \(N_q(g)\) is the maximum value of \(|X(\mathbb F_q)|\) for curves \(X\) of genus \(g\), defined over \(\mathbb F_q\). Results of Drinfeld-Vladuts, Ihara and the construction of Tsfasman-Vladuts-Zink lead to \(A(q)=\sqrt q-1\), for \(q\) a square. The exact computation of \(A(q)\) for \(q\) non-square is still an open question, and some lower bounds by Serre, Temkine, Niederreiter-Xing and Hajir-Maire are discussed.
The paper reviews then some improvements of the Hasse-Weil bound due to Ihara, Serre, Oesterlé and Kresh-Wetherell-Zieve. It contains some open questions concerning the numbers \(N_q(g)\), like finding the exact value of \(\liminf_{g\to\infty}N_q(g)/g\) or asking for more precise bounds for the maximum value of \(|X(\mathbb F_q)|\) for curves with fixed genus and gonality.
Another section deals with the problem of determining all maximal curves, that is, curves attaining the upper Hasse-Weil bound. An example of such a curve is the Hermitian curve defined over \(\mathbb F_{q^2}\) by the equation \(x^{q+1}+y^{q+1}+z^{q+1}=0\), which has genus \(g=q(q-1)/2\). Stichtenoth has conjectured that every maximal curve is dominated by this Hermitian curve. Another maximal curve over \(\mathbb F_{q^2}\) (for \(q\) odd) is given by \(y^q+y=x^{(q+1)/2}\), which has genus \(g=(q-1)^2/4\). By the work of Rück-Stichtenoth, Fuhrmann-Torres and Korchmaros-Torres any other maximal curve has genus \(g\leq[(q^2-q+4)/6]\). To determine the possible values of \(g\) inside this interval is still an open question.
Finally, the author points out that some of the questions of Algebraic Geometry inspired by Coding Theory are related to stratifications of the moduli space \({\mathcal M}_g\) of curves of genus \(g\). These stratifications are either directly defined on \({\mathcal M}_g\), like the stratification by gonality, or obtained by pull-back under the Torelli map, \(t\colon {\mathcal M}_g\rightarrow {\mathcal A}_g\), of natural stratifications of the moduli space \({\mathcal A}_g\) of principally polarized abelian varieties of dimension \(g\). For example, the stratifications by the Newton polygon or by the structure as a group scheme of the kernel of multiplication by \(p\). At this level, one may study the variation of the number of points in families of curves or ask for other essential questions concerning the locus of Jacobians inside \({\mathcal A}_g\). The open problems on these questions are numerous, deep and extremely interesting. algebraic curve; finite field; linear code; zeta function; moduli space; Jacobian variety van der Geer G.: Coding theory and algebraic curves over finite fields: a survey and questions. In: Applications of Algebraic Geometry to Coding Theory, Physics and Computation, NATO Sci. Ser. II Math. Phys. Chem., vol. 36, pp. 139--159. Kluwer, Dordrecht (2001). Curves over finite and local fields, Algebraic coding theory; cryptography (number-theoretic aspects), Geometric methods (including applications of algebraic geometry) applied to coding theory, Arithmetic codes, Applications to coding theory and cryptography of arithmetic geometry Coding theory and algebraic curves over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(A\) be an abelian variety defined over a number field \(k\). In this short note we give a characterization of the endomorphisms that preserve the height pairing associated to a polarization. We also give a functorial interpretation of this result. endomorphisms of abelian variety; height pairing associated to a polarization Valerio Talamanca, A note on height pairings on polarized abelian varieties, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat.10 (1999), p. 57-60 Algebraic theory of abelian varieties, Arithmetic varieties and schemes; Arakelov theory; heights A note on height pairings on polarized 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 Let \(\mathcal{C}\) be a projective, smooth, absolutely integral curve of genus \(g \in \{2,3\}\) over a finite field \(\mathbf{F}_{\!q}\).
Denote by \(\ell\) an odd prime different from the characteristic of \(\mathbf{F}_{\!q}\). Let \(\mathcal{V}\) be a maximal isotropic subgroup of the \(\ell\)-torsion in the Jacobian \(J_{\mathcal{C}}\) of \(\mathcal{C}\) with respect to the Weil pairing. The author describes algorithms for computing the equation of a curve \(\mathcal{D}\) such that the quotient \(J_{\mathcal{C}}/\mathcal{V}\) is equal to the Jacobian \(J_{\mathcal{D}}\) of \(\mathcal{D}\). Rational fractions expliciting the quotient isogeny \(f : J_{\mathcal{C}} \rightarrow J_{\mathcal{D}}\) are also given.
Actually this work is based on a paper by \textit{J.-M. Couveignes} and \textit{T. Ezome} [LMS J. Comput. Math. 18, 555--577 (2015; Zbl 1333.14038)] which relates how to efficiently evaluate eta and theta functions on Jacobian varieties with dimension \(\ge 2\), and compute \((\ell, \ell)\)-isogenies between Jacobians of genus 2 curves. Milio starts by recalling Couveignes-Ezome algorithm for computing isogenies between \(2\)-dimensional Jacobians. It happens that this algorithm requires the equation of the Kummer surface associated to \(\mathcal{D}\), however Milio shows that there is actually no need for this equation to output the results in dimension 2. Also he optimized the computation of the equation of the curve \(\mathcal{D}\), and the computation of the rational functions describing \(f\). Furthermore he extended the Couveignes-Ezome algorithm to the case of hyperelliptic curves of genus 3. But the generalization faces obstacles concerning genus 3 non-hyperelliptic curves: there are some uncertainties to recover the equation of the curve \(\mathcal{D}\) from data about its Kummer surface. To avoid this, the author uses theta based formulas and the reconstruction of plane quartics from bitangents. hyperelliptic curve; non-hyperelliptic curve; Jacobian varieties; eta function; theta functions; isogeny; Kummer variety Isogeny, Theta functions and abelian varieties, Computational aspects of algebraic curves, Effectivity, complexity and computational aspects of algebraic geometry Computing isogenies between Jacobians of curves of genus 2 and 3 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) denote a finite group and \(M\subset N\subset G\) subgroups. If \(G\) acts on an algebraic curve \(X\) as a group of automorphisms, then the isotypical decomposition of the Jacobian \(JX\) with respect to the given action carries over to the intermediate Prym variety \(P(X/M\to P(X/N)\). It may happen that two such Prym varieties have the same decomposition and are thus isogenous without the corresponding pairs of subgroups being conjugate in \(G\). The paper shows that this occurs in the case of the dihedral group of order 8. Moreover the degree of the isogeny is computed. As a special case the authors obtain a different proof of the result of \textit{S. Pantazis} on the bigonal construction [Math. Ann. 273, 297--315 (1986; Zbl 0566.58028)]. Prym variety; isotypical decomposition of the Jacobian Jacobians, Prym varieties, Isogeny Prym varieties and fourfold covers. II: The dihedral case | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Period domains are defined by a semistability condition on weighted filtrations of a vector space. In the case of a finite ground field this yields Zariski-open subsets of generalized flag varieties, whereas for a local field one obtains admissible open subsets in the sense of rigid-analytic geometry.
One purpose of the present paper is to give a brief exposition of the construction and the basic structure of period domains. In the first section we explain the notion of semistability of weighted flags of a vector space and its variants. In the second section we introduce the period domains. We mostly restrict ourselves to a finite ground field in order to minimize technicalities. We also introduce the stratification of the flag varieties according to the degree of instablity which shares many properties with the Harder-Narasimhan stratification of the space of vector bundles over a Riemann surface. In the third section we try to bring out the special nature of Drinfeld's period space (the complement of all rational hyperplanes in projective space). This is the period domain that was discovered first and is the best-known; it also is a particular example of a complement of a hyperplane arrangement in affine space.
The depth of the period domains lies, however, in their connection with Fontaine's conjectures [cf. \textit{J.-M. Fontaine}, Astérisque 65, 3-80 (1979; Zbl 0429.14016)] and their generalizations (to families of abelian varieties respectively of \(p\)-adic Galois representations). Therefore the other aim of this paper is to explain the arithmetic significance of \(p\)-adic period domains. The fourth section explains on concrete examples some work by \textit{M. Rapoport} and \textit{Th. Zink} [``Period spaces for \(p\)-divisible groups'', Ann. Math. Stud. 141 (1996; Zbl 0873.14039)] in this direction. In the work under review we construct étale covers of some \(p\)-adic period domains and integral models of them, and use them to establish non-archimedean uniformization theorems of Shimura varieties. characteristic \(p\); semistability condition; vector spaces over a non-archimedean field; \(p\)-adic period domains; generalized flag varieties; rigid-analytic geometry; stratification of the flag varieties; non-archimedean uniformization theorems of Shimura varieties Michael Rapoport, Period domains over finite and local fields, Algebraic geometry --- Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 361 -- 381. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Variation of Hodge structures (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations, \(p\)-adic cohomology, crystalline cohomology, Modular and Shimura varieties, Formal groups, \(p\)-divisible groups, Local ground fields in algebraic geometry, Finite ground fields in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Period domains over finite and local fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0644.00009.]
The main result of the paper is a construction of a reduced simplicial Kan-complex AZ(R), called Azumaya complex of a commutative ring R, such that \(\pi_ 1(AZ(R))\) is the Brauer group Br(R) of equivalence classes of Azumaya algebras over R, \(\pi_ 2(AZ(R))\) is the Picard group Pic(R) of isomorphism classes of invertible modules over R, \(\pi_ 3(AZ(R))\) is the multiplicative group \(G_ m(R)\) of units of R and \(\pi_ n(AZ(R))=0\) for \(n>3\). The complex AZ(R) has unique 0-simplex R. 1- simplices are Azumaya R-algebras A. 2-simplices are triplets of Azumaya R-algebras \((A_ 0,A_ 1,A_ 2)\) together with an invertible left \(A_ 1\otimes (A_ 2\otimes A_ 0)^ 0\)-module M (denoted by M: \(A_ 1\Rightarrow A_ 2\otimes A_ 0)\). 3-simplices are formed by families of 2-simplices \(M_{123}: A_{13}\Rightarrow A_{12}\otimes A_{23}\), \(M_{023},: A_{03}\Rightarrow A_{02}\otimes A_{23}\), \(M_{013}: A_{03}\Rightarrow A_{01}\otimes A_{13}\), \(M_{012}: A_{02}\Rightarrow A_{01}\otimes A_{12}\), which satisfy some natural compatibility condition. The definition of 4-simplices is similar but compatibility conditions are complicated. An alternative description of the complex AZ(R) given by R. Street is outlined. bimodule; homotopy groups; Brauer group of equivalence classes of Azumaya algebras; Picard group of isomorphism classes of invertible modules; reduced simplicial Kan-complex; Azumaya complex of a commutative ring J W Duskin, The Azumaya complex of a commutative ring (editor F Borceux), Lecture Notes in Math. 1348, Springer (1988) 107 Simplicial sets, simplicial objects (in a category) [See also 55U10], Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Brauer groups of schemes, Monoidal categories (= multiplicative categories) [See also 19D23] The Azumaya complex of a commutative ring | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors prove that minimal number of generators of endomorphism rings of Kronecker modules of degree at least 3 over an algebraic closed field is exactly 3 (Theorem 4.7). The authors reduce this to an algebraic geometric result, which they also prove (Corollary 4.5): If a smooth point of a (in a non-standard sense) hyperelliptic curve is not fixed by the hyperelliptic involution then the curve minus this point is not embeddable into the affine plane. The definitions are too involved to recall here. The paper splits into two parts: a mainly algebraic part consisting of the above mentioned reduction followed by the algebraic geometry part entirely in the realm of algebraic curves. The proofs are easy to follow. However, the article is minimalistic in the sense that it recalls only the parts of the notions necessary to understand the proofs. endomorphism ring of a Kronecker module; locally planar hyperelliptic curve; minimal number of generators Mckinnon, D.; Roth, M.: Curves arising from endomorphism rings of Kronecker modules, Rocky mountain J. Math. 37, 879-891 (2007) Plane and space curves, Endomorphism rings; matrix rings Curves arising from endomorphism rings of Kronecker modules | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 almost all curves \(D\) in \(\mathbb{C}^2\) given by an equation of the form \(g(x,y)^a +h(x,y)^b=0\), where \(a>1\) and \(b>1\) are coprime integers, the fundamental group of the complement of the curve has presentation \(\pi_1(\mathbb{C}^2 \setminus D) \cong \langle x_1,x_2 \mid x_1^a =x_2^b\rangle\), that is, it coincides with the group of the torus knot \(K_{a,b}\). In the projective case, for almost every curve \(\overline D\) in \(\mathbb{P}^2\) which is the projective closure of a curve in \(\mathbb{C}^2\) given by an equation of the form \(g(x,y)^a +h(x,y)^b=0\), the fundamental group \(\pi_1 (\mathbb{P}^2 \setminus \overline D)\) of the complement is a free product with amalgamated subgroup of two cyclic groups of finite order. In particular, for the general curve \(\overline D\subset \mathbb{P}^2\) given by the equation \(l^a_{bc} (z_0,z_1,z_2) +l^b_ {ac} (z_0,z_1, z_2)=0\), where \(l_q\) is a homogeneous polynomial of degree \(q\), we have \(\pi_1 (\mathbb{P}^2 \setminus \overline D) \cong \langle x_1,x_2 \mid x^a_1=x_2^b, x_1^{ac} =1\rangle\). fundamental group of the complement of a curve; torus knot Coverings of curves, fundamental group, Homotopy theory and fundamental groups in algebraic geometry, Special algebraic curves and curves of low genus, Fundamental group, presentations, free differential calculus, Knots and links in the 3-sphere On the fundamental groups of complements of toral curves | 0 |
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