text stringlengths 571 40.6k | label int64 0 1 |
|---|---|
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Authors' abstract: ``In this paper, we show that the reduction of divisors in the Jacobian of a curve \(C\) can be performed by considering the intersections of a suitable projective model of \(C\) with quadrics in projective space. We apply this idea to a certain projective model of elliptic and hyperelliptic curves on one hand, and to the canonical model of \(C_{ab}\) curves on the other hand, and we find well known algorithms.'' \(C_{ab}\) curves; Jacobian variety; addition of divisors; reduction algorithms; intersections in projective space Special algebraic curves and curves of low genus, Jacobians, Prym varieties, Computational aspects of algebraic curves, Abelian varieties of dimension \(> 1\), Curves over finite and local fields, Algebraic coding theory; cryptography (number-theoretic aspects) A geometric interpretation of reduction in the Jacobian of \(C_{ab}\) 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 Studies of Riemann surfaces of infinite genus have been started in mathematics some years ago [see e.g. \textit{P. Myrberg}, Acta Math. 76, 185-224 (1945; Zbl 0060.21602)]. According to McKean and Trubowitz (1976) there exists a one-to-one correspondence between divisors of Riemann surfaces of infinite genus and smooth almost-periodic solutions of the integrable evolution equations in terms of theta functions similar to what had been proved before for the compact Riemann surfaces by Its and Matveev (1975). This approach is summarized in the books [\textit{B. M. Levitan}, ``The Sturm-Liouville problems'', Moscow: Nauka (1984; Zbl 0575.34001) and \textit{J. Pöschel} and \textit{E. Trubowitz}, ``Inverse spectral theory'', Boston Academic Press (1987; Zbl 0623.34001)].
In his memoir, the author proposes new and more geometrical methods to describe this correspondence based on an appropriate generalization of the Picard group (i.e., the set of all equivalence classes of holomorphic line bundles with their tensor product) for the Riemann surfaces of infinite genus. With this aim he adds in a special way to the spectral curve of the Lax operator (which is in this case the ordinary differential matrix operator with periodic coefficients) the points corresponding to infinite value of the spectral parameter. The resulting object is no longer a Riemann surface in the usual sense but is quite similar to the compact one. It allows to generalize in natural way all basic tools of the theory of compact Riemann surfaces to this spectral curve and thus describe explicitly for the infinite genus the structure of complete integrability: 1) the eigenbundles define holomorphic line bundles on the spectral curve which completely determine potentials; 2) the line bundles are described by divisors of the same degree as the genus; 3) these divisors give the Darboux coordinates, and the Serre duality leads to the symplectic form; 4) the isospectral sets may be identified with open dense subsets of the Jacobian varieties in accordance with the Riemann-Roch theorem; 5) the real parts of the isopectral sets are infinite-dimensional tori etc. integrable systems; Picard group; Riemann surfaces of infinite genus; complete integrability; eigenbundles; holomorphic line bundles; spectral curve; divisors; Darboux coordinates; Serre duality Schmidt M U 1996 \textit{Integrable Systems and Riemann Surfaces of Infinite Genus}\textit{(Memoirs of the American Mathematical Society vol 122)} (Providence, RI: American Mathematical Society) pp 1--111 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Singularities of curves, local rings, Theta functions and curves; Schottky problem, NLS equations (nonlinear Schrödinger equations) Integrable systems and Riemann surfaces of infinite genus | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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.]
Let \(J_ d(C)\) be the Jacobian variety of line bundles of degree \(d\) over a smooth curve \(C\) of genus \(g\). If \(d\geq 2g+1\), the Picard bundle of degree \(d\) is a bundle \(P_ d\) of rank \(d+1-g\) over \(J_ d(C)\) whose fiber over any line bundle \(L\) is \(\Gamma(C,L)\). The authors prove that \(P_ d\) is stable with respect to the principal polarization given by the theta divisor. The first case \(d=2g-1\) was already done by \textit{G. R. Kempf} [Am. J. Math. 112, No. 3, 397-401 (1990; Zbl 0719.14010)]. Two different proofs are given and as an application the authors prove the semistability of higher conormal bundles for elliptic curves. These are defined, for \(i\leq d-2\), as kernel of the map from \(\Gamma(C,L)\otimes{\mathcal O}_ C\) to the bundle of \(i\)-th order principal parts of \(L\). stability of Picard bundle; Jacobian variety; theta divisor; semistability of higher conormal bundles for elliptic curves Ein, L.; Lazarsfeld, R., Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves, complex projective geometry, \textit{London Math. Soc. Lect. Note Ser.}, 179, 149-156, (1992) Jacobians, Prym varieties, Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0744.00029.]
The following theorem is proved: Let \(S\) be a complex smooth projective surface and let \(L\) be a linear system which defines a birational map from \(S\) onto its image \(F\); assume that \(F\) is not a scroll, nor has rational hyperplane sections. Then, the ring of endomorphisms of the Jacobian of the normalization of a general member of \(| L|\) is isomorphic to \(\mathbb Z \times \text{End}(\text{Alb}(S))\). In particular, if \(S\) is regular, the Jacobian of the normalization of a general member of \(| L|\) is simple.
To prove this theorem, one reduces immediately to the case where \(| L|\) has no base point. A general member \(C\) of \(| L|\) is then smooth, and its Jacobian is isogenous to the product of \(\text{Alb}(S)\) and the identity component \(K(C,S)\) of the kernel of the surjective morphism \(\text{Alb}(C) \to \text{Alb}(S)\). One proves then that \(\text{End}(K(C,S))\) is isomorphic to \(\mathbb{Z}\) by degenerating to an irreducible curve in \(| L|\) with a single node. The fact that there are no nonzero homomorphisms from \(K(C,S)\) to \(\text{Alb}(S)\) then follows easily.
It should be mentioned that the results of this article are entirely contained in an article by \textit{S. Mori} [cf. Jap. J. Math., New Ser. 2, 109--130 (1976; Zbl 0339.14016)] who deals with the same problems in a more general setting and in all characteristics, with similar methods. However, the reader interested only in the characteristic zero case will probably find the article under review more easily accessible. Jacobian of a hyperplane section of a surface; endomorphisms of abelian varieties; Albanese variety; linear system Ciliberto, C., van~der Geer, G.: On the Jacobian of a hyperplane section of a surface. In: Classification of Irregular Varieties (Trento, 1990). Lecture Notes in Mathematics, vol. 1515, pp. 33-40, Springer, Berlin (1992) Jacobians, Prym varieties, Picard schemes, higher Jacobians, Divisors, linear systems, invertible sheaves On the Jacobian of a hyperplane section of a surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be an integral curve with one unibranch singularity of embedding dimension greater than two. We study decomposition of the compactified Picard variety \((\text{Pic}^ 0C)^ =\) into irreducible components for a unibranch singularity. We give a criterion for determining the beginning of a decomposition, defined by the number of generators of a module locally at the singularity. We discuss some particular cases where it is possible to give more precise descriptions. In the case of curves characterised by ``maximal ideal of \(M\) of the singularity equal to the conductor of \(C\)'' the basic decomposition given in section two actually gives components of \((\text{Pic}^ 0C)^ =\). We deal with the obvious first extension of this, namely, \(\text{rk} (M/C) = 1\). In this case the basic decomposition of \((\text{Pic}^ 0C)^ =\) given in section 1 does not give irreducible components but is only a step towards giving the irreducible components, which we do for the case of Gorenstein curves with \(\text{rk} (M/C) = 1\). compactified Jacobian; decomposition of Picard variety; unibranch singularity; Gorenstein curves Jacobians, Prym varieties, Singularities of curves, local rings, Picard groups, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Compactified Jacobian of some unibranch 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 Consider the set \({\mathcal S} = \{ (\tilde S,\tilde L ) | \tilde S\) is a smooth projective surface and \(\tilde L \in \hbox{Pic}(\tilde S)\) is very ample\(\}\). An adjunction process on the pair \((\tilde S, \tilde L )\) in \(\mathcal S\) is the construction of a new pair \((S,L)\) in the following way: first contract all the \(-1\) rational curves \(C\) on \(\tilde S\) with \(\tilde L \cdot C = 1\), getting a new surface \(S\) and a reduction morphism \(\pi : \tilde S \rightarrow S\); then consider on \(S\) the line bundle \(H = K_ S \otimes L\), where \(L = \pi_ * \tilde L\). The pair \((S,L)\) is called the reduction of \((\tilde S,\tilde L)\). It is known that if \((\tilde S,\tilde L)\) is not in a certain subset \(\mathcal E\) of \(\mathcal S\) then \((S,H)\) is again in \(\mathcal S\).
Therefore there is a map \({\mathcal RA} : {\mathcal S} - {\mathcal E} \rightarrow {\mathcal S}\) which associates to a pair \((\tilde S,\tilde L)\) the pair \((S,H)\). It is not hard to check that \(\mathcal RA\) is not a surjection. This paper considers the question of characterizing the pairs in \(\mathcal S\) which are in the image of \(\mathcal RA\). Answers are given in the following cases: \smallskip \noindent (a) surfaces in \({\mathbf P}^ 4\) (i.e.\ \(h^ 0 (H) \leq 5\));
\noindent (b) surfaces of degree \(\leq 9\) (i.e.\ \(H^ 2 \leq 9\));
\noindent (c) surfaces for which \(K_ S^{\otimes -1}\) is nef. \smallskip The work relies heavily on adjunction theory and related classification results for surfaces of small sectional genus. It also uses Castelnuovo's bound for the genus of a curve in projective space. Picard group; adjunction process; surfaces of small sectional genus; Castelnuovo's bound for the genus of a curve Special surfaces, Divisors, linear systems, invertible sheaves, Picard groups, Projective techniques in algebraic geometry On projective surfaces arising from an adjunction process | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(M\) be a compact affine nonsingular real algebraic variety of dimension \(m\). Denote by \(H^{\text{alg}}_{m-1} (M, \mathbb{Z}/2)\) the subgroup of the homology group \(H_{m-1} (M, \mathbb{Z}/2)\) which consists of all homology classes represented by Zariski closed real algebraic hypersurfaces of \(M\). Let \(H^ 1_{\text{alg}} (M,\mathbb{Z}/2)\) be the image of \(H^{\text{alg}}_{m-1} (M, \mathbb{Z}/2)\) under the Poincaré duality isomorphism \(H_{m-1} \to H^ 1\). The group \(H^ 1_{\text{alg}} (M,\mathbb{Z}/2)\) is an important invariant with numerous interesting applications.
In the paper under review one investigates the group \(H^ 1_{\text{alg}} (M, \mathbb{Z}/2)\) where \(M\) is the underlying real algebraic variety of a complex projective nonsingular variety. Every complex projective variety \(V\) can be, in the obvious way, considered as a real algebraic variety and, as such, will be denoted by \(V_ \mathbb{R}\). Of course, \(\dim V_ \mathbb{R} = 2 \dim V\). Moreover, \(V_ \mathbb{R}\) is an affine real algebraic variety and if \(V\) is nonsingular, then \(V_ \mathbb{R}\) is also nonsingular. Define \(d(V) = \dim_{\mathbb{Z}/2} H^ 1_{\text{alg}} (V_ \mathbb{R}, \mathbb{Z}/2)\). In many cases, \(d(V)\) is strictly less than \(\dim_{\mathbb{Z}/2} H^ 1(V_ \mathbb{R}, \mathbb{Z}/2)\), which makes \(d(V)\) an interesting invariant. The paper contains the computation (or at least estimates) of \(d(V)\) for a large class of complex projective nonsingular varieties \(V\). Here are samples of results. Let \(\text{Alb} (V)\) be the Albanese variety of \(V\) and \(\text{End(Alb} (V))\) its ring of endomorphisms. If \(\dim V = 1\), \(\text{Alb}(V)\) is just the jacobian \(\text{Jac} (V)\) of \(V\).
Theorem 1. One has \(d(V) \leq \text{rank End(Alb} (V))\). Let us recall that for a generic abelian variety \(A\) (resp. generic complex projective nonsingular curve \(V)\), one has \(\text{End} (A) \approx \mathbb{Z}\) (resp. End (Jac\((V)) \approx \mathbb{Z})\). It follows that for a generic abelian variety \(A\) (resp. generic projective curve \(V)\), one has \(d(A) \leq 1\) (resp. \(d(V) \leq 1)\).
Theorem 2. Let \(V\) be a complex projective irreducible nonsingular curve of positive genus, defined over \(\mathbb{R}\), and let \(V(\mathbb{R})\) be its real part. If \(V_ \mathbb{R} \backslash V (\mathbb{R})\) is disconnected or \(V(\mathbb{R}) = \emptyset\), then \(d(V) \leq \text{rank End(Jac}(VI))-1\).
Corollary 3. For \(V\) as in theorem 2, if \(\text{End(Jac} (V)) \cong \mathbb{Z}\), then \(d(V) = 1\) if \(V_ \mathbb{R} \backslash V (\mathbb{R})\) is connected and \(V(\mathbb{R}) \neq \emptyset\), \(d(V) = 0\) otherwise.
Theorem 4. For every complex elliptic curve \(E\) without complex multiplication, one has \(d(E)\leq 1\). If, moreover, the \(j\)- invariant \(j(E)\) of \(E\) is in \(\mathbb{R}\), then \(d(E) = 0\) if \(j(E) > 1728\), \(d(E)=1\) if \(j(E) < 1728\).
Proposition 5. For \(k = 0, 1\) or 2 let \(B_ k = \{j \in \mathbb{C} \mid j = j(E)\), \(E\) is a complex elliptic curve, \(d(E) = k\}\). Then each of these sets is dense in \(\mathbb{C}\), the sets \(B_ 0\) and \(B_ 1\) are uncountable, and \(B_ 2\) is countable.
Theorem 6. Let \(V\) be a complex projective irreducible nonsingular curve and let \(\text{Jac} (V)\) be its jacobian variety. Then \(d(V) = d (\text{Jac} (V))\). dimension of real algebraic homology group; real algebraic variety; Zariski closed real algebraic hypersurfaces; Albanese variety; endomorphisms; complex elliptic curve; jacobian variety Bochnak J., Kucharz W.: Real algebraic hypersurfaces in complex projective varieties. Math. Ann. 301, 381--397 (1995) Real algebraic sets, (Co)homology theory in algebraic geometry, Hypersurfaces and algebraic geometry, Projective techniques in algebraic geometry, Jacobians, Prym varieties Real algebraic hypersurfaces in complex projective varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(R\) be a complete discrete valuation ring, with quotient field \(K\) and residue field \(k\), algebraically closed and with characteristic \(p\). To any abelian variety of dimension \(g\) over \(K\), one can associate its Néron model, \({\mathcal A}/R\), whose special fiber, \({\mathcal A}_k\), is an extension of a finite abelian group \(\Phi_K\), the group of components, by a smooth connected group scheme \({\mathcal A}^0_k\), the connected component of \(0\) in \({\mathcal A}_k\). In particular, there is a canonical reduction map \(A(K)\to{\mathcal A}_k(k)\), and we denote by \(\pi\) the composition of the reduction map with the projection on the component group \(\Phi_K\).
If \(X\) is a smooth, proper, geometrically connected variety of dimension \(1\) over \(K\) (a curve) and \(A/K\) denotes its Jacobian, given two points \(P\) and \(Q\) of \(X(K)\), the divisor \(P-Q\) is a point of \(A(K)\) and one can consider its reduction, by means of \(\pi\), to a point of the component group \(\Phi_K\) of the Néron model of the Jacobian. In a previous paper [cf. \textit{D. Lorenzini}, J. Reine Angew. Math. 445, 109-160 (1993; Zbl 0871.14029)] the author introduced two functorial filtrations of the prime-to-\(p\) part of the group \(\Phi_K\), which have been used for a complete description of all the possible groups occurring as prime-to-\(p\) part of a component group [cf. \textit{B. Edixhoven}, Compos. Math. 97, 29-49 (1995; Zbl 0863.14023)]. In the present paper, the author studies the reduction \(\pi(P-Q)\) in terms of the reduction of the points \(P\) and \(Q\) in a regular model of the curve \(X\). In this way the author is able to give a sufficient condition on the special fiber of the model of \(X\), under which the image of \(P-Q\) belongs to one of the functorial filtrations of the component group and to provide a formula for the order of this image. We refer to section 1 of the paper for a detailed description of the content of this article. varieties over a local field; Néron models; arithmetical graphs; discrete valuation ring; Jacobian; abelian variety; group of components Lorenzini, D., Reduction of points in the group of components of the Néron model of a Jacobian, J. Reine Angew. Math., 527, 117-150, (2000) Algebraic theory of abelian varieties, Jacobians, Prym varieties Reduction of points in the group of components of the Néron model of a Jacobian | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Dans cet article les AA. étudient les Jacobiennes des courbes de Hurwitz. Ils appliquent le calcul du réseau des périodes et celui de la Jacobienne qui est fait par les AA. dans Manuscr. Math. 84, No. 2, 163-175 (1994; Zbl 0842.14022). Ils démontrent que la Jacobienne de la quintique de Snyder est isogène au cube d'une variété Abélienne qui est elle-même \(R\)-isomorphe au produit de deux courbes elliptiques. Ils donnent aussi un exemple où la Jacobienne ne se décompose pas en produit de courbes elliptiques. Jacobian of Hurwitz curve; decomposition of a Jacobian as product of elliptic curves Bennama, H.; Carbonne, P.: Périodes et jacobiennes des courbesxm+Ym+Zm=0. Bull. Polish acad. Sci. 44 (1996) Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Picard schemes, higher Jacobians Periods and Jacobians of the curves \(X^ mY+Y^ mZ+Z^ mX=0\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author investigates Weierstrass points on a closed Riemann surface by extending the study of \textit{J. Lewittes} [Acta Math. 111, 37-61 (1964; Zbl 0125.318)] of the role of the choice of base point for the canonical imbedding of the surface into its Jacobian variety. Characterizations of the variety of Weierstrass points with given gap sequences are obtained in terms of conditions on the thetafunction of the Jacobian variety. Weierstrass points on a closed Riemann surface; gap sequences; thetafunction of the Jacobian variety Riemann surfaces; Weierstrass points; gap sequences, Compact Riemann surfaces and uniformization, Jacobians, Prym varieties, Theta functions and abelian varieties, Special algebraic curves and curves of low genus A description of the Weierstrass gap sequence by means of the Riemann theta function | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper the author gives an interesting, elementary proof of Clifford's theorem which describes the relationship between dim \(| D|\) and deg D for special divisors D on a nonsingular projective algebraic curve over an algebraically closed field. The proof is obtained by combining the Riemann-Roch formula with the following result, for which the author gives an elementary proof, and which he calls Clifford's lemma: Let A,B,C be vector spaces over an algebraically closed field and let \(\phi: A\times B\to C\) be a bilinear map. Assume that \(\phi (a,b)=0\) implies a or b is zero. Then \(\dim C\geq \dim A+\dim B-1.\) Clifford theorem; divisors on a nonsingular projective algebraic curve; dimension of product of vector spaces; Riemann-Roch formula; bilinear map Divisors, linear systems, invertible sheaves, Vector spaces, linear dependence, rank, lineability, Riemann-Roch theorems, Curves in algebraic geometry A linear algebra proof of Clifford's theorem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth projective curve defined over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(M_X(r)\) be the moduli space of semi-stable rank r vector bundles with fixed trivial determinant. Relative Frobenius \(X \rightarrow X_1\) induces by pull-back a rational map called Verschiebung \(V:M_{X_1}(r) \rightarrow M_X(r)\). The authors study in detail the properties of this map. One result is that \(V\) has base points for any triple \((g,r,p)\). In case \(g=r=2\), explicit equations are given for \(V\) in the following two cases:
1. \(p=2\) and \(X\) is nonordinary with Hasse-Witt invariant equal to 1. Here use is made of equations defining a Kummer's quartic relating to the Verschiebung.
2. \(p=3\), any \(X\). Here use is made of polar equations of the Kummer's surface. The article is written in a very clear and precise manner. moduli of rank 2 vector bundles over a curve; relative Frobenius Laszlo Y. and Pauly C. (2004). The Frobenius map, rank 2 vector bunbles and Kummer's quartic surface in characteristic 2 and 3. Adv. Math. 185: 246--269 Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 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 In this paper, five new families of Calabi-Yau threefolds without maximal unipotent monodromy are constructed. For these families, the variation of their Hodge structures and their parametrizing spaces are described, and their Picard-Fuchs differential equations are explicitly determined. Since these families have no boundary points with maximal unipotent monodromy, the mirror families cannot be readily defined within the usual mirror symmetry framework.
The construction of these new families of Calabi-Yau threefolds are done as follows: Start with \(K3\) surfaces and elliptic curves admitting non-symplectic automorphisms (of order \(4\)). Consider a product of such a \(K3\) surface \(S\) and such an elliptic curve \(E\). Next take the quotient of the product \(E\times S\) by the non-symplectic automorphism of order \(4\). Resolving singularities, one obtains a family of Calabi-Yau threefolds.
The \(K3\) surfaces used in this construction are rather special. Indeed, a \(K3\) surface is realized as the quotient of the product of an elliptic curve \(E\) (admitting a non-symplectic automorphism \(\alpha_E\) of order \(4\)) with a curve \(C\) of genus \(g\in\{1,2,3\}\) admitting the automorphism \(\alpha_C\) with a specific property. Then the quotient of the product \(E\times C\) by the automorphism \(<\alpha_E, \alpha_C>\) admits a desingularization \(S\), which is a \(K3\) surface with non-symplectic automorphism \(\alpha_S\) of order \(4\). This yields five families of \(K3\) surfaces with non-symplectic automorphisms \(\alpha_S\) of order \(4\).
Next consider the product \(E\times S\) of an elliptic curve \(E\) and a K3 surface \(S\), where \(S\) and \(E\) are one of the \(K3\) surfaces and elliptic curves, respectively, considered above. Then the desingularization of its quotient by the automorphism \(\alpha_E^3\times \alpha_S\) gives rise to five families of Calabi-Yau threefolds with Hodge numbers \((h^{1,1},h^{2,1}) \in\{(90,0), (73,1), (56,2), (61,1), (39,3)\}\).
For these five families of Calabi-Yau threefolds, the parametrizing spaces are also determined. Also the Picard-Fuchs differential equations are computed, based on the fact that the variation of the Hodge structures depends only on the variation of the Hodge structures of the elliptic curve component. In particular, this results in the assertion that these families of Calabi-Yau threefolds admit no maximal unipotent monodromy. Calabi-Yau threefolds; unipotent monodromy; variation of the Hodge structure; Picard-Fuchs differential equation; K3 surfaces; elliptic curves; product of a \(K3\) surface and an elliptic curve Garbagnati, A, New families of Calabi-Yau threefolds without maximal unipotent monodromy, Manuscripta Math., 140, 273-294, (2013) Calabi-Yau manifolds (algebro-geometric aspects), \(K3\) surfaces and Enriques surfaces, Mirror symmetry (algebro-geometric aspects), Structure of families (Picard-Lefschetz, monodromy, etc.), Variation of Hodge structures (algebro-geometric aspects), Families, moduli, classification: algebraic theory, Automorphisms of surfaces and higher-dimensional varieties New families of Calabi-Yau threefolds without maximal unipotent monodromy | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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{B. Mazur} [Publ. Math., Inst. Hautes Étud. Sci. 47 (1977), 33-186 (1978; Zbl 0394.14008)] defined the Eisenstein ideal for a Fermat curve to be the endomorphism of the jacobian which annihilates the cuspidal group. The cuspidal group of a Fermat jacobian was determined by \textit{D. E. Rohrlich} [Invent. Math. 39, 95-127 (1977; Zbl 0357.14010)]. One difficulty in determining the Eisenstein ideal is that the full endomorphism ring of a Fermat jacobian is not known in general.
The authors determine the endomorphism ring of the degree 5 Fermat curve, and compute the Eisenstein ideal and the action of endomorphism ring on the cuspidal divisors. Let \(F_ 5\) denote the Fermat curve of degree 5 defined by \(X^ 5+Y^ 5+Z^ 5=0\), and let \(J_ 5\) be its Jacobian. Let \(\zeta=\exp(2\pi i/5)\). The automorphism group \(G\) of \(F_ 5\) is generated by
\[
\sigma:(X:Y:Z)\to(\zeta X:Y:Z),\;\tau:(X:Y:Z)\to(X:\zeta Y:Z)
\]
\[
\iota:(X:Y:Z)\to(Y:X:Z),\;\rho:(X:Y:Z)\to(Z:X:Y).
\]
Then using the map \(\Phi:\mathbb{Q}[G]\to\text{End}(J_ 5)\otimes\mathbb{Q}\) induced from the canonical injection \({G\hookrightarrow\Aut(J_ 5)}\), the authors describe explicitly the endomorphism ring of \(J_ 5\). With the structure of the endomorphism ring at their disposal, they are able to determine the Eisenstein ideal of \(F_ 5\):
Theorem. The Eisenstein ideal \({\mathcal I}\) is the two-sided ideal of \(\text{End}(J_ 5)\) generated by \(5 \text{End}(J_ 5)\), \(\mathbb{Z}[\sigma,\tau]\cap{\mathcal I}\) and \(\omega:=(\tau-\tau^ 2-2\sigma- 2\sigma\tau-\sigma\tau^ 2)(1+\rho+\rho^ 2)\).
The proof is computational. Eisenstein ideal for a Fermat curve; endomorphism of the jacobian; Fermat curve of degree 5 Global ground fields in algebraic geometry, Arithmetic ground fields for curves, General ternary and quaternary quadratic forms; forms of more than two variables, Computational aspects of algebraic curves, Jacobians, Prym varieties The Eisenstein ideal of a Fermat curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory -- especially through the concepts of curvature and positivity which are central themes in Kodaira's contributions to mathematics. The aim of these lectures is to present recent results concerning the geometric side of the problem. The Green-Griffiths-Lang conjecture stipulates that for every projective variety \(X\) of general type over \(\mathbb{C} \), there exists a proper algebraic subvariety \(Y\) of \(X\) containing all entire curves \(f : \mathbb{C} \rightarrow X\). Using the formalism of directed varieties and jet bundles, we show that this assertion holds true in case \(X\) satisfies a strong general type condition that is related to a certain jet-semi-stability property of the tangent bundle \(T_X\). It is possible to exploit similar techniques to investigate a famous conjecture of Shoshichi Kobayashi (1970), according to which a generic algebraic hypersurface of dimension \(n\) and of sufficiently large degree \(d \geqslant d_n\) in the complex projective space \(\mathbb{P}^{n+1}\) is hyperbolic: in the early 2000's, Yum-Tong Siu proposed a strategy that led in 2015 to a proof based on a clever use of slanted vector fields on jet spaces, combined with Nevanlinna theory arguments. In 2016, the conjecture has been settled in a different way by Damian Brotbek, making a more direct use of Wronskian differential operators and associated multiplier ideals; shortly afterwards, Ya Deng showed how the proof could be modified to yield an explicit value of \(d_n\). We give here a short proof based on a substantial simplification of their ideas, producing a bound very similar to Deng's original estimate, namely \({d_n} = \left\lfloor{\frac{1}{3}{{(en)}^{2n + 2}}} \right\rfloor \). Kobayashi hyperbolic variety; directed manifold; genus of a curve; jet bundle; jet differential; jet metric; Chern connection and curvature; negativity of jet curvature; variety of general type; Kobayashi conjecture; Green-Griffiths conjecture; Lang conjecture Hyperbolic and Kobayashi hyperbolic manifolds, Value distribution theory in higher dimensions, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Global differential geometry of Hermitian and Kählerian manifolds, \(n\)-folds (\(n>4\)) Recent results on the Kobayashi and Green-Griffiths-Lang conjectures | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a real algebraic curve of genus \(g\geq 1\) having at least \(g\) real components. We show that there is an embedding of \(C\) into \(\mathbb{P}^{2g}\) as a curve of degree \(3g\) which induces a group structure on a connected component \(X\) of the set of effective divisors on \(C\) of degree \(g\). Moreover, after having chosen a base point \(0\in X\), there is a natural isomorphism of \(X\) onto the neutral real component of the Jacobian of \(C\). This furnishes an explicit description of the group structure on the neutral real component of the Jacobian of a real algebraic curve of genus \(g\geq 1\) having many real components. If \(g=1\), one recovers the geometric description of the group structure on the neutral real component of a real elliptic curve. real algebraic curve; effective divisors; group structure on the neutral real component of the Jacobian Huisman, J.: On the neutral component of the Jacobian of a real algebraic curve having many components. Indag. math. 12, No. 1, 73-81 (2001) Real algebraic sets, Plane and space curves, Group actions on varieties or schemes (quotients) On the neutral component of the Jacobian of a real algebraic curve having many components | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(g>0\) be a fixed integer. The authors prove that the problem of computing the structure of the group of rational points \(J(\mathbb{F}_ q)\) on the Jacobian \(J\) of a hyperelliptic curve \(X\) over a finite field \(\mathbb{F}_ q\) of cardinality \(q\) is in \(NP\cap co-NP\). The certificate is a set of independent generators of the group of prime power order. The independence is checked by means of the Weil pairing. To prove that the points generate the entire group, the authors employ Pila's algorithm [\textit{J. Pila}, Mth. Comput. 55, No. 192, 745-763 (1990; Zbl 0724.11070)] to compute the number of points in \(J(\mathbb{F}_ q)\). This last part is not really necessary; it suffices to use the fact that \((\sqrt q- 1)^{2g}\leq\#J(\mathbb{F}_ q)\leq(\sqrt q+1)^{2g}\). Jacobian of a hyperelliptic curve; group of rational points; finite field; Weil pairing Jacobians, Prym varieties, Computational aspects of algebraic curves, Cryptography, Analysis of algorithms and problem complexity, Finite ground fields in algebraic geometry, Elliptic curves, Complexity classes (hierarchies, relations among complexity classes, etc.) Efficient algorithms for the construction of hyperelliptic cryptosystems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 double cover of \({\mathbb{P}}^ 3\) branched along a sextic surface S. Using a method of Clemens and Letizia, in this paper we show that, for general X, the Abel-Jacobi map associated to the surface F of curves contained in X which are preimages of conics ''totally tangents'' to S, induces an isomorphism between the Albanese variety of F and the intermediate Jacobian of X. threefold; double cover of projective 3-space branched along a sextic surface; Abel-Jacobi map; Albanese variety; intermediate Jacobian \(3\)-folds, Projective techniques in algebraic geometry The Abel-Jacobi isomorphism for the sextic double solid | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(Z=(Z_ 1,...,Z_ n)\) be a holomorphic vectorfield in a neighbourhood U of \(0\in {\mathbb{C}}^ n\) which has 0 as an isolated singularity. \({\mathcal F}_ z\) denotes the complex one-dimensional foliation with singularity at 0 defined by the integral curves of Z. A special case which should be kept in mind is the gradient vectorfield of a holomorphic function with isolated singularity at the origin.
The aim of the article is to define certain topological invariants of isolated singularities of holomorphic functions in the more general situation described above and to show that these invariants are indeed topological invariants of the foliation. E.g. if \(n\geq 2\), the Milnor number \(\mu (Z,0)=\dim_{{\mathbb{C}}} {\mathcal O}_{{\mathbb{C}}^ n,0}/(Z_ 1,...,Z_ n)\) is a topological invariant of \({\mathcal F}_ z\) as well as the index at 0 of the vectorfield \(Z|_ D\) where D is the intersection of a small ball with an irreducible complex curve V such that V-\(\{\) \(0\}\) is a leaf of \({\mathcal F}_ z.\)
In the second part the authors consider vectorfields Z in \({\mathbb{C}}^ 2\). They show that after finitely many quadratic transformations at singular points, the foliation \({\mathcal F}_ z\) is transformed into a foliation \(\tilde {\mathcal F}_ z\) with finitely many singularities of a very special kind which the authors call ''simple''. These simple singularities persist after further quadratic transformations and \(\tilde {\mathcal F}_ z\) is called a desingularization of \({\mathcal F}_ z\). A ''generalized curve'' is a vectorfield Z such that all simple singularities of \(\tilde {\mathcal F}_ z\) have nonvanishing eigenvalues (which is the case for the gradient vectorfield of a plane curve singularity). It is shown that two generalized curves have isomorphic desingularizations. From this the authors deduce that the algebraic multiplicity of a generalized curve is a topological invariant, which was of course well known for ''true'' plane curve singularities. holomorphic vectorfield; foliation; topological invariants of isolated singularities of holomorphic; functions; Milnor number; desingularization; algebraic multiplicity of a generalized curve; topological invariants of isolated singularities of holomorphic functions César Camacho, Alcides Lins Neto & Paulo Sad, ``Topological invariants and equidesingularization for holomorphic vector fields'', J. Differ. Geom.20 (1984) no. 1, p. 143-174 Complex singularities, Modifications; resolution of singularities (complex-analytic aspects), Deformations of complex singularities; vanishing cycles, Local complex singularities, Moduli, classification: analytic theory; relations with modular forms Topological invariants and equidesingularization for holomorphic vector 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 \(k\) be an algebraically closed field, \(R = k[x, y]\) a polynomial ring in two variables, and \(I\) an ideal of \(R\) minimally generated by homogeneous forms \(h_1, h_2, h_3\) of the same degree \(d > 0\). Let \(C\) be the curve parametrized by \(h_1, h_2, h_3\) , we can assume that the curve \(C\) has degree \(d\). Let \(\psi\) the syzygy matrix of the ideal \(I\).
The two aspects, the curve C parametrized by the forms \(h_1, h_2, h_3\) and the syzygy matrix \(\psi\), are mediated by the Rees algebra \(R[It]\) of \(I\). \(R[It]\) becomes a standard bi-graded \(k\)-algebra if one sets \(\deg x = \deg y = (1, 0)\) and \(\deg t = (-d, 1)\), which gives \(\deg h_i t = (0, 1)\). The symmetric algebra \(\mathrm{Sym}(I)\), the Rees algebra \(R[It]\), and the ideal \(A\) of \(\mathrm{Sym}(I)\) that defines \(R[It]\) all are naturally equipped with two gradings. Let \(A_i\) be the \(S\)-submodule of \(A\) which consists of all elements homogeneous in \(x\) and \(y\) of degree \(i\), we can view \(A\) as a sum of the \(A_i\). The aim of this article is to study this ideal \(A\).
The purpose is more clear in the case \(d = 6\), the case of a sextic curve. The authors show that there is, essentially, a one-to-one correspondence between the bi-degrees of the defining equations of \(R[It]\) and the types of the singularities on or infinitely near the curve \(C\). bi-graded structures; duality; elimination theory; generalized zero of a matrix; generator degrees; Hilbert-Burch matrix; infinitely near singularities; Koszul complex; local cohomology; linkage; matrices of linear forms; Morley forms; parametrization; rational plane curve; rational plane sextic; Rees algebra; Sylvester form; symmetric algebra 10.1016/j.jalgebra.2016.08.014 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Plane and space curves, Singularities of curves, local rings, Rational and birational maps The bi-graded structure of symmetric algebras with applications to Rees rings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a proper smooth variety over \(\mathbb{C}\) of dimension \(n\) and let \(Y\) be an effective divisor on \(X\). Let \(I\) be the ideal of \(\mathcal{O}_X\) which defines \(Y\), let \(I_1\) be the ideal of \(\mathcal{O}_X\) which defines the reduced part of \(Y\), and let \(J=II_1^{-1}\). Let \(\mathcal{D}_{X,Y}(r)\) denote the kernel of the surjective homomorphism of the Deligne complexes \(\mathcal{D}_X(r)\rightarrow\mathcal{D}_Y(r)\). In this paper the authors introduce a generalized Albanese variety \(\text{Alb}(X,Y)\) of \(X\) of modulus \(Y\) and shows that it fits into the two exact sequences \(0\rightarrow \text{Alb}(X,Y)\rightarrow H^{2n}(X,\mathcal{D}_{X,Y}(n))\rightarrow\mathbb{Z}\rightarrow 0\), and \(H^{n-1}(X,{\Omega}_X^n)\rightarrow H_c^{2n-1}(X-Y,\mathbb{C}/\mathbb{Z}(n))\bigoplus H^{n-1}(X,{\Omega}_X^n/J{\Omega}_X^n)\rightarrow \text{Alb}(X,Y)\rightarrow 0\). Furthermore they notice that in case \(Y=0\) these become the usual presentations of the Albanese variety \(\text{Alb}(X)\). generalized Albanese variety; modulus of a rational map; generalized mixed Hodge structure Kato, K.; Russell, H., \textit{Albanese varieties with modulus and Hodge theory}, Ann. Inst. Fourier (Grenoble), 62, 783-806, (2012) Group varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Motivic cohomology; motivic homotopy theory Albanese varieties with modulus and Hodge theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Take an algebraically closed field k with \(ch(k)=p>2\). A smooth curve X of genus g is said to be ordinary (resp. very special) if the p-rank of its Jacobian is g (resp. 0). Most curves are ordinary and the first examples of such curves where given by \textit{L. Miller} [Math. Ann. 197, 123-127 (1972; Zbl 0235.14009)]. Fix a polynomial \(f_ t\), deg f\({}_ t=t\), with t different roots. Here the author gives many examples of ordinary and very special hyperelliptic curves. He proves the following theorem: Let \(a(z)=\sum^{t}_{m=0}c_ mz^{p^ m}, c_ 0\neq 0\), be an additive separable polynomial, \(X_ 1\) (resp. \(X_ 2)\) the curve with equation \(y^ 2=f_{2g+1}(x)\) (resp. \(y^ 2=f_{2g+2}(x))\) and X'\({}_ 1\) (resp. X'\({}_ 2)\) the curve whose equation is obtained from that of \(X_ 1\) (resp. \(X_ 2)\) by the substitution \(x=a(z)\); assume \(X_ 1\) very special and \(X_ 2\) ordinary; then X'\({}_ 1\) is very special and X'\({}_ 2\) ordinary. The proof uses elementary properties of the Cartier operator. ordinary curve; holomorphic differential; Hasse-Witt matrix; p-division points; very special curve; p-rank of Jacobian; characteristic p; hyperelliptic curves; Cartier operator Jacobians, Prym varieties, Finite ground fields in algebraic geometry, Special algebraic curves and curves of low genus, Algebraic theory of abelian varieties An iterative construction for ordinary and very special hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians These notes are based upon my lectures at the Tata Institute from November 1975 to March 1976 and further oral communication between me and the note taker, \textit{Balwant Singh}. The notes are divided into two parts. In \S 8 of part one we prove the fundamental theorem on the structure of the coordinate ring of a meromorphic curve and its value group. We then give some applications of the fundamental theorem, the principal one among them being the epimorphism theorem. The proof of the main lemmas (\S 7) presented here is a simplified version of the original proof of the author and \textit{T.-T. Moh} [J. Reine Angew. Math. 260, 47--83 and 261, 29--54 (1973; Zbl 0272.12102)].
In part two we record some progress on the Jacobian problem, which is as yet unsolved. The results presented here were obtained by me during 1970/71. Partial notes on these were prepared by \textit{M. van der Put} and \textit{W. Heinzer} at Purdue University in 1971.
[For an updated version of the book under review including these notes see the author, Proc. Indian Acad. Sci., Math. Sci. 104, No. 3, 515--542 (1994; Zbl 0812.13013)]. coordinate ring of a meromorphic curve; epimorphism theorem; Jacobian problem S.S. Abhyankar , '' Expansion technics in Algebraic Geometry ,'' Tata Institute of Fundamental Research, Bombay, 1977. Relevant commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra Lectures on expansion techniques in algebraic geometry. With notes by Balwant Singh | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review is a review (survey) article on Néron models which is a written version of a series of lectures delivered at the Sino-French Conference in Algebraic and Arithmetic Geometry in Bordeaux, France, May 23--27, 2016.
The author of the paper under review started by mentioning some historical background of Néron models in the Introduction. ``Néron models were introduced by André Néron (1922--1985) in his seminar at the IHÉS in 1961''. ``Néron's own full account of his theory is found in [\textit{A. Neron}, Publ. Math., Inst. Hautes Étud. Sci. 21, 128 (1964; Zbl 0132.41403)]. Unfortunately, this article is not completely written in the modern language of algebraic geometry.'' Then he mentions some major references in the language of schemes, including ``the 1990 book [Néron models. Berlin etc.: Springer-Verlag (1990; Zbl 0705.14001)] by \textit{S. Bosch} et al.,'' which ``remains the most detailed source of information.'' After the Introduction, the remaining sections consist of ``Models, (Strong) Néron models, (Weak) Néron models, The Néron models of an abelian variety, The Néron model of a Jacobian'', and ``The group of components'', where in the last section, a conjecture was posed regarding the size of a quotient of the group of components of some abelian variety. This short survey consists of 24 pages in the main text and 4 pages listing 111 references including the past and more recent results. The survey also provides many examples contributing to better understanding for readers new in the area. Néron model; weak Néron model; abelian variety; group scheme; elliptic curve; semi-stable reduction; Jacobian; group of components Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\), Group schemes Néron models | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0588.00015.]
The derivative of the Abel-Jacobi mapping \(\phi: S\to J(V)=(H^{3,0}(V)+H^{2,1}(V))^*/H_ 3(V_ 1{\mathbb{Z}})\) of a versal family \({\mathcal C}\subset (S\times V)\) of curves on a smooth hypersurface V of \({\mathbb{P}}_ 4({\mathbb{C}})\) is computed in terms of the obstruction to split the normal bundle sequence of the triple \((C_ s,V,{\mathbb{P}}_ 4({\mathbb{C}}))\). The non-rationality of the general cubic hypersurface V of \({\mathbb{P}}_ 4({\mathbb{C}})\) is then shown, considering the family of rational cubic curves on \(V: \Phi\) (S) is the theta divisor of J(V), the above computation is used to show that V is isomorphic to the tangent cone to \(\Phi\) (S) at the singular point \(\Phi\) (s), where s is a plane section of V. This shows that J(V) is not the Jacobian of a curve. intermediate Jacobian; Abel-Jacobi mapping; non-rationality of the general cubic hypersurface; not the Jacobian of a curve \(3\)-folds, Rational and unirational varieties, Picard schemes, higher Jacobians The infinitesimal Abel-Jacobi mapping for hypersurfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Jacobian variety of a smooth plane cubic \(f=0\) over a field can be given in the form \(f^*=0\) with \(f^*\) an explicit Weierstrass form whose coefficients are homogeneous polynomials with integral coefficients in the coefficients of \(f\). \textit{A. Weil} treated the case of fields of characteristic not \(2\) and \(3\) [Arch. Math. 5, 197--202 (1954; Zbl 0056.03402)].
In the present paper this is done in the wider context of a family \(X\) of cubics in the projective plane \(\mathbb P^2\) over a base scheme \(S\). The authors give an explicit Weierstrass cubic equation \(f^*=0\) such that the smooth locus of the associated scheme represents \(\text{Pic}^0_{X/S}\). The proof uses that \(\text{Pic}^0_{X/S}\) satisfies a certain characterization of algebraic spaces that are commutative groups given by a Weierstrass equation. cubic curve; Jacobian; relative Picard scheme M. Artin, F. Rodriguez-Villegas and J. Tate, \textit{On the Jacobians of Plane Cubics}, \textit{Adv. Math.}\textbf{198} (2005) 366. Picard schemes, higher Jacobians, Jacobians, Prym varieties, Elliptic curves over global fields, Elliptic curves On the Jacobians of plane cubics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We establish the basic properties of the Mal'tsev completion of a discrete group relative to a Zariski dense representation of the group into an affine algebraic group. We then study the completion of a mapping class group with respect to its standard representation on the first homology of the corresponding Riemann surface. In particular, we prove that the pro-unipotent completion of the corresponding Torelli group is a central extension of the pro-unipotent radical of this completion of the mapping class group and that the kernel is one-dimensional, provided the genus is sufficiently large. The central extension is related to the normal function of the algebraic 1-cycle \(C-C^-\) in the Jacobian of an algebraic curve \(C\). Mal'tsev completion of a discrete group; affine algebraic group; mapping class group; pro-unipotent completion; Torelli group; algebraic 1-cycle; Jacobian of an algebraic curve Hain, R., Completions of mapping class groups and the cycle \(C - C^-\), Contemp. math., 150, 75-105, (1993) Topology of Euclidean 2-space, 2-manifolds, General low-dimensional topology, Families, moduli of curves (algebraic), Fundamental groups and their automorphisms (group-theoretic aspects), Algebraic cycles, Affine algebraic groups, hyperalgebra constructions, Infinite-dimensional Lie (super)algebras, Infinite-dimensional Lie groups and their Lie algebras: general properties, Rational homotopy theory, Differential topological aspects of diffeomorphisms Completions of mapping class groups and the cycle \(C-C^ -\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Here the author describes all the irreducible components of the variety of special divisors on a general \(5\)-gonal curve. His work is based the corresponding work for \(4\)-gonal curves and general \(k\)-gonal curves, \(k \geq 5\), made respectively in [\textit{M. Coppens} and \textit{G. Martens}, Math, Nachr. 213, 35--55 (2000; Zbl 0972.14021), and Abh. Math. Sem. Univ. Hamburg 69, 347--371 (1999; Zbl 0957.14018)]. gonality; variety of special divisors; Brill-Noether theory; \(k\)-gonal curve; pentagonal curve Park S.-S.: On the variety of special linear series on a general 5-gonal curve. Abh. Math. Sem. Univ. Hamburg 72, 283--291 (2002) Special divisors on curves (gonality, Brill-Noether theory) On the variety of special linear series on a general 5-gonal 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 topic of this paper is the development of some general approach to the study of fibration by holomorphic curves over arbitrary compact Kähler manifolds \(X\). The authors' method is based on harmonicity properties of the period map to a compactified moduli variety. They prove by this method that there exist at most finitely many nontrivial fiber spaces over \(X\) for which the fibers are curves of genus \(> 1\) outside some divisor \(R \subset X\) with simple normal crossings. This follows from the Arakelov theorem proved by purely algebraic-geometrical methods. Also the authors prove a version of the Mordell conjecture for the sections of such fiber spaces.
Then they apply their method to the following problem. Let \(f : M \to N\) be a smooth topological map of the Kähler manifolds \(M,N\) such that \(f\) has a maximal rank outside some divisor \(R \subset N\) as above and with two-dimensional fibers. When is this map homotopic to a holomorphic map with the same properties? -- If \(M\) is a complex surface and \(N\) is a curve then the authors show that a sufficient condition for this is nontriviality of the map on the second homology. In particular, if a compact Kähler surface \(M\) is fibrated over a curve then the same is true for any surface homeomorphic to \(M\).
The author, also study variations of the singular set of families over a curve \(C\). Let \(B_ i\), \(i = 1,2\), be two fibrations over \(C\) with singular sets \(S_ i\) which are algebraically equivalent on \(C\). Assume that \(C - S_ 1\) is not conformly equivalent to \(C - S_ 2\) and assume that there are two diffeomorphisms of \(C\) and between \(B_ 1\) and \(B_ 2\) which make these fibrations topologically equivalent. Then there exists a holomorphic family of holomorphic fibrations which include these two. At the end a new proof of Faltings' rigidity theorem for abelian varieties is proved by the new method. harmonic maps; finiteness problems; topological map homotopic to a holomorphic map; fibration by holomorphic curves; period map; compactified moduli variety; Arakelov theorem; Mordell conjecture; Kähler manifolds; variations of the singular set of families over a curve Jürgen Jost and Shing-Tung Yau, Harmonic mappings and algebraic varieties over function fields, Amer. J. Math. 115 (1993), no. 6, 1197 -- 1227. Families, fibrations in algebraic geometry, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Global differential geometry of Hermitian and Kählerian manifolds, Algebraic functions and function fields in algebraic geometry, Singularities of curves, local rings Harmonic mappings and algebraic varieties over 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 In this paper we continue the study of curves \(C\) on a smooth cubic surface \(S\) of \(\mathbb{P}^ 3\) begun by the first author [cf. Mathematiche 40, No. 1/2, 252-266 (1985; Zbl 0699.14005) and Ann. Mat. Pura Appl., IV. Ser. 153, 275-292 (1988; Zbl 0695.14015)]. We obtain a complete description of the graded Betti numbers of the ideal sheaf \({\mathcal I}_ C\) of \(C\) in terms of the seven integers which describe \(C\) as a divisor on \(S\). A consequence of our results is that the graded Betti numbers of a reduced and irreducible curve \(C\), lying on a smooth cubic (or quadric) surface \(S\), do not change within the same linear equivalence class, i.e., they are determined by the class of \(\text{Pic} S\) to which \(C\) belongs. cubic surface; space curve; Picard group; graded Betti numbers of a curve S. Giuffrida and R. Maggioni, On the resolution of a curve lying on a smooth cubic surface in \(\mathbf P^3\), Trans. Amer. Math. Soc., 331 (1992), 181-201. Plane and space curves, Picard groups, Divisors, linear systems, invertible sheaves, Special surfaces On the resolution of a curve lying on a smooth cubic surface in \(\mathbb{P}^ 3\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We give a description of the singular locus of a Schubert variety X, in the flag variety G/B, where G is a classical group and B is a Borel subgroup. The singular locus is determined by using standard monomial theory as developed in ''Geometry of G/B'' [by the authors and \textit{C. Musieli}; part I-IV in Collect. Publ. C. P. Ramanujan and Papers in his Mem., Tata Inst. Fundam. Res., Stud. Math. 8, 207-239 (1978); Proc. Indian Acad. Sci., Sect. A 87, No.2, 1-54 (1978); ibid. 88, No.2, 93-177 (1979); ibid. 88, No.4, 279-362 (1979; Zbl 0447.14010-14013); part V (to appear)]. A consequence of this theory is the determination of the ideal defining X in G/B, using which, we are able to write the Jacobian matrix Jw,\(\tau\) (here w is given by \(X=X(w)\) and \(e_{\tau}\) is the T-fixed point in G/B corresponding to \(\tau\), \(\tau\leq w\), T being a maximal torus contained in B) in the affine neighborhood \(U^-_{\tau}\cdot\tau \) of \(e_{\tau}\), where \(U^-_{\tau}=\tau U^-\tau^{-1}\), \(U^-\) being the unipotent part of the Borel subgroup of G opposite to B. Evaluating Jw,\(\tau\) at \(e_{\tau}\), we obtain the dimension of \(Z_{w,\tau}\), the Zariski tangent space to X(w) at \(e_{\tau}\). Denoting by \(\{\) \(p(\lambda\),\(\mu)\}\) the weight vectors as given by standard monomial theory, let \(R(w,\tau)=\{\beta\in \tau (\Delta^+)\quad there\quad exists\quad a\quad p(\lambda,\mu),\quad such\quad that\quad\quad w\ngeq\lambda \quad and\quad X_{- \beta}p(\lambda,\mu)=cp(\tau),c\in k^*\}.\) Then we have the main theorem \(\dim Z_{w,\tau}=N-\# R(w,\tau)\) where \(N=\# \Delta^+=\# \{positive\quad roots\}).\) In particular we have X(w) is smooth at \(e_{\tau}\) if and only if \(N-\# R(w,\tau)=\upharpoonright (w),\) the length of w. dimension of Zariski tangent space; singular locus of a Schubert variety; flag variety; standard monomial theory; Jacobian matrix; weight vectors C.S. Seshadri : Normality of Schubert variety . Proceeding de ''Algebraic Geometry'' (Bombay, Avril 1984). Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry Singular locus of a Schubert variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\widetilde A\) denote a smooth compactification of the \(k\)-fold fiber product of the universal family \hbox {\(A^ 1 \to M\)} of elliptic curves with level \(N\) structure. The purpose of this paper is to completely describe the algebraic cycles in and the Hodge structure of the Betti cohomology \(H^* (\widetilde {A}, \mathbb{Q})\) of \(\widetilde A\). For by doing so we are able
(a) to verify both the usual and generalized Hodge conjectures for \(\widetilde A\);
(b) to describe both the kernel and the image of the Abel-Jacobi map from algebraic cycles algebraically equivalent to zero (modulo rational equivalence) into the Griffiths intermediate Jacobian; and
(c) to verify Tate's conjecture concerning the algebraic cycles in \(H_{\text{et}}^* (\widetilde {A}\otimes \overline {\mathbb{Q}}, \mathbb{Q}_ \ell)\), the étale cohomology.
The methods used lead also to a complete description of the Hodge structure of the Betti cohomology \(H^*(E^ k, \mathbb{Q})\) of the \(k\)-fold product of an elliptic curve \(E\) without complex multiplication, and a verification of the generalized Hodge conjecture for \(E^ k\).
The main result concerning \(H^*(E^ k, \mathbb{Q})\) is that as a rational Hodge structure it is the direct sum of a number of rational sub-Hodge structures which in turn are products of irreducible rational Hodge structures of type \(\{(\nu,0)\), \((\nu - 1,1), \dots, (0, \nu)\}\) with one of type \(\{(\mu, \mu)\}\), where the product is actually realized by the cup product in \(H^* (E^ k, \mathbb{Q})\), and \(2\mu + \nu = m\). The major ingredients of the proof are that the special Mumford-Tate group, or Hodge group, of \(E\) is \(\text{SL}_ 2\), that the \(\text{SL}_ 2\)- invariant subspaces of \(H^* (E^ k),\mathbb{Q})\) are precisely the rational sub-Hodge structures, that these subspaces can be determined by classical invariant theory, and that by the usual Hodge conjecture, the one- dimensional invariants are all algebraic cycles.
The main result concerning \(H^* (\widetilde{A}, \mathbb{Q})\) is that as a rational Hodge structure it is a direct sum of sub-Hodge structures which are products of algebraic cycles with irreducible sub-Hodge structures of type \(\{(\nu,0)\) \((0, \nu)\}\). Moreover, the proof shows that there are three kinds of classes which generate \(H^* (\widetilde{A}, \mathbb{Q})\) as an algebra:
(i) divisors in \(\widetilde A\) which are the closures in \(\widetilde A\) of divisors in a generic fiber \(A_ \eta\);
(ii) the classes of algebraic cycles contained in the compactifying divisor at infinity; and
(iii) classes which correspond to holomorphic cusp forms on \(\Gamma (N)\) and their complex conjugates. As a result, strong forms of both the usual and generalized Hodge conjectures are true for \(\widetilde A\): all of \(H^{d,d} (\widetilde A)\) is spanned (over \(\mathbb{C})\) by algebraic cycles; and the coniveau filtration is exactly equal to the Hodge filtration intersected with rational cohomology. Then Tate's conjecture follows immediately from this strong form of the usual Hodge conjecture and the main theorem of \(p\)-adic Hodge theory, while the validity of the generalized Hodge conjecture for \(\widetilde A\) implies that Abel-Jacobi equivalence and incidence equivalence are isogenous on algebraic cycles algebraically equivalent to zero. Kuga variety; intermediate Jacobian; cusp forms; generalized Hodge conjecture; Abel-Jacobi map; algebraic cycles; elliptic curve; rational Hodge structure; Tate's conjecture 10.2307/2154385 Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic cycles, Geometric invariant theory Algebraic cycles and the Hodge structure of a Kuga fiber 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 One can try to study the fundamental group of a smooth, complex projective variety \(S\) by looking at spaces of representations of \(\pi_ 1 (S)\). If the space of representations is a union of isolated points, then this structure reduces to the discrete structure of the set of representations. So a natural problem is to try to find examples of varieties \(S\) where there exist nontrivial continuous families of representations (or, equivalently, local systems on \(S) \). It is not too hard to see that if \(S\) is a projective algebraic curve of genus \(g \geq 2\), then the moduli space of representations of rank \(r \geq 1\) has dimension \((2g - 2) r^ 2 + 2\); or, for example, if \(S\) is a variety with \(\dim H^ 1(S, \mathbb{C}) = a\), then the space of representations of rank 1 has dimension \(a\). Taking tensor products of pullbacks of families of local systems arising in these ways, we obtain some more families. To pursue the problem we ask: Do families other than these exist?
The idea in this article is to use the next natural construction, taking higher direct images of local systems. Suppose that \(f:X \to S\) is a smooth projective morphism and \(\{W_ t\}\) is a family of local systems on \(X\), chosen in a simple way. Let \(V_ t = R^ i f_ * (W_ t)\). This is a collection of local systems, and if the ranks are constant, then it is a continuous family. We can hope that \(\{V_ t\}\) will be an interesting family of local systems on \(S\). The principal question that needs to be addressed is whether, if the family \(\{W_ t\}\) varies nontrivially, the family of direct images \(\{V_ t\}\) varies nontrivially. We are also interested in constructing examples where we can show that the family \(\{V_ t\}\) does not, by some miracle, arise from a simpler construction such as the one described before.
This article consists essentially of two parts. The first (sections 1-5) is devoted to answering our principal question in a fairly general situation. For this we develop the technique of taking the direct image of a harmonic bundle and its associated Higgs bundle. We give a way to calculate the spectral varieties of the Higgs bundles associated to \(V_ t\), as a way of verifying that the \(V_ t\) vary nontrivially. -- The second part (sections 6-8) is concerned with the construction of a particular class of examples and the verification of some additional properties about monodromy groups and possible factorization through morphisms to algebraic curves; these properties serve to show that our examples do not come from tensor products of pullbacks. The methods used in this part are all fairly well known. representations of fundamental group; local systems on complex projective variety; moduli space of algebraic curve; families of representations; direct image of a harmonic bundle; Higgs bundle; spectral varieties of the Higgs bundles Simpson C.: Some families of local systems over smooth projective varieties. Ann. Math. 138, 337--425 (1993) Homotopy theory and fundamental groups in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Variation of Hodge structures (algebro-geometric aspects) Some families of local systems over smooth projective varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We propose three ideas to speed up the computation of the group operation in the Jacobian of a hyperelliptic curve: 1. Division of polynomials without inversions in the base field, and an extended gcd algorithm which uses only one inversion in the base field. 2. The omission of superfluous calculations in the reduction part. 3. Expressing points on the Jacobian in a slightly different form. Jacobian of a hyperelliptic curve Nagao, K.: Improving group law algorithms for Jacobians of hyperelliptic curves, Lecture notes in comput. Sci. 1838, 439-448 (2000) Symbolic computation and algebraic computation, Jacobians, Prym varieties, Applications to coding theory and cryptography of arithmetic geometry, Software, source code, etc. for problems pertaining to algebraic geometry Improving group law algorithms for Jacobians of hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The level of a field~\(K\) is defined as the minimal number of terms that are needed to write \(-1\) as a sum of squares. If~\(K\) is the function field of a real algebraic variety without real points then the level of~\(K\) is finite, and, in fact, a power of~\(2\) according to a theorem of \textit{A. Pfister} [Invent. Math. 4, 229-237 (1967; Zbl 0222.10022)]. Using equivariant cohomology, the author establishes a topological criterion for~\(K\) to be of level~\(2\). As a consequence, he shows that the function field of a non-trivial principal homogeneous space under the action of a real abelian variety is of level~\(2\).
Summary of the paper: The image of the Picard group of a non-singular projective variety \(X\) over \(\mathbb{R}\) into the Galois-invariant part of the Picard group of \(X\otimes \mathbb{C}\) is described in terms of cohomological invariants of the analytic manifold \(X(\mathbb{C})\) with its antiholomorphic involution. This description gives for example a link between the level of the function field of \(X\) and the topological level of (Zariski-open subsets of) \(X(\mathbb{C})\); it easily follows that a non-trivial principal homogeneous space over a real abelian variety has a function field of level~2. function field of homogeneous space; level of a field; equivariant cohomology; real abelian variety; Picard group Van Hamel, J., \textit{divisors on real algebraic varieties without real points}, Manuscripta Math., 98, 409-424, (1999) Real algebraic sets, Picard groups, Forms over real fields, Other nonalgebraically closed ground fields in algebraic geometry, Divisors, linear systems, invertible sheaves, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) Divisors on real algebraic varieties without real points | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is the second volume of the book, the first volume of which consists of two chapters I, II (1983; Zbl 0509.14049). Chapter I is devoted to the theta functions \(\theta\) of genus one, or geometric and arithmetic theory of elliptic curves; in Chapter II \(\theta\) is generalized to the case of several variables and the author develops the theory of abelian varieties, especially Jacobian varieties associated to compact Riemann surfaces mainly following Riemann's lines, with applications to modular forms.
This volume II includes three parts III a-III c, and the third part III c is written by \textit{H. Umemura}, where he proves that any algebraic equation is solved by using hyperelliptic theta functions and hyperelliptic integrals (instead of radicals in the case of abelian polynomials). Here Thomae's formula (theorem 8.1 in III a) plays an essential role. III a and III b are concerned only with theta functions \(\theta\) (z,\(\Omega)\) in the case, where \(\Omega\) are symmetric matrices arising as period matrices of Riemann surfaces C; C is hyperelliptic in III a, and is an arbitrary Riemann surface in III b. In both cases the author lays stress on producing solutions of some important non-linear partial differential equations from these special \(\theta\) 's. Now we look over the outline of each part.
III a is the main part of this volume and a beautiful combination of the cassical geometry of hyperelliptic curves and dynamical systems, with applications to non-linear PDE's. After the reader reviews some algebraic geometric background in {\S} 0, he learns a parameter variety of effective divisors D of degree \(\nu\), on a hyperelliptic curve C (due to Jacobi) in {\S} 1. In this case D is not arbitrary, but this parameter variety and the coordinate corresponding to the divisor D (which we call ''Jacobi coordinate'' though it is not named in the paper), seems more convenient for our purpose than the Chow variety and coordinate which are more generally applicable. When C is of genus g and \(\nu =g\), the parameter variety is used for algebraic construction of the Jacobian variety Jac C of C ({\S} 2), and the Jacobi coordinate corresponding to D is very explicitly determined later ({\S} 5, {\S} 7).
Under this model of Jacobi the translation invariant vector fields on Jac C are given by explicit formulae ({\S} 3), and these models and formulae are used to solve the Neumann dynamical system ({\S} 4). {\S} 5 links the above theory with the analytic and Riemann theory of Jac C as a complex torus \({\mathbb{C}}^ g/L_{\Omega}\) in chapter II, and the Jacobi coordinates are determined (up to a scalar) as meromorphic functions on \({\mathbb{C}}^ g/L_{\Omega}\) using theta functions. These sections {\S}{\S} 0-5 cover good fundamental theory of hyperelliptic curves, which enables us to prove that hyperelliptic thetas have the fundamental vanishing property (corollary 6.7) and that this property characterizes hyperelliptic period matrices \(\Omega\) among all matrices in the Siegel upper half space (theorem 9.1). These are the main results in {\S} 6 and {\S} 9.
On the other hand between these two sections, Frobenius' theta formula ({\S} 7) and Thomae's formula ({\S} 8) are given; the former formula has two applications: (1) evaluating the scalar part of the Jacobian coordinates and (2) giving explicitly via thetas the solutions of Neumann's dynamical system; the latter formula determines the affine ring of the moduli space of hyperelliptic curves and is also used in III c as we mentioned before. The application (1) in {\S} 7 and the theory in {\S} 4 (corollary 4.9) are essential in the proof of characterization of ''hyperelliptic'' in {\S} 9.
In {\S} 10, on a hyperelliptic Jacobian Jac C an analogy of Weierstrass'\(\wp P\)-function, which is also called the ''\(\wp\)-function'', is given; and the main point of this section is to discuss affine embedding of an open subset of Jac C by using P(z) and its derivatives, and to write down explicitly the equations defining the image variety. The final section {\S} 11 of III a starts with rather long general discussion of the co-symplectic structure and presents KdV as a completely integrable dynamical system in an infinite dimensional space of pseudo-differential operators for the purpose of describing all the differential identities satisfied by hyperelliptic thetas. Actually the KdV dynamical system is solved by using \(\wp\)-function in {\S} 10.
The second part III b of the chapter takes up general Jacobian theta functions (i.e., \(\theta\) (z,\(\Omega)\) for \(\Omega\) the period matrix of an arbitrary Riemann surface C), except in {\S} 5. The fundamental identity between such thetas is the ''trisecant'' identity, due to \textit{J. Fay} [''Theta functions on Riemann surfaces'', Lect. Notes Mat. 352 (1973; Zbl 0281.30013)], whose geometric interpretation tells us that the Kummer variety (corresponding to Jac C) has \(\infty^ 4\) trisecants. This identity is proved in {\S} 2 after a preliminary tool, the ''prime form'' on \(C\times C\), is discussed in {\S} 1. Specializing the identity one gets several formulae involving derivatives of theta functions ({\S} 3) and constructs special solutions to many non-linear PDE's occuring in mathematical physics: the KP equation (in the general case) and KdV, Sine-Gordon (in the hyperelliptic case) in {\S} 4. This part III b concludes with showing how to use the generalized Jacobian in the simplest case and its theta functions to describe and explain the soliton solutions to KdV as limits of the thetas. The contents of III c is already reviewed above.
At the end of the introduction, the author says there are two striking unsolved problems: (1) to find the differential identities in z satisfied by \(\theta\) (z,\(\Omega)\) for general \(\Omega\), and (2) the ''Schottky problem''; and gives four forthcoming papers as reference for the second problem. The reviewer refers to a new paper of \textit{H. Morikawa} for the first problem [''A decomposition theorem on differential polynomials of theta functions'', Nagoya Math. J. 96, 113-124 (1984)]. Finally we mention that the book is stimulating to both young students and specialists. Korteweg-de Vries dynamical system; theta functions; period matrices of Riemann surfaces; dynamical systems; effective divisors D; Jacobi coordinate; Jacobian variety; hyperelliptic curves; hyperelliptic thetas; Weierstrass'\(\wp P\)-function Mumford, D.: Tata Lectures on Theta II. Birkhäuser, Boston (1984) Theta functions and abelian varieties, Picard schemes, higher Jacobians, Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Elliptic functions and integrals, Jacobians, Prym varieties, Relations of PDEs on manifolds with hyperfunctions, Partial differential equations of mathematical physics and other areas of application Tata lectures on theta. II: Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0752.00024.]
Fuchsian systems of partial differential equations of two variables with monodromy groups acting on the complex two-dimensional unit ball \(\mathbb{B}\) have been first studied in more detail by \textit{E. Picard} [see e.g. Acta Math. 2, 114-135 (1883)]. \textit{P. Deligne} and \textit{G. D. Mostow} [Publ. Math., Inst. Hautes Étud. Sci. 63, 5-89 (1986; Zbl 0615.22008)] proved that a lot of them are sublattices of the group \(\text{Aut}(\mathbb{B})=PU((2,1),\mathbb{C})\) of biholomorphic automorphisms of \(\mathbb{B}\), in several cases they are of arithmetic nature. They appeared also in the work of Shimura as modular groups in connection with polarized abelian threefolds which have an additional (complex) multiplication with elements of an imaginary quadratic number field \(E\). In a series of papers, \textit{J. Feustel} and the reviewer [around the reviewer's book: ``Geometry and arithmetic. Around Euler partial differential equations'' (1986; Zbl 0595.14017)] classified from a geometric point of view ball quotient surfaces \(\Gamma\backslash\mathbb{B}\) where \(\Gamma\) is a sublattice of the arithmetic group \(U((2,1),{\mathfrak O})\), \({\mathfrak O}={\mathfrak O}_ E\) the ring of integers of \(E\). They called these surfaces and their compactifications ``Picard modular surfaces'' in analogy to the Hilbert modular surfaces.
Both types of surfaces belong to a broad class of varieties called ``Shimura varieties''. They have been studied in- and extensively by Shimura in a series of papers from an arithmetic viewpoint. \textit{P. Deligne} [in Semin. Bourbaki 1970/71, Exposé 389, Lect. Notes Math. 244, 123-165 (1971; Zbl 0225.14007)] provided a clear introduction adding at the same time a fertile new concept [in: Automorphic forms, representations and \(L\)-functions, Proc. Symp. Corvallis/Oregon 1977, Proc. Symp. Pure Math. 33, No. 2, 247-290 (1979; Zbl 0437.14012)]. Especially for the Picard surfaces the basic arithmetic properties, such as their reduction are presented as aspects of the solution of a moduli problem.
Great advanced techniques are required in order to understand finer arithmetic properties of Shimura varieties: representation theory, harmonic analysis, Hecke algebra, trace formula and endoscopy, intersection cohomology and \(L^ 2\)-cohomology, abelian varieties and their reductions. They all take active part on recent developments around the central problem of understanding the Zeta functions of these varieties. Most of the essential difficulties arise already in the arithmetic study of Picard modular surfaces, but fortunately, in this case one disposes additionally on an intuitive geometric appeal helping to overcome them.
The book of the reference title contains 16 lectures presented by the following authors:
B. B. Gordon, M. J. Larsen, M. Goresky, J. D. Rogawski, R. E. Kottwitz, M. Rapoport, J. S. Milne, R. P. Langlands, D. Ramakrishnan, Th. C. Hales, D. Blasius, V. K. Murty, R. MacPherson, L. Saper and M. A. Stern.
We start with a series of references, one for each lecture, the first given by \textit{B. B. Gordon}.
Besides of the introduction the article is divided into six sections: 1. Picard modular surfaces over \(\mathbb{C}\); 2. Adelic construction; 3. Moduli space interpretation; 4. Canonical models; 5. Compactifications; 6. Generalizations to a CM field.
Let \(V\) be a 3-dimensional \(E\)-vector space with an \({\mathfrak O}\)-lattice \(L\) in \(V\) and \(J:V\times V\to E\) a non-degenerate hermitian form on \(V\), such that \(J(L,L)\subset{\mathfrak O}\). \(G':=SU(J,V)\) is a semisimple algebraic group over \(\mathbb{Q}\), \(G'(\mathbb{R})\cong SU((2,1),\mathbb{C})\), \(G'(\mathbb{Z})=SU(J,L)\) is the special Picard modular group acting properly discontinuously on \(\mathbb{B}\) as any \(\Gamma\subset U((2,1),\mathbb{C})\) commensurable with \(G'(\mathbb{Z})\) does. The author considers only Picard modular surfaces \(S_ \Gamma(\mathbb{C}):=\Gamma\backslash\mathbb{B}\) with \(\Gamma\subseteq G'\) forgetting in this way some geometrically important cases as investigated carefully in the reviewer's book cited above. The normal projective Satake-Baily-Borel compactification \(\overline S_ \Gamma(\mathbb{C})\) is obtained from \(S_ \Gamma(\mathbb{C})\) by adjoining a finite number of cusps. These are simple elliptic singularities if \(\Gamma\) acts neatly on \(\mathbb{B}\). The crucial points of this construction with references to the origins is given in the first section. Especially, the compactification part is described by
\[
\overline S^ \infty_ \Gamma(\mathbb{C})=\Gamma\backslash\mathbb{B}^ \infty\cong\Gamma\backslash G'(\mathbb{Q})/P'(\mathbb{Q}),
\]
where \(P'\subset G'\) is a parabolic subgroup stabilizing a rational line of the boundary set \(\mathbb{B}^ \infty\subset Gr_ 1(V(\mathbb{R}))\) of \(J\)-isotropic lines in \(V(\mathbb{R})\).
From section 2 on the author applies consequently Deligne's concept of Shimura varieties to Picard modular surfaces. For this purpose the reductive algebraic group \(G:=GU(J,V)_ \mathbb{Q}\) of unitary similitudes is introduced. Let \(\mathbb{A}\) be the ring of \(\mathbb{Q}\)-adeles, \(\mathbb{A}_ f\) its finite part and \(K\) a compact open subgroup of \(G(\mathbb{A}_ f)\). The Picard modular surface of \(K\) is the Shimura variety over \(\mathbb{C}\)
\[
S_ K(G,\mathbb{B})_ \mathbb{C}(\mathbb{C}):=G(\mathbb{Q})\backslash G(\mathbb{A})/K_ \infty K=G(\mathbb{Q})\backslash \mathbb{B}\times G(\mathbb{A}_ f)/K,
\]
where \(K_ \infty\) is the centralizer in \(G(\mathbb{R})\) of a complex structure morphism \(h:S\to G_ \mathbb{R}\), \(S=R_{\mathbb{C}/\mathbb{R}}(G_{m\mathbb{C}})\); \(S_ K(G,\mathbb{B})(\mathbb{C})\) splits into a finite disjoint set of connected components of geometric Picard surfaces of type \(S_ \Gamma(\mathbb{C})\) described in the first section. It is proved that these components form a principal homogeneous space for the group \(\pi_ 0(T(\mathbb{A})/T(\mathbb{Q})\nu(K))\), where \(T:=R_{E/\mathbb{Q}}G_{mE}\), \(\nu=\text{det}/\mu\) a special fixed homomorphism from \(G\) to \(T\).
In section 3 the following result is proved: Proposition. For any compact open \(K\subset G(\hat\mathbb{Z})\), there is a one-to-one correspondence between the set of points of \(S_ K(G,\mathbb{B})(\mathbb{C})\) and the set of isomorphism classes of quadruples \((A,p,m,\overline\varphi)\) where \((A,p)\) is a 3-dimensional homogeneously polarized abelian variety over \(\mathbb{C}\) with complex multiplication \(m:{\mathfrak O}\to\text{End}(A)\) by \({\mathfrak O}\), signature (2,1), and level-\(K\) structure \(\overline\varphi\).
Section 4 remembers to the definition of reflex fields. It is proved that the reflex field of \((G,\mathbb{B})\) coincides with \(E\). Then it is verified that the Picard surfaces \(S_ K(G,X)_ \mathbb{C}\) have a ``canonical model'' \(S_ K(G,X)\) over \(E\). The same is true for the projective limit
\[
S_ \mathbb{C}=S(G,\mathbb{B})_ \mathbb{C}=\varprojlim_ KS_ K(G,\mathbb{B})_ \mathbb{C}=G(\mathbb{Q})\backslash G(\mathbb{A})/K_ \infty=G(\mathbb{Q})\backslash\mathbb{B}\times G(\mathbb{A}_ f)
\]
over all possible \(K\) (proposition 4.6, Miyake's theorem 4.9). The reciprocity law is explained as an action of \(\text{Gal}(\overline E/E)\) on the set of connected components of \(S\) or \(S_ K\).
The close connection of the number of cusps of a Picard surface \(\overline S_ K(\mathbb{C})\) with the class number of \(E\) is explained in section 5. Moreover it is proved that there are models of \(\overline S_ K\) and \(\tilde S_ K\) defined over \(E\) as normal projective schemes. This also holds for the infinite parts \(\overline S_ K^ \infty\) respectively \(\tilde S_ K^ \infty\). Their components are defined over \(E^{ab}\) (theorem 5.2, proposition 5.5).
In the last section the author outlines the transfer of the results to arbitrary CM fields \(E\) over totally real fields \(F\). In this generality one deals with arithmetic quotients of finite products \(B^ r\),
\[
G'(\mathbb{R})=\prod_ \lambda G_ \lambda'(\mathbb{R})\cong SU(2,1)^ r\times SU(3)^{g-r},
\]
where signature\((V_ \lambda,J_ \lambda)=(2,1)\), for \(1\leq\lambda\leq r\) and signature\((V_ \lambda,I_ \lambda)=(3,0)\), for \(r\leq\lambda\leq g\). Picard modular surfaces; Shimura varieties; Fuchsian systems of partial differential equations; polarized abelian threefolds; polarized abelian variety; complex multiplication; number of cusps of a Picard surface; class number; CM fields B. Gordon, Canonical models of Picard modular surfaces, The zeta functions of Picard modular surfaces, Centre de Recherches Mathématiques, Université de Montréal, Montréal (1992), 1-29. Modular and Shimura varieties, Global ground fields in algebraic geometry, Special surfaces, Complex multiplication and abelian varieties, Arithmetic ground fields for abelian varieties, Families, moduli, classification: algebraic theory Canonical models of Picard modular surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In general, algorithms for computing the Selmer group of the Jacobian of a curve have relied on either homogeneous spaces or functions on the curve. We present a theoretical analysis of algorithms which use functions on the curve, and show how to exploit special properties of curves to generate new Selmer group computation algorithms. The success of such an algorithm will be based on two criteria that we discuss. To illustrate the types of properties which can be exploited, we develop a \((1-\zeta_p)\)-Selmer group computation algorithm for the Jacobian of a curve of the form \(y^p= f(x)\), where \(p\) is a prime not dividing the degree of \(f\). We compute Mordell-Weil ranks of the Jacobians of three curves of this form. We also compute a 2-Selmer group for the Jacobian of a smooth plane quartic curve using bitangents of that curve, and use it to compute a Mordell-Weil rank.
Added in 2007: The erratum concerns the second half of Proposition 2.4 which is true only under certain conditions. quartic Diophantine equations; computation of Selmer group; Jacobian of a curve; algorithms; Mordell-Weil ranks E. Schaefer, ''Computing a Selmer Group of a Jacobian Using Functions on the Curve,'' Math. Ann. 310, 447--471 (1998); ''Erratum,'' Math. Ann. 339, 1 (2007). Abelian varieties of dimension \(> 1\), Curves of arbitrary genus or genus \(\ne 1\) over global fields, Jacobians, Prym varieties, Cubic and quartic Diophantine equations Computing a Selmer group of a Jacobian using functions on the curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a connected semisimple affine algebraic group defined over \(\mathbb{C}\). We study the relation between stable, semistable \(G\)-bundles on a nodal curve \(Y\) and representations of the fundamental group of \(Y\). This study is done by extending the notion of (generalized) parabolic vector bundles [see \textit{Usha N. Bhosle}, Ark. Math. 30, No. 2, 187-215 (1992; Zbl 0773.14006)] to principal \(G\)-bundles on the desingularization \(C\) of \(Y\) and using the correspondence between them and principal \(G\)-bundles on \(Y\).
We give an isomorphism of the stack of generalized parabolic bundles on \(C\) with a quotient stack associated to loop groups.
We show that if \(G\) is simple and simply connected then the Picard group of the stack of principal \(G\)-bundles on \(Y\) is isomorphic to \(\bigoplus_m \mathbb{Z}\), \(m\) being the number of components of \(Y\). \(G\)-bundles on a nodal curve; stack of principal \(G\)-bundles; fundamental group; stack of generalized parabolic bundles; loop groups Vector bundles on curves and their moduli, Coverings of curves, fundamental group, Group actions on varieties or schemes (quotients), Singularities of curves, local rings Principal \(G\)-bundles on nodal 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 we consider certain abelian varieties defined over \(\mathbb{Q}\) which arise as simple factors of the Jacobian variety \(J_ 0(N)\) of the modular curve \(X_ 0(N)\). These abelian varieties with `extra twist' were investigated fully by \textit{K. A. Ribet} [Math. Ann. 253, 43-62 (1980; Zbl 0421.14008)] and \textit{F. Momose} [J. Fac. Sci., Univ. Tokyo, Sect. IA 28, 89-109 (1981; Zbl 0482.10023)]. Here we shall be interested in the case where the variety \(A\) is two-dimensional and the extra twist is given by the character attached to a quadratic field \(k\); the \(L\)-series of \(A\) then turns out to be the Mellin transform \(L(F,s)\) of a cuspidal automorphic form \(F\) of weight 2 over \(k\), with rational integer coefficients. Such cusp forms, for imaginary quadratic \(k\), have been studied by the author \((\dots)\) and others \((\dots)\), seeking a Weil-Taniyama-type correspondence between cusp forms of weight 2 and elliptic curves over \(k\). Our main result \((\dots)\) is that if \(A\) remains simple over \(k\) then \(L(F,s)\) is \textit{not} the \(L\)-series of an elliptic curve defined over \(k\).''
Numerical examples are given in the range \(N\leq 300\). Actually, 12 relevant abelian varieties \(A\) are listed, 2 of which (one with CM, one without) fail to split over the twisting field \(k\). The final section is devoted to a discussion of the Weil-Taniyama conjecture over imaginary quadratic fields in relation to the author's results. modular curve; abelian varieties with twist; \(X_ 0\); simple factors of the Jacobian variety; correspondence between cusp forms of weight 2 and elliptic curves; Taniyama-Weil conjecture J.E. Cremona, Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields, J. London Math. Soc. (2), 45 (1992), 404-416. MR 93h:11056 Arithmetic ground fields for abelian varieties, Modular and Shimura varieties, Abelian varieties of dimension \(> 1\), Elliptic curves over global fields, Holomorphic modular forms of integral weight, Quadratic extensions Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic 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 \(C\) be an integral curve in \(\mathbb{P}^ 3\) lying on a smooth quartic surface \(S\). The purpose of this paper is to determine the degrees of the elements in a minimal generating set of the homogeneous ideal of \(C\), continuing work of the same authors for curves on a smooth quadric or cubic surface. The technique is to study certain cohomology groups which arise in a natural way from \(C\) and depend only on the element of Pic\(S\) associated to \(C\). Here the problem is somewhat more complicated than it was in the case of a cubic surface since there are several non-isomorphic smooth quartic surfaces, having different Picard groups; hence the results are not as detailed as they are in the cubic case. Bounds are given on the number of generators, and in particular it is asked whether gaps can occur in the degrees of the generators. These results are related to the geometry of \(C\) and of \(S\) in a nice way, and involve liaison. Slightly more detailed results are obtained at the end of the paper for the case where \(S\) is a so-called \textit{Mori quartic}, since in this case more is known about the geometry of \(S\). minimal generating set of the homogeneous ideal of integral space curve; curves on a surface; Picard groups; liaison; Mori quartic DOI: 10.1016/0022-4049(91)90140-W Plane and space curves, Relevant commutative algebra, Picard groups Generators for the ideal of an integral curve lying on a smooth quartic 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 The author presents a collection of algebraic equivalences between tautological cycles on the Jacobian \(J\) of a curve, i.e., cycles in the subring of the Chow ring of \(J\) generated by the classes of certain standard subvarieties of \(J.\) These equivalences are universal in the sense that they hold for all curves of given genus. He show also that they are compatible with the action of the Fourier transform on tautological cycles and compute this action explicitly. Let \(J\) be the Jacobian of a smooth projective complex curve \(C\) of genus \(g \geq 2.\) For every \(d, 0 \leq d \leq g,\) consider the morphism \(\sigma_{g} : \text{Sym}^{d}C \rightarrow J : D \mapsto O_{C}(D - d p),\) where \(p\) is fixed point on \(C.\) Let us set \(p_{k} = -N^{k}(w) \in \text{CH}^{k}(J)\) for \(k \geq 1,\) where \(N^{k}(w)\) are the Newton polynomials on the classes \(w_{1},\dots , w_{g}.\)
Theorem 0.1. (i) Let us define the differential operator \(D\) acting on polynomials in infinitely many variables \(x_{1}, x_{2},\dots \) by setting
\[
D = -g \partial_{1} + \frac{1}{2} \sum_{m,n \geq 1} C^{m + n}_{m}x_{m+n-1} \partial_{m} \partial_{n},
\]
where \(\partial_{i} = \frac{d}{dx_{i}}.\) Then for every polynomial \(F\) of the form \(F(x_{1}, x_{2},\dots ) = D^{d}(x_{1}^{m_{1}}\dots x_{k}^{m_{k}}),\) where \(m_{1} + 2 m_{2} +\dots + k m_{k} = g,\) \(m_{1} < g\) and \(d \geq 0,\) one has \(F(p_{1}, p_{2},\dots ) = 0,\) in \(\text{CH}^{g-d}(J)_{{\mathbb Q}}/(\text{alg}).\)
(ii) Here is another description of the same collection of relations. For every \(k \geq 1,\) every \(n_{1},\dots , n_{k}\) such that \(n_{i} > 1,\) and every \(d\) such that \(0 \leq d \leq k - 1,\) one has
\[
\sum_{[1,k]=I_{1}\sqcup I_{2}\sqcup I_{m}} C^{m-1}_{d+m-k} b(I_{1}) \dots b(I_{m}) \times p_{1}^{[g-d-m+k-\sum_{i=1}^{k}n_{i}]} p_{d(I_{1})}\dots p_{d(I_{m})} = 0
\]
in \(\text{CH}^{g-d}(J)_{{\mathbb Q}}/(\text{alg}),\) where the summation is over all partitions of the set \([1,k] = \{1,\dots , k\}\) into the disjoint union of nonempty subsets \(I_{1},\dots , I_{m}\) such that \(-d + k \leq m \leq g - d + k - \sum_{i=1}^{k}n_{i}\) (two partitions differing only by the ordering of the parts are considered to be the same); for a subset \(I = \{i_{1},\dots , i_{s} \subset [1,k]\}\) we denote \(b(I) = \frac{(n_{i_{1}} + \dots + n_{i_{s}})!}{n_{i_{1}}!\dots n_{i_{s}}!},\) \(d(I) = n_{i_{1}} + \dots + n_{i_{s}} - s + 1.\) Jacobian of a curve; Chow ring; smooth projective complex curve A. Polishchuk, Universal algebraic equivalences between tautological cycles on Jacobians of curves,Math. Z. 251 (2005), 875--897. Jacobians, Prym varieties, Theta functions and curves; Schottky problem, (Equivariant) Chow groups and rings; motives Universal algebraic equivalences between tautological cycles on Jacobians of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main purpose of this paper is to establish a formula for the Cartier operator on a plane algebraic curve in terms of a differential operator in polynomials in two variables. This formula is very useful for computations and is also of theoretical interest. To illustrate the latter we make an analysis of the ideals defining canonical curves in characteristic 2 and also give an elementary proof of a theorem of Manin on the number, modulo p, of rational points of an algebraic curve defined over a finite field of characteristic p. Cartier operator on a plane algebraic curve; computations; ideals defining canonical curves; rational points of an algebraic curve K.-O. Stöhr, A formula for the Cartier operator on plane algebraic curves, J. reine angew. Math., 377, 49, (1987) Curves in algebraic geometry, Finite ground fields in algebraic geometry, Rational points A formula for the Cartier operator on 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 \(S\) be a connected Dedekind scheme with field of rational functions \(K\) and let \(X_ K\) be a smooth and separated \(K\)-scheme of finite type. A Néron model of \(X_ K\) is a smooth separated \(S\)-scheme \(X\) of finite type with generic fibre \(X_ K\) satisfying the following universal property: for each smooth \(S\)-scheme \(Y\) and each \(K\)-morphism \(u_ K: Y_ K\to X_ K\) there exists a unique \(S\)-morphism \(u: Y\to X\) extending \(u_ k\).
This book is devoted to the construction of the Néron models and to the study of their properties. In particular, in the case of a relative curve \(X\to S\), the Néron model of the Jacobian \(J_ K\) of the generic fibre \(X_ K\) is studied and its relationship with the relative Picard functor is explained. The book contains also an ample exposition of the main tools and methods used so that, for example, chapter 8 and chapter 9 are a useful reference for questions related to the Picard functor and to its representability. Néron model of the Jacobian; relative Picard functor Bosch, S.; Lütkebohmert, W.; Raynaud, M., Néron models, (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, (1990), Springer-Verlag: Springer-Verlag Berlin), x+325 pp Schemes and morphisms, Picard schemes, higher Jacobians, Group schemes, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Picard groups Néron models | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 the seminal work of Farkas and Rauch, the modern theory of Prym varieties has been developed notably by Mumford and Beauville. The present paper is at once a very readable and instructive introduction to this theory, and an exposition of the author's main result:
The generalized Prym variety introduced by \textit{A. Beauville} [Invent. Math. 41, 149-196 (1977; Zbl 0333.14013)] is isomorphic, as principally polarized abelian variety, to a sum of Jacobians of nonsingular curves if and only if the quotient curve is of one of the following types: hyperelliptic, trigonal, quasi-trigonal, plane quintic (and the Beauville pair is odd.) - As an application new components are found in the Andreotti-Mayer variety of principally polarized abelian varieties of dimension g whose theta-divisors have singular loci of dimension \(\geq g- 4\), and a rationality criterion for conic bundles over a minimal rational surface is obtained in terms of the intermediate Jacobian. Prym varieties; sum of Jacobians of nonsingular curves; Andreotti-Mayer variety; principally polarized abelian varieties; theta-divisors; rationality; intermediate Jacobian Шокуров, В. В., Многообразия прима: теория и приложения, Изв. АН СССР. Сер. матем., 47, 4, 785-855, (1983) Picard schemes, higher Jacobians, Jacobians, Prym varieties, Theta functions and abelian varieties, Rational and unirational varieties, Families, moduli of curves (algebraic) Prym varieties: Theory and applications | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author presents a number of results on supersingular curves and abelian varieties. First he classifies \(\ell\)-adic and rational polarizations, in particular principal polarizations on a product of supersingular elliptic curves. To this end he introduces mass functions on groupoids and proves some formulas for the mass of certain sets of polarizations.
In the second part the author shows that a necessary condition for the Jacobian of a curve of genus \(g\) over a field of characteristic \(p\) to be a product of supersingular elliptic curves is 2g\(\leq p(p-1)\). He also finds a sufficient condition for this to happen (involving group representations), and gives some examples.
In the last section families of genus two curves with supersingular generic fibre are characterized. This is applied to show that the total space of a family of curves over \({\mathbb{P}}^ 1\) is of general type if its Jacobian has good reduction everywhere, the only exception being a product of two supersingular elliptic curves in characteristic 2. crystals; mass of polarizations; semistable family of curves; principal polarizations on a product of supersingular elliptic curves; Jacobian of a curve [5] T. Ekedahl, `` On supersingular curves and abelian varieties {'', \(Math. Scand.\)60 (1987), no. 2, p. 151-178. &MR 9 | &Zbl 0641.} \(p\)-adic cohomology, crystalline cohomology, Jacobians, Prym varieties, Arithmetic ground fields for abelian varieties, Arithmetic ground fields for curves On supersingular curves and 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 The author studies conic bundles X over a nonsingular surface S. Let \(A^ 2(X)\) denote the group of codimension 2 cycles which are algebraically equivalent to zero, and let J(X) be the intermediate Jacobian. The main result is that \(A^ 2(X)\) is naturally isogenous to \(A^ 2(S)\oplus A^ 1(S)\oplus P_ X\), where \(P_ X\) is the (generalized) Prym variety associated to X [this extends a result of \textit{A. Beauville}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 309-391 (1977; Zbl 0368.14018) where it is assumed that \(S={\mathbb{P}}^ 2]\). It is also shown that J(X) is isogenous to \(Alb(S)\oplus Pic^ 0(S)\oplus P_ X\). Chow group; conic bundles over a nonsingular surface; group of codimension 2 cycles; intermediate Jacobian; Prym variety M. Beltrametti,On the Chow group and the intermediate Jacobian of the conic bundle, Annali Mat. Pura Appl.,141 (1985), pp. 331--351. Parametrization (Chow and Hilbert schemes), Picard schemes, higher Jacobians On the Chow group and the intermediate Jacobian of a conic 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 These lecture notes are a very stimulating introduction to abelian integrals with modern sheaf-theoretic methods; moreover, the author succeeds in leading the reader to current research in special divisors.- The first two sections describe the classical background to abelian integrals, e.g. Abel's theorem and its converse. The classical inversion technique is carried out in detail for the lemniscate.- In the remaining chapters, abelian integrals are discussed from the point of view of the Picard variety. Before introducing this variety, the author recollects some cohomology of sheaves, e.g. finiteness theorems and Riemann-Roch. The author shows how to compute the cohomology of an invertible sheaf L on a curve C with an 'approximating homomorphism'. Roughly spoken, this means the canonical homomorphism \(h:H^ o(C,L(D)\to H^ 1(C,L(D)|_ D)\) where D is an effective divisor on C with \(H^ 1(C,L(D))=0;\) in fact, h describes the polar part of a section of L(D) at each point of the support of D and has \(H^ o(C,L)\) as kernel and \(H^ 1(C,L)\) as cokernel. A technical refinement of this procedure and a down- to-earth discussion of the cup-product is the basis for the study of the variation of cohomology in families of invertible sheaves. The author proves semi-continuity results, studies first order variations, determinantal varieties and its infinitesimal properties, which are approached using first order variational methods. In the next sections, the author constructs the Picard variety and the varieties of effective divisors following the functorial approach. Then he treats some local geometry on the Picard variety, e.g. the theorem of Riemann on the multiplicities of the theta divisor. Further, the author proves that the abelian integral morphism \(Div(C)\to Pic(C)\) is a well-presented family of projective spaces and deduces some information about the geometry on the Picard variety. Finally he reproves a result of E. Arbarello and J. Harris to the effect that the quadrics of rank \(\leq 4\) containing a canonical curve are contained in the linear span of those quadrics coming from the tangent cones to the theta divisor in its double points [cf. \textit{E. Arbarello} and \textit{J. Harris}, Compos. Math. 43, 145-179 (1981; Zbl 0494.14011), theorem 1.7]. Recently, M. L. Green has shown that those quadrics actually span the linear space of all quadrics containing the canonical curve [cf. \textit{M. L. Green}, Invent. Math. 75, 85-104 (1984)]. effective divisors; families of sheaves; abelian integrals; Picard variety; cohomology of sheaves; variation of cohomology; determinantal varieties; theta divisor Kempf, George, Abelian integrals, Monografías del Instituto de Matemáticas [Monographs of the Institute of Mathematics] 13, vii+225 pp., (1983), Universidad Nacional Autónoma de México, México Algebraic functions and function fields in algebraic geometry, Jacobians, Prym varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Picard schemes, higher Jacobians, Algebraic theory of abelian varieties, Theta functions and abelian varieties Abelian 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 In this third lecture of these proceedings the author introduces the Picard functor of a morphism \(f:X\to Y\) of schemes and discusses the usual representability criteria due to Grothendieck, Serre, Murre, Oort and others. The case of the Picard functor of a proper flat curve \(f:X\to S\), locally of finite presentation, is discussed in details: its neutral component, its unipotent part and its torical and abelian component. Picard functor; Picard scheme; relative effective divisors; flat curve Picard groups, Arithmetic ground fields for curves, Local ground fields in algebraic geometry The Picard functor | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The subject of this paper is the Picard group of the moduli variety \(M(r,c_ 1,c_ 2)\) of semi-stable algebraic sheaves on \({\mathbb{P}}_ 2({\mathbb{C}})\), of \(rank\quad r\) and Chern classes \(c_ 1, c_ 2\). The first result is that if \(M(r,c_ 1,c_ 2)\) is locally factorial, so \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to the group of linear equivalence classes of Weil divisors of \(M(r,c_ 1,c_ 2)\). There is a unique map \(\delta: {\mathbb{Q}}\to {\mathbb{Q}}\) such that \(\dim (M(r,c_ 1,c_ 2))>0\) if and only if \((c_ 2-(r-1)c^ 2_ 1/2r)/r\geq \delta (c_ 1/r)\). Then if one has equality, \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to \({\mathbb{Z}}\), and if the inequality is strict, \(Pic(M(r,c_ 1,c_ 2))\) is isomorphic to \({\mathbb{Z}}^ 2\). A description of \(Pic(M(r,c_ 1,c_ 2))\) is given, using a subgroup of the Grothendieck group \(K({\mathbb{P}}_ 2)\) of \({\mathbb{P}}_ 2\). It is then possible to compute the canonical bundle \(\omega _ M\) of \(M(r,c_ 1,c_ 2)\). Chern classes; Weil divisors; Picard group of moduli variety; factoriality of moduli variety Drezet, J.-M. , Groupe de Picard des varietés de modules de faisceaux semi-stables sur P2(C) , Ann. Inst. Fourier 38 (1988), 105-168. Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Characteristic classes and numbers in differential topology Groupe de Picard des variétés de modules de faisceaux semi-stable sur \({\mathbb{P}}_ 2({\mathbb{C}})\). (Picard group of the moduli varieties of semi- stable sheaves on \({\mathbb{P}}_ 2({\mathbb{C}}))\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 result of Faltings and Hriljac states that the Néron- Tate pairing on the Jacobian of a curve defined over a number field can be constructed using Arakelov intersection theory on the arithmetic surface obtained from the curve (at least this is true for curves which are the generic fibre of a regular arithmetic surface). On such a Jacobian (in fact more generally on abelian varieties over local fields) one also has other (\(p\)-adic) height pairings; hence the question arises whether one can interpret this in a similar way using an analogue of the classical Arakelov theory involving \(p\)-adic local intersection numbers.
This question is not new, as can e.g. be seen from work of R. Rumely and also of E. Kani. The work under review is a Ph.D.-thesis in which especially the \(p\)-adic pairing constructed by P. Schneider is considered. The main result is that essentially such an approach via Arakelov theory is possible, although as can be expected the fibers at the infinite primes cause some difficulty. It would have been interesting to see whether the aforementioned older literature, notably the Potential theory on curves, could have been used here [see \textit{E. Kani}, in Théorie des nombres, C. R. Conf. Int., Quebec 1987, 475-543 (1989; Zbl 0724.31006) and other references given there]. intersection numbers; biextensions; \(p\)-adic height pairings; Néron- Tate pairing on the Jacobian of a curve; Arakelov intersection theory Arithmetic varieties and schemes; Arakelov theory; heights, Algebraic theory of abelian varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry General Arakelov theory on arithmetical 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 classical Torelli theorem says that the Jacobian \(J(X)\) of a curve, together with the polarization given by the theta divisor, determines the curve \(X\); that is, if \(J(X)\) and \(J(X')\) are isomorphic as polarized varieties, then \(X\) is isomorphic to \(X'\). A similar result holds for \(SU_{X}^{n,\xi},\) the moduli space of stable vector bundles on \(X\) with rank \(n\) and fixed determinant \(\xi\) with \(n\) and \(\deg(\xi)\) coprime (assuming \(g > 1)\): namely, the isomorphism class of \(SU_{X}^{n,\xi}\) determines the isomorphism class of \(X\) (note that this moduli space has a unique generator of polarization). In this paper under review the authors consider the same question for the moduli space of Higgs bundles with fixed determinant of degree coprime to the rank and with traceless Higgs field. Let \(X\) be a smooth projective curve over \({\mathbb C}\) and \(M_{X}^{n,\xi}\) be the moduli space of stable Higgs bundles on \(X\) (with genus \(g > 1)\) of rank \(n\) and fixed determinant \(\xi\) with \(n\) and \(\deg(\xi)\) coprime. Let \(X'\) and \(\xi'\) be another such curve and line bundle. The authors prove that if \(M_{X}^{n,\xi}\) and \(M_{X^{'}}^{n,\xi'}\) are isomorphic as algebraic varieties, then \(X\) and \(X'\) are isomorphic. classical Torelli theorem; Jacobian of a curve with polarization; moduli space of stable Higgs bundles Biswas I., Gómez T.L.: A Torelli theorem for the moduli space of Higgs bundles on a curve. Quart. J. Math. 54, 159--169 (2003) Vector bundles on curves and their moduli, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Torelli problem, Jacobians, Prym varieties A Torelli theorem for the moduli space of Higgs bundles on a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let C be a curve of genus g. Denote by \(W\) \(r_ d(C)\subset Pic\) d(C) the scheme parametrizing line bundles of degree d on C which have at least \(r+1\) independent sections. Consider the following problem: characterize the subsets g \(s(W\) \(r_ d(C))\) of the Jacobian of C formed by those a such that \(a+W\) \(r_ d(C)\subset W\) \(s_ d(C)\). Two particular cases of this question had been previously dealt with: \((a)\quad For\quad r=s=0,\) \textit{A. Weil} showed that g \(0(W\) \(0_ d(C))=0\) provided \(\emptyset \neq W\) \(0_ d(C)\neq Pic\) d(C) [Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., II a 1957, 32-53 (1957; Zbl 0079.370)]. \((b)\quad For\quad d=g-1,\) \(r=1\), \(s=0\), \textit{G. E. Welters} proved that g \(0(W\) \(1_{g-1}(C))=C-C\) [``The surface C-C on Jacobi varieties and second order theta functions'', Acta Math. 157, 1-22 (1986)].
In this paper we consider the case \(r=s\). We find that, if C is generic in the moduli space of curves of genus \(g\) and \(\emptyset \neq W\) \(r_ d(C)\neq Pic\) d(C), then g \(r(W\) \(r_ d(C))=0\). On the other hand, for particular curves C and suitable d one may obtain that g \(r(W\) \(r_ d(C))\) may have any dimension between O and r. The cases \(r=1\) and \(d=g- 1\) and g-2 are thoroughly studied. invariance subsets of the Jacobian; moduli space; line bundles of degree d on a curve of genus g Montserrat Teixidor i Bigas, On translation invariance for \?^{\?}_{\?}, J. Reine Angew. Math. 385 (1988), 10 -- 23. Families, moduli of curves (algebraic), Jacobians, Prym varieties, Theta functions and abelian varieties On translation invariance for \(W\) \(r_ d\) | 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 There are many ways to construct curves in a Jacobian variety, whereas it is difficult to construct interesting curves in most other abelian varieties. The present paper investigates methods to construct such curves by deforming curves in Jacobians out of the Jacobian locus. First a cohomological criterion is given for a curve in a Jacobian to deform in the direction of an infinitesimal deformation of the Jacobian. Given a Jacobian \(JC\) of a curve \(C\) admitting a pencil \(g^1_d\), this criterion is applied the curve \(X= X_2(g^1_d)\) parametrizing pairs of points of the \(g^1_d\). The main result is the following theorem: Suppose \(C\) is non-hyperelliptic and \(d\geq 4\). Suppose that the curve \(X\) deforms in a direction out of the Jacobian locus. Then either \(d= 4\) or \(d= 5\), \(h^0(g^1_5)= 3\) and \(C\) is of genus \(4\) with some additional properties. If \(d= 4\), the curve \(X\) is a Prym-embedded curve and hence deforms out of the Jacobian locus. For \(d= 5\) this is not clear. In [Geom. Dedicata 115, 33--63 (2005; Zbl 1117.14045)] the author proves an analogous result for the curves parametrizing triples of points of a \(g^1_d\). abelian variety; deformation; symmetric powers of a curve E. Izadi, Deforming curves in Jacobians to non-Jacobians. I. Curves in \?\?²\?, Geom. Dedicata 116 (2005), 87 -- 109. Subvarieties of abelian varieties, Algebraic cycles, Infinitesimal methods in algebraic geometry, Jacobians, Prym varieties Deforming curves in Jacobians to non-Jacobians. I: Curves in \(C^{(2)}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a non-singular projective curve of genus \(g\) defined over an algebraically closed field of characteristic different from 2, and let \(JC\) denote its Jacobian variety.
Let \((A,D)\) be a principally polarised abelian variety of dimension \(g.\) An isogeny \(f : JC \to A\) is called a Richelot isogeny if \(2\Theta\) and \(f^*(D)\) are algebraically equivalent, where \(\Theta\) is the canonical principal polarisation of \(JC.\)
The article under review provides a necessary and sufficient condition under which there exists a so-called \textit{decomposed} Richelot isogeny of \(JC\), in the case that the genus of \(C\) is three. This condition is given in terms on the existence of a special kind of involution of \(C\), termed \textit{long}.
The arguments and the explicit decompositions split naturally into two cases, according to whether or not the curve is hyperelliptic. algebraic curve of genus 3; Jacobian variety; abelian variety; Richelot isogeny; isogeny decompositions Isogeny, Jacobians, Prym varieties, Special algebraic curves and curves of low genus Decomposed Richelot isogenies of Jacobian varieties of curves of genus 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 \(f : \mathcal{C} \to B\) be a smoothing of a stable curve \(C\) and \(S_f^*\) be the moduli space of theta characteristics on the smooth fibers of \(f\). We describe the Néron model \(N(S_f^*)\) in terms of combinatorial invariants of the dual graph of \(C\). Furthermore, we provide a modular description of \(N(S_f^*)\) and we construct an immersion \(\psi_f : N(S_f^*) \to J_{\mathcal{E}}^{\sigma}\) where \(J_{\mathcal{E}}^{\sigma}\) is a suitable relative compactified Jacobian. We show that \(\psi_f\) factors through the locus of \(J_{\mathcal{E}}^{\sigma}\) parametrizing locally free rank-\(1\) sheaves. Jacobian of a curve; spin curve; Néron model M. Pacini, On Néron models of moduli spaces of theta characteristics, J. Algebra (accepted for publication) Theta functions and curves; Schottky problem On Néron models of moduli spaces of theta characteristics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0539.00015.]
This paper gives explicit equations for the Kummer surface of a curve of genus 2 presented as \(y^ 2=f(x).\) When everything is defined over a finite field \({\mathbb{F}}_ q\) counting points on the Kummer surface leads to a bound for the number of \({\mathbb{F}}_ q\)-rational points on the Jacobian of the curve, and hence, by reduction mod p, to a bound for the possible torsion of a divisor on the curve, when this is defined over an algebraic number field. In any specific case the bound thus obtained is better than the bound obtained from the Riemann hypothesis. Some examples are given. The relevance is to computational aspects of the theory of integration in finite terms of differentials on the curve. algebraic integration; Kummer surface of a curve; rational points on the Jacobian of the curve; integration in finite terms of differentials on the curve Arithmetic ground fields for curves, Divisors, linear systems, invertible sheaves, Symbolic computation and algebraic computation, Special surfaces, Algebraic functions and function fields in algebraic geometry, Finite ground fields in algebraic geometry, History of algebraic geometry Detecting torsion divisors on curves of genus 2 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is the author's doctoral thesis. It consists of an introduction and five chapters, the first three of which can be considered as the first part, then the fourth as the second part, and the last, called ``Miscellanea'', as the third part. Almost all results are proved by explicit, and often, ingeneous manipulations with correspondences between abelian schemes or varieties.
The first part is concerned with Chow motives of abelian schemes over a fixed smooth, quasi-projective, connected base scheme \(S\) defined over an arbitrary field \(k\). It is close to a paper by \textit{C. Deninger} and \textit{J. Murre} [J. Reine Angew. Math. 422, 201-219 (1991; Zbl 0745.14003)]. A detailed exposition of the Fourier transform for such abelian schemes is given, thereby generalizing results of \textit{A. Beauville} [see Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982, Lect. Notes Math. 1016, 238-260 (1983; Zbl 0526.14001)] and of Bloch for abelian varieties. For an abelian scheme \(A\) over \(S\) with dual abelian scheme \(\hat A\), the Fourier transform \(F=F_ A\in CH^*(A\times_ S\hat A,\mathbb{Q})\), where for any smooth projective \(S\)-scheme \(X\), \(CH^*(X,\mathbb{Q})\) denotes the Chow ring (with respect to intersection product) tensored with \(\mathbb{Q}\), is defined as the relative correspondence \(F=\text{ch}(L)=\exp(\ell)=1+\ell+\ell^ 2/2!+\dots\) . Here \(\ell=c_ 1({\mathcal L})\in CH^ 1(A\times_ S\hat A,\mathbb{Q})\) is the divisor class of a Poincaré bundle \({\mathcal L}\) on \(A\times_ S\hat A\) with class \(L\in\text{Pic}(A\times_ S\hat A)\) rigidified along the zero sections as in the cited paper by \textit{C. Deninger} and \textit{J. Murre}. The correspondence \(F\) gives rise to a homomorphism \(F_{CH}:CH^*(A,\mathbb{Q})\to CH^*(\hat A,\mathbb{Q})\) by \(F_{CH}(\alpha)=F\circ\alpha\), where \(\circ\) denotes composition of correspondences. The biduality of \(A\) implies that one can define \(\hat F_{CH}:CH^*(\hat A,\mathbb{Q})\to CH^*(A,\mathbb{Q})\), with \(\hat F_{CH}\) coming from \(\hat F=F_{\hat A}={^ tF_ A}\). The behaviour of the intersection product and the Pontryagin product under the Fourier transforms \(F_{CH}\) and \(\hat F_{CH}\) remains valid for abelian schemes. Also, for any smooth projective \(S\)-scheme \(X\) and a cycle \(\alpha\in CH^*(X,\mathbb{Q})\) a Lefschetz operator \(L_ \alpha\in CH^*(X\times_ SX,\mathbb{Q})\) is defined as \(L_ \alpha=\Delta_ *(\alpha)\), where \(\Delta:X\to X\times_ SX\) denotes the diagonal morphism. In part two, this Lefschetz operator \(L_ d\) for an ample symmetric divisor \(d\) on an abelian variety will play the major role. For an abelian scheme \(A\) over \(S\) with fiber dimension \(g\) and a cycle \(\alpha\in CH^*(A,\mathbb{Q})\), the Lambda-operator \({^ c\Lambda_ \alpha}\) is defined by
\[
{^ c\Lambda_ \alpha}=(-1)^ g[{^ t\Gamma_ \sigma}]\circ \hat F\circ L_{CH(\alpha)}\circ F\in CH^*(A\times_ SA,\mathbb{Q}),
\]
where \(\sigma:A\to A\) is multiplication by \(-1\) and where \([\Gamma_ f]\in CH^*(X\times_ SY,\mathbb{Q})\) denotes the class of the graph of the morphism \(f:X\to Y\) for any smooth projective \(S\)-schemes \(X\) and \(Y\). For suitable \(\alpha\), written \(\alpha=c\), determined by the ample symmetric divisor \(d\), the \(L_ d\) and \({^ c\Lambda_ c}\), for an abelian variety \(A\), play the same role as the corresponding symbols in Kähler geometry.
The categories \({\mathcal M}(S)\) and \({\mathcal M}^ o(S)\) of relative Chow motives with respect to ungraded, resp. graded correspondences, are introduced as by \textit{C. Deninger} and \textit{J. Murre} (loc. cit.). Their basic properties such as realizations, multiplicative structures and Chow-Künneth decompositions, are recalled. Of special importance are the exterior products \(\bigwedge^ iM\), \(i\geq 0\), of the motive \(M=(X,p,m)\). They are defined by \(\bigwedge^ iM=(X^ i,s_ i\circ(p\otimes_ S\dots\otimes_ Sp),m\cdot i)\), where \(s_ i={1\over i!}\sum_{\tau\in{\mathfrak S}_ i}[{^ t\Gamma_ \tau}]\in CH^ g(X^ i\times_ SX^ i,\mathbb{Q})\) and where the \(\tau:X^ i=X\times_ S\cdots\times_ SX\to X^ i\) are permutations of the factors. According to \textit{C. Deninger} and \textit{J. Murre} (loc. cit.), for an abelian scheme \(A\) over \(S\), there is a Chow-Künneth decomposition \([\Delta]=\sum^{2g}_{i=0}\pi_ i\), canonical with respect to \(n\)- multiplication on \(A\). Here an elegant explicit form of the projectors \(\pi_ i\) is found: \(\pi_ i={1\over(2g- i)!}\log([\Gamma_{id}])^{*(2g-i)}\in CH^*(A\times_ SA,\mathbb{Q})\), where \(^*(2g-i)\) denotes the \((2g-i)\)-fold Pontryagin product on the abelian scheme \(A\times_ SA@>pr_ 1>>A\). Here the generalization of Bloch's result assures the convergence of the series for the logarithm. Using this explicit form of the \(\pi_ i\)'s and writing \(R^ 1(A/S)\) for the relative motive \((A,\pi_ 1)\) determined by the abelian scheme \(A\) over \(S\), the main result of the first part can be stated:
Let \(R(A/S)\in{\mathcal O}b({\mathcal M}^ 0(S))\) be the relative Chow motive corresponding to the abelian scheme \(A\) over \(S\), then there is a canonical functorial isomorphism \(\bigwedge^ \bullet R^ 1(A/S)@>\sim>>R(A/S)\). This isomorphism is compatible with the canonical multiplicative structures on both sides. This clarifies a result originally due to Shermenev and Manin for abelian varieties over an algebraically closed field.
The fourth chapter deals with abelian varieties \(A\) defined over a field \(k\). According to the first part or to the cited paper by \textit{C. Deninger} and \textit{J. Murre} the motive \(h(A)=R(A/\text{Spec}(k))\) of \(A\) admits a canonical Chow-Künneth decomposition \(h(A)=\bigoplus^{2g}_{i=0}h^ i(A)\). Let \(A\) be polarized by the ample divisor \(d\) and write \(v(d)\) for the Euler-Poincaré characteristic of the associated line bundle. It is assumed that \(d\in CH^ 1(h^ 2(A),\mathbb{Q})\), e.g. \(d\) is symmetric. Define the curve \(c\) on \(A\) by \(c={d^{g-1}\over v(d)(g-1)!}\in CH^{g-1}(h^{2g- 2}(A),\mathbb{Q})\). This \(c\) is to be considered as the dual of \(d\). By means of the Fourier transformation \(F_{CH}\) the duality (with intersection product vs. Pontryagin product) is given by a formula in \(CH^ p(A,\mathbb{Q})\), \(0\leq p\leq g:\) \({d^ p\over v(d)p!}={c^{*(g-p)}\over(g- p)!}\). Using the explicit form of the \(\pi_ i\)'s and the paper by \textit{A. Beauville} cited above, a somewhat tedious calculation leads to the commutator \([{^ c\Lambda_ c,L_ d}]=\sum^{2g}_{i=0}(g-i)\pi_ i\), where the \(L_ d\) is the Lefschetz operator for \(d\) and \({^ c\Lambda_ c}\) is the Lambda-operator for \(c\) as defined above. Now the \(\pi_ i\)'s, \(L_ d\) and \({^ c\Lambda_ c}\) with their relations form a so-called Lefschetz algebra, which implies, in particular, that one is able to define an `inverse' \(\Lambda_ c\) to \(L_ d\), in terms of combinations of products of the \(\pi_ i\)'s, \(L_ d\) and \({^ c\Lambda_ c}\). The final result is the hard Lefschetz theorem for Chow motives of abelian varieties: For \(i\in\{0,\dots,g\}\), the morphism \(L_ d^{g-i}:h^ i(A)\to h^{2g-i}(A)(g-i)\) is an isomorphism in \({\mathcal M}^ 0(k)\) with inverse \(\Lambda_ c^{g-i}\). One can also give a `primitive decomposition' of the motives \(h^ i(A)\) with the usual properties.
The last chapter contains some special results. First, for an abelian variety \(A\) defined over a finite field, it is shown that \(CH^ p(h^ i(A),\mathbb{Q})=0\) for \(i\neq 2p\). The proof relies on the main theorem of part one and a weight argument due to Soulé. For a suitable definition of `pure weight' for a motive, one shows that \(h^ 1(A)\) is pure of weight one and then applies the main theorem. As a corollary, using the hard Lefschetz result, one obtains Soulé's hard Lefschetz theorem for Chow groups of abelian varieties over finite fields. For an abelian variety \(A\) over an arbitrary field \(k\), \textit{A. Beauville} [loc. cit. and in Math. Ann. 273, 647-651 (1986; Zbl 0566.14003)] conjectured that \(CH^ p(A,\mathbb{Q})=0\) whenever \(i>2p\). It is shown that the truth of this conjecture implies the theorem of the hypercube, a generalization of the theorem of the cube for line bundles to cycles of higher codimension. Next, for an abelian variety \(A\) over a field \(k\), with polarization \(\varphi:A\to\hat A\) given by an ample symmetric divisor \(d\), the Fourier transformation is modified to get an endomorphism \(D\) of the motive \(h(A)\). This \(D\) is defined by \(D={1\over v(d)}[{^ t\Gamma_ \varphi}]\circ F\). This \(D\) plays the role fo the Hodge \(*\)-operator in Kähler geometry. More precisely, for a complex abelian variety \(A\), the realization of \(D\) in singular cohomology coincides, up to sign, with the Hodge \(*\)-operator. All this is demonstrated by explicit calculations. abelian variety over a finite field; Chow motives of abelian schemes; Fourier transforms; Lefschetz operator; relative Chow motive; hard Lefschetz theorem for Chow motives of abelian varieties; theorem of the hypercube Generalizations (algebraic spaces, stacks), Algebraic theory of abelian varieties, Parametrization (Chow and Hilbert schemes), Drinfel'd modules; higher-dimensional motives, etc., Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, Arithmetic ground fields for abelian varieties Chow motives of abelian schemes and Fourier transform | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 us have two integers \(p, n \geq 2\). A closed Riemann surface \(S\) is a generalized Fermat curve of type \((p, n)\) if it admits a group of conformal automorphisms \(H\approx (\mathbb{Z}/p\mathbb{Z})^n\), such that the orbifold \(S/H\) is the Riemann sphere with \(n+1\) branch points of order \(p\). In the paper under review, the authors obtain an isogenous decomposition of the Jacobian variety \(JS\) of generalized Fermat curves \(S\) of type \((p, n)\), as a product of the Jacobian varieties of certain cyclic \(p\)-gonal curves. Besides, they give explicit equations of these curves in terms of the equations of \(S\). Jacobian variety; isogenous decomposition; generalized Fermat curve Wiman, A.: Über die algebraischen Curven von den Geschlechtern \(p = 4, 5\) und \(6\), welche eindeutige transformationen in sich besitzen. Svenska Vet.-Akad. Handlingar Bihang till Handlingar \textbf{21}(I-3), 1-41 (1895) Jacobians, Prym varieties, Compact Riemann surfaces and uniformization Jacobian variety of generalized Fermat curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0725.13004. Euclidean domains; algebras of finite type over a field; diophantine geometry; integral points on curves; Euclidean algorithm; generalized Jacobian varieties Commutative Artinian rings and modules, finite-dimensional algebras, Euclidean rings and generalizations, Rational points Euclidean rings of affine curves. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be a smooth complex Fano variety (i.e. the anticanonical divisor \(- K_ X\) is ample) and consider its pseudoindex p, defined as the smallest intersection index \(-K_ X\cdot C\), when C runs among the rational curves on X. Let r stand for the index of X, i.e. for the largest integer dividing \(-K_ X\) in Pic(X).
By using Mori's theory and, in particular, a construction of a family deformation of a rational curve, the author proves that, if \(p>\dim (X)+1\), then \(Pic(X)={\mathbb{Z}}\). Since \(p\geq r\) obviously, the above result can be looked at as a refined version of a conjecture by Mukai claiming that \(Pic(X)={\mathbb{Z}}\) as soon as \(r>\dim (X)+1.\)
Furthermore, the result is shown to remain true if \(r=\dim (X)+1\) unless X is isomorphic to \({\mathbb{P}}^{r-1}\times {\mathbb{P}}^{r-1}\). In a more recent paper the author considers the case \(r=\dim (X)+\) and proves that the Picard number is still 1, apart from three exceptions (``On Fano manifolds of large index'', preprint). Fano variety; anticanonical divisor; pseudoindex; Mori's theory; family deformation of a rational curve Wiśniewski, J., On a conjecture of Mukai, Manuscripta Math., 68, 135-141, (1990) Fano varieties, Picard groups On a conjecture of Mukai | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(P_ 4 (X)\) be a quartic polynomial with complex algebraic coefficients and without multiple roots, \(C: Y^ 3= P_ 4 (X)\) the corresponding complex algebraic curve of genus 3, \(J\) the jacobian of \(C\) admitting multiplications by the ring \({\mathfrak O}\) of integers in \(\mathbb{Q} (\sqrt {-3})\). Let us choose 3 loops in \(C\), whose homology classes are \({\mathfrak O}\)-linearly independent in \(H_ 1 (C, \mathbb{Z})\) and let \(I_ 0\), \(I_ 1\), \(I_ 2\) be the period integrals of \(dX/Y\) along these loops. The author proves that the point \((I_ 0: I_ 1: I_ 2)\) of the projective plane \(\mathbb{P}^ 2\) is transcendental if and only if \(J\) is not an abelian variety of CM-type. Picard integrals; transcendental point; jacobian; abelian variety of CM- type R.-P. Holzapfel, Transcendental Ball Points of Algebraic Picard Integrals, \textit{Math. Nachr.}, \textbf{161} (1993), 7-25. Analytic theory of abelian varieties; abelian integrals and differentials, Complex multiplication and abelian varieties Transcendental ball points of algebraic Picard 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 [For the entire collection see Zbl 0661.00019.]
From the author's abstract: Let \(C=(F(x_ 0,x_ 1,x_ 2)=0)\) be an algebraic curve of degree \(n\) in \({\mathbb{P}}^ 2({\mathbb{C}})\). By a determinantal representation of C one means a matrix \(U=U(x_ 0,x_ 1,x_ 2)\) of order \(n\) whose entries are linear in x's: \(U=\sum^{2}_{i=0}x_ iA_ i \), \(A_ i\in M_ n({\mathbb{C}})\), and such that \(\det (U(x))=cF(x)\), \(c\in {\mathbb{C}}\), \(c\neq 0\). The author performs a thorough investigation of determinantal representation of a smooth irreducible curve. Using the notion of the corresponding vector bundle and its class of divisors he obtains for them a complete classification and clarifies the topological structure of the set they form. determinantal representation of a smooth irreducible curve; divisors; topological structure Vinnikov, V.: Determinantal representations of algebraic curves. Linear algebra in signals, systems, and control (1988) Curves in algebraic geometry, Topological properties in algebraic geometry, Determinantal varieties Determinantal representations 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 C be a projective irreducible smooth non-hyperelliptic curve of even genus 2n, \(n\geq 2\) over an algebraically closed field of characteristic zero which is generic in the sense of Brill-Noether. Let \(W_ d\) and \(W_ d^ 1\) be the image of the positive divisors of degree \( d\) on C and the image of special linear series \(g_ d^ 1\) under the Abel- Jacobi map, respectively. The authors show how the curve C and all its \((2n)!/(n+1)!n!\) special linear series can be recovered from the Gauss map G on \(W_ n.\)
Let B be the analytic subset of the Grassmannian \({\mathbb{G}}r(n-1,2n-1)\) consisting of the points where the fibre of G has more than one point. The authors show that there is a bijection between the components of B and \(W^ 1_{n+1}\), each one of the curves \(G^{-1}(g^ 1_{n+1})\) is birational to C and the restriction of G to \(G^{-1}(g^ 1_{n+1})\) gives the rational map associated to the linear series \(g^ 1_{n+1}\). Torelli theorem; Jacobian variety; non-hyperelliptic curve of even genus; Abel-Jacobi map; special linear series; Gauss map Jacobians, Prym varieties, Divisors, linear systems, invertible sheaves A note on curves of even genus | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Gauss, hypergeometric functions were studied by many mathematicians. Recently they are studied and generalized by \textit{K. Aomoto} [J. Fac. Sci., Univ. Tokyo, Sect. IA 22, 271-297 (1975; Zbl 0339.35021)]\ and \textit{I. M. Gel'fand}, \textit{A. V. Zelevinskij} and \textit{M. M. Kapranov} [Funkts. Anal. Appl. 23, No. 2, 94-106 (1989); translation from Funkts. Anal. Prilozh. 23, No. 2, 12-26 (1989; Zbl 0787.33012)]. These hypergeometric functions are expressed as integrals of complex powers of rational functions on varieties. They reflect a geometry of configurations of poles and zeros of these rational functions. These integrals regarded as functions on the space of configurations satisfy a regular holonomic system. In this paper, we are interested in generalized hypergeometric functions and their determinant structure.
We explain briefly the contents of this paper. First two sections are considered in the purely algebraic category. The first section is devoted to the definition of the determinants for modules with connections. In this section, we define the determinant of a connection for a finitely generated module or more generally for a perfect complex with a connection. In this section, we also define the notion of quasi- isomorphism. A quasi-isomorphism is not assumed to be compatible with connections but it is assumed to induce the isomorphism as modules with connections on their cohomologies. We prove the proper base change theorem for the determinants of modules with connections in this section. In section 2, we apply them to the geometric situation. The first main theorem is the linearity theorem (theorem 2.2.1) for the determinants of Gauss-Manin connections with indeterminate residue. By this theorem, the character of the Gauss-Manin connection comes to be a linear form on the residue of the logarithmic connection. The coefficients of this linear form satisfy an inductive relation by using the proper base change theorem. We apply the theorem of linearity to a more analytic situation in section 3. In this section we define hypergeometric functions for relative divisors. By the theorem of linearity, the determinant of hypergeometric function is expressed as a product of (analytic) Kummer characters and a function of the residue. These Kummer characters relate to the leading coefficients of the characters of the determinant of the Gauss-Manin connection obtained in theorem 2.2.1.
The last two sections are spent to obtain this Kummer character for configurations of hyperplane sections on hypersurfaces. In section 4, we construct a base of hypergeometric functions. To obtain a base, we use three basic relations for differential forms. In the last section, we treat the case where the number of the hyperplane sections is equal to the dimension of the hypersurfaces plus two. The main theorem is theorem 5.2.3. In this case the factor of the determinant which depends on the residue of the connection is expressed by Gamma functions explicitly. hypergeometric functions of hypersurfaces; modules with connections; determinant of a connection; proper base change theorem; character of the Gauss-Manin connection; logarithmic connection; hypergeometric functions for relative divisors; theorem of linearity; Kummer characters Terasoma, T.: On the determinant of Gauss-Manin connections and hypergeometric functions of hypersurfaces. Invent. Math.110, 441-471 (1992) Singularities of surfaces or higher-dimensional varieties, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Connections of hypergeometric functions with groups and algebras, and related topics, Appell, Horn and Lauricella functions On the determinant of Gauss-Manin connections and hypergeometric functions of hypersurfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The classical Abel map for smooth projective curves was extended to singular integral curves in \textit{A. Altman} and \textit{S. Kleiman} [Adv. Math. 35, No. 1, 50--112 (1980; Zbl 0427.14015)] and considered in detail in several papers [see. e.g. \textit{E. Esteves} and \textit{S. Kleiman}, Adv. Math. 198, No. 2, 484--503 (2005; Zbl 1105.14060)]. Here the authors consider non-integral curves: the stable ones. Their work is essential for the extension of the Abel map to the moduli space of stable curves. They want that Abel maps extends the definition in the smooth case related to the universal line bundle on \(X^d \times X\) and that it varies well in families. They use the coarse moduli spaces for equivalence classes of degree \(d\) semibalanced line bundles on \(X\) or the moduli space of stable genus \(g\) curves [\textit{L. Caporaso}, Am. J. Math. 130, No. 1, 1--47 (2008; Zbl 1155.14023)]. After the construction of the Abel map (which is an essential contribution of the paper) they use their construction to study it. If \(d=1\) they show that the Abel map is as close as possible to be injective and determines the few non-trivial fibers. In the case \(d \geq 2\) they say in the introduction what (at that time) was open. stable curve; moduli space of stable curves; Abel map; reducible curve; Néron model; compactified Jacobian; compactified Picard functor Melo, M., Rapagnetta, A., Viviani, F.: Fourier-Mukai and autoduality for compactified Jacobians. I. Journal für die reine und angewandte Mathematik (Crelles J.) (2017). 10.1515/crelle-2017-0009 Families, moduli of curves (algebraic), Jacobians, Prym varieties, Divisors, linear systems, invertible sheaves, Picard schemes, higher Jacobians On Abel maps of stable curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For an overview over the entire collection see Zbl 0811.14019.]
This paper is devoted to some consequences of Faltings' theorem (formerly Mordell's conjecture), mainly with a view towards arithmetic questions on the number of points of degree \(d\) on smooth irreducible algebraic curves over a number field. In a recent paper, \textit{D. Abramovich} and \textit{J. Harris} [Compos. Math. 78, 227--238 (1991; Zbl 0748.14010)] raised several questions and conjectures concerning the set of points of degree less than or equal to \(d\) in an arithmetic curve. These problems are related to the geometry of the Brill-Noether loci in the Jacobian of such a curve, in particular to the question of the existence of abelian subvarieties in the Brill-Noether loci \(W_ d(C)\), and the combination with Faltings' finiteness theorem for rational points then allows quantitative statements about the sets of points of degree \(\leq d\) in the curve \(C\). The author explains this link in detail, and gives various comments on the problems raised by Abramovich and Harris, in particular with regard to the recent progress made by \textit{O. Debarre} and \textit{R. Fahlaoui} [Compos. Math. 88, No 3, 235--249 (1993; Zbl 0808.14025)] and related results by \textit{O. Debarre} and \textit{M. J. Klassen} [J. Reine Angew. Math. 446, 81--89 (1994; Zbl 0784.14014)], \textit{A. Alzati} and \textit{G. Pirola} [Contemp. Math. 162, 1--17 (1994; Zbl 0818.14013)], and \textit{P. Vojta} [Arithmetic Algebraic Geometry, Proc. Conf., Texel 1989, Prog. Math. 89, 359--376 (1991; Zbl 0749.14018)]. rational points of bounded degree on a curve; Faltings' theorem; Mordell's conjecture; Brill-Noether loci; Jacobian Global ground fields in algebraic geometry, Curves of arbitrary genus or genus \(\ne 1\) over global fields Points of degree \(d\) on curves over number fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians ``On the moduli spaces of logarithmic connections over smooth complex projective curves'' one studies two actions: ``one given by the automorphisms of the base curve and the other given by the torsion points of suitable order of the Jacobian of the curve''.
From the authors' abstract: ``Firstly, we establish a dictionary between logarithmic orbifold connections and parabolic logarithmic connections over the quotient curve. Secondly, we prove that fixed points on the moduli space of connections under the action of finite order line bundles are exactly the push-forward of logarithmic connections on a certain unramified Galois cover of the base curve. In the coprime case, this action of finite order line bundles on the moduli space is cohomologically trivial.'' complex projective curve; Jacobian of a curve; Galois cover; logarithmic connection; moduli space of connections Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the logarithmic connections over curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth irreducible curve of genus \(g\geq 1\) and denote by \(C^{(2)}\) its second symmetric product. On \(C^{(2)}\) one can consider ``natural'' divisors: For \(P\in C\), define \(X_P=\{P+Q :Q\in C\}\). Let \(x\) and \(\delta\) be the classes of \(X_P\) and \(\Delta=\{2Q: Q\in C\}\) respectively. For a curve \(C\) with general moduli, it is known that the Néron-Severi group is generated by \(x\) and \(\delta\).
This paper studies the cone of the effective divisors on \(C^{(2)}\) in the \(x,\delta/2\)-plane, by considering a degeneration to a rational curve with \(g\) nodes, first given by Franchetta. In this way the problem is reduced to a ``Nagata type'' problem for plane curves containing preassigned singularities at general points of the plane.
In particular the authors describe the boundary of the cone of effective divisors when \(g\) is a perfect square \(\geq 9\). second symmetric product of a curve; Nagata type problem; Néron-Severi group; cone of the effective divisors; degeneration Ciliberto C., Kouvidakis A.: On the symmetric product of a curve with general moduli. Geom. Dedicata 78(3), 327--343 (1999) Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, \(n\)-folds (\(n>4\)) On the symmetric product of a curve with general moduli | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Pythagoras number of the field \(\mathbb{R}(X,Y)\) is known to be 4 (i.e., every sum of squares is a sum of at most 4 squares, and there are sums of squares that are not sums of 3 squares). Every positive semi-definite polynomial in \(\mathbb{R}[X,Y]\) is a sum of squares in the quotient field, hence is a sum of at most four squares. The authors exhibit several new families of positive semi-definite polynomials in \(\mathbb{R}[X,Y]\) that cannot be written as sums of fewer than 4 squares in \(\mathbb{R}(X,Y)\). Their polynomials are monic of degree 4 in the variable \(Y\), the coefficients are polynomials in the variable \(X\). The coefficients are used to define certain elliptic curves over the field \(\mathbb{R}(X)\). The analysis of these curves yields criteria to decide whether or not a polynomial is a sum of 3 squares. rational functions; sums of squares; Pythagoras number; elliptic curve; Jacobian variety; divisor DOI: 10.1007/s00229-004-0535-0 Forms over real fields, Sums of squares and representations by other particular quadratic forms, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Elliptic curves Sums of three squares of fractions of two variables | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0527.00002.]
Let \(f(x,y)=0\) be a germ of a plane curve. The concept of ''a distinguished deformation'' of f is introduced and its characterizations by the star-sequence and jacobian ideals are discussed. Newton polyhedron; germ of a plane curve; distinguished deformation; star-sequence; jacobian ideals Singularities of curves, local rings, Jacobians, Prym varieties, Deformations of complex singularities; vanishing cycles, Deformations of singularities Distinguished deformations of isolated singularities of 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 This paper relates two different approaches to extending families of Jacobian varieties. If \(X_{0}\) is a smooth projective curve of genus \(g,\) then the associated Jacobian variety is a \(g-\)dimensional smooth projective variety \(J_{0}\) that can be described in two different ways: as the Albanese variety and as the Picard variety. If \(X_{U} \rightarrow U\) is a family of smooth, projective curves, then the Jacobians of the fibers fit together to form a family \(J_{U} \rightarrow U.\) Let \(U\) will be an open subset of a smooth curve \(B\) (or a Dedekind scheme). The author investigates extending \(J_{U}\) to a family over \(B.\)
Viewing the Jacobian as the Picard variety is to extend \(J_{U} \rightarrow U\) as family of moduli spaces of sheaves (the approach of Mayer and Mumford). One first extends \(X_{U} \rightarrow U\) to a family of curves \(X \rightarrow B\) and then extends \(J_{U}\) to a family \(\overline{J} \rightarrow B\) with the property that the fiber over a point \(b \in B\) is a moduli space of sheaves on \(X_{b}\) parametrizing certain line bundles, together with their degenerations. In this paper, the author showed that the line bundle locus \(J\) in \(\overline{J}\) is canonically isomorphic to the Néron model for some schemes \(\overline{J}.\) The main result is
Theorem 1. Fix a Dedekind scheme \(B.\) Let \(f: X \rightarrow B\) be a family of geometrically reduced curves with regular total space \(X\) and smooth generic fiber \(X_{\eta}.\) Let \(J \subset \overline{J}\) the locus of line bundles in one of the following moduli spaces:
the Esteves compactified Jacobian \(\overline{J}^{\sigma}_{{E}};\)
the Simpson compactified Jacobian \(\overline{J}^{0}_{{L}}\) associated to an \(f\)-ample line bundle \({L}\) such that slope semistability coincides with slope stability.
The \(J\) is the Néron model of its generic fiber. projective curve; Jacobian varieties; Albanese variety and Picard variety; Néron model Chiodo, A.: Néron models of \(Pic^0\) via \(Pic^0\). arXiv:1509.06483 (Preprint) Jacobians, Prym varieties, Group schemes, Fine and coarse moduli spaces, Families, moduli of curves (algebraic) Two ways to degenerate the Jacobian are the same | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Jacobian problem asks whether a polynomial endomorphism of complex affine \(n\)-space with non-vanishing Jacobian determinant is an isomorphism. Such a morphism is étale and surjective modulo codimension two. The generalized Jacobian problem asks whether an étale morphism from a simply connected complex variety of dimension \(n\) to complex \(n\)- space which is surjective modulo codimension two is an isomorphism. If the generalized problem had an affirmative answer, so would the original one. In this paper, the author constructs a counterexample to the generalized problem. His method is to show the equivalence between the generalized problem and a condition on fundamental groups of complex affine plane curve complements, namely whether such can have proper subgroups of finite index generated by geometric generators. He shows that if \(\overline D\) is a degree 4 projective plane curve with three cuspidal singularities and \(L\) is a projective line transversally intersecting \(\overline D\) in four points then the fundamental group of the plane curve \(D-L\) has a geometrically generated proper finite index subgroup.
He also considers the related local question, namely whether the fundamental group of the complement of a germ of an analytic curve in a two dimensional complex ball can have a proper finite index subgroup generated by geometric generators, and shows that this is equivalent to the question of whether such a ball can be the image by the germ of an étale surjective holomorphic map of degree more than one from a simply connected analytic surface. He shows that this second question has a negative answer, which thus settles the first negatively in the analytic case also.
The paper also establishes two additional equivalent formulations of the Jacobian problem: first, that injective Lie endomorphisms of the set of derivations of the polynomial ring are automorphisms, and the second in terms of properties of the differential equations associated to sets of certain derivations of complex \(n+1\) space. generalized Jacobian problem; étale morphism; germ of an analytic curve Kulikov, V.S.: Generalized and local Jacobian problems, (Russian) Izv. Ross. Akad. Nauk Ser. Mat. \textbf{56}(5), 1086-1103 (1992); translation in Russian Acad. Sci. Izv. Math. \textbf{41}(2), 351-365 (1993) Automorphisms of curves, Polynomial rings and ideals; rings of integer-valued polynomials, Germs of analytic sets, local parametrization Generalized and local Jacobian problems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let C be a smooth irreducible projective curve of genus g (over an algebraically closed field of characteristic \(\neq 2)\) and let \(\pi: \tilde C\to C\) be an irreducible, étale double cover of C. Associated to \(\pi\) we have the norm map \(Nm: Pic^{2g-2}(\tilde C)\to Pic^{2g- 2}(C).\) If \(\omega_ C\) denotes the canonical bundle of C, one can consider for any integer r the subvarieties \(V^ r\) of \(Nm^{- 1}(\omega_ C)\) formed by all line bundles L such that \(h^ 0(\tilde C,L)\geq r+1\), \(h^ 0(\tilde C,L)\equiv r+1\quad (mod 2).\) A known fact about these varieties \(V^ r\) is that \(\dim (V^ r)\geq g-1-r(r+1)/2\) if \(V^ r\neq \emptyset\). A recent theorem by \textit{G. E. Welters} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 671-683 (1985; Zbl 0628.14036)] states that \(V^ r\), if not empty, has exactly the expected dimension \(g-1-r(r+1)/2\) and is smooth away from \(V^{r+2}\) if C is a general curve of genus \(g\) and \(\pi: \tilde C\to C\) is any irreducible, étale double covering. - Moreover Welters also conjectures that an existence result of Kempf's and Kleiman-Laksov's type should hold: namely that \(V^ r\neq \emptyset\) as soon as \(g-1-r(r+1)/2\geq 0\) for any curve C of genus \(g\) and any covering \(\pi: \tilde C\to C\) (in fact he proves a weaker form of this conjecture, namely that \(V^ r\neq \emptyset\) when \(g\geq (r+1)^ 2+1).\)
The present note answers Welter's conjecture in the affirmative. The (easy and elegant) proof is based on the results on excess linear series on an algebraic curve contained in a paper by \textit{W. Fulton}, \textit{J. Harris} and \textit{R. Lazarsfeld} [Proc. Am. Math. Soc. 92, 320-322 (1984; Zbl 0549.14004)]. theta-divisor of a Prym variety; norm map; linear series on an algebraic curve A. Bertram,An existence theorem for Prym special divisors, Invent. Math.90 (1987), 669--671. Divisors, linear systems, invertible sheaves, Arithmetic ground fields for curves An existence theorem for Prym special divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\phi : X \rightarrow Y\) be a spectral curve; the first aim in this paper is the study of the correspondence between isomorphism classes of torsion free sheaves of rank 1 on \(X\) and classes of pairs \((E,u)\), where \(E\) is a locally free sheaf on \(Y\) whose rank is \(r=\text{ deg}\phi\) and \(u:E \rightarrow E\otimes {\mathcal L}\) is a homomorphism whose characteristic polynomial defines \(X\), while \({\mathcal L}\) is the invertible sheaf on \(Y\) associated to the spectral curve.
It was known that such correspondence is bijective when \(X\) is integral; the first main result in the paper is to show that this is true for any spectral curve.
A criterion for stability is given for the pairs \((E,u)\), and it is used to study the morphism \(H: M(r,d,{\mathcal L}) \rightarrow \bigoplus _{i=0}^r H^0(Y,{\mathcal L^{\otimes^i}})\), where \(M(r,d,{\mathcal L})\) is the (coarse) moduli space for pairs \((E,u)\). What is shown is that \(\text{Pic}^0(X)\) is open in a fiber \(H^{-1}(P)\), thus proving that \(H^{-1}(P)\) can be viewed as a compactification of the generalized Jacobian of \(X\). spectral curve; stability; torsion free sheaves; compactification of the generalized Jacobian Daniel Schaub, ''Courbes spectrales et compactifications de jacobiennes'', Math. Z.227 (1998) no. 2, p. 295-312 | Vector bundles on curves and their moduli, Jacobians, Prym varieties, Families, moduli of curves (algebraic) Spectral curves and compactification of Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a reduced, irreducible, non-singular, projective curve of genus \(g\). Assume \(\chi: X\to P^{g-1}\) be the canonical imbedding and \(C=\chi (X)\). Let \(Q\) denote the subscheme of \(P^{g-1}\) consisting of the intersection of all quadrics through \(C\). The aim of this paper is the description of \(Q\) in connection with some old ideas of \textit{M. Noether} [Math. Ann. 17, 263--284 (1880; JFM 12.0323.02)] and \textit{F. Enriques} [Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. Vol. III (Italian) Bologna: Zanichelli (1924; JFM 51.0508.03), 97-109]. One of main results is:
Theorem 1:
(a) \(Q\) is a projective variety in \(P^{g-1}\), reduced and irreducible and \(\dim Q\) is \(1\) or \(2\);
(b) if \(\dim Q=1\) then \(Q=C\);
(c) if \(\dim Q= 2\) and \(g\neq 4\) then: \(Q\) is a smooth surface of degree \(g-2\).
Suppose in addition: (i) \(Q= P^2\); if \(g=3\) then \(P^{g-1}=Q\), \(\mathcal O_Q(C) =\mathcal O_{P^2}(4)\); if \(g=6\) then \(\mathcal O_Q(1) = \mathcal O_{P^2}(2)\), \(\mathcal O_Q(C)=\mathcal O_{P^2} (5)\) (this means that \(C\) appears as a curve of degree 5 in \(P^2\) and \(Q\) is the Veronese image of \(P^2)\).
These are the only possibilities because if \(g\neq 3,6\), then \(Q\neq P^2\); (ii) \(Q=F_n\), where \(n\) satisfies:
\[
0\leq n\leq \min \left\{\frac{g+2}{3}, g-4\right\},\quad n\equiv g\pmod 2,
\]
then
\[
\mathcal O_Q(1) = \mathcal O_{F_n} \left(b_n + \frac{g+n-2}{2}s_n\right), \quad \mathcal O_Q(C)= \mathcal O_Q(C)\otimes \Omega_Q^{-1}= \mathcal O_{F_n}\left(3b_n + \frac{g+3n+2}{2}s_n\right)
\]
[here the rational ruled surfaces \(F_n\), and \(b_n\) and \(s_n\) are defined as by \textit{M. Nagata} in [Mem. Coll. Sci., Univ. Kyoto, Ser. A 33, 271--293 (1960; Zbl 0100.16801)];
(d) if \(g=4\) then \(Q\) is \(F_0\) or a cone which has as basis a nonsingular curve of degree \(2\).
A curve \(X\) such that \(\dim Q=2\) is called special. The author gives a criterium of speciality (theorem 2) and an existence theorem for special curves of any \(g\). There are partial results -- nice by themselves -- as to be pointed out: the computation of the cohomology of \( I(n)\) \((I\) being the ideal of \(C\) in \(P^{g-1})\) and the following description of an intersection of quadrics: ``If \(M\) is a scheme obtained as an intersection of \((g-2)(g-3)/2\) independent quadrics in \(P^{g-2}\) and \(M\) contains \(2g-2\) closed points in general position, then \(M_{\text{red}} = M\) and moreover, if \(\#(M) > 2g-2\), then \(M\) is just an irreducible non-singular curve of genus \(0\) and degree \(g-2\).'' intersection of quadrics passing through a canonical curve; reduced variety V. V. Shokurov, ?The Nöther-Enriques theorem on canonical curves,? Mat. Sb.,86, No. 3, 367?408 (1971). Families, moduli of curves (algebraic) The Noether-Enriques theorem on 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 For a positive integer \(N\), let \(\mathscr{C}(N)\) be the subgroup of \(J_0(N)\) generated by the equivalence classes of cuspidal divisors of degree 0 and \(\mathscr{C}(N)(\mathbb{Q}) : = \mathscr{C}(N) \cap J_0(N)(\mathbb{Q})\) be its \(\mathbb{Q} \)-rational subgroup. Let also \(\mathscr{C}_{\mathbb{Q}}(N)\) be the subgroup of \(\mathscr{C}(N)(\mathbb{Q})\) generated by \(\mathbb{Q} \)-rational cuspidal divisors. We prove that when \(N = n^2 M\) for some integer \(n\) dividing 24 and some squarefree integer \(M\), the two groups \(\mathscr{C}(N)(\mathbb{Q})\) and \(\mathscr{C}_{\mathbb{Q}}(N)\) are equal. To achieve this, we show that all modular units on \(X_0(N)\) on such \(N\) are products of functions of the form \(\eta(m \tau + k / h), m h^2 | N\) and \(k \in \mathbb{Z}\) and determine the necessary and sufficient conditions for products of such functions to be modular units on \(X_0(N)\). modular units; Jacobian of a modular curve; Dedekind eta function Arithmetic aspects of modular and Shimura varieties, Elliptic and modular units, Modular and Shimura varieties, Rational points Modular units and cuspidal divisor classes on \(X_0(n^2M)\) with \(n|24\) and \(M\) squarefree | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\pi: \widetilde {C}\to C\) be an étale double covering of a smooth curve \(C\) of genus \(g+1\). The Prym variety \(P(\widetilde {C},C)\) associated to this covering is the connected component containing 0 of the kernel of the norm map \(\text{Nm}_ \pi: J(\widetilde {C})\to J(C)\). \(P(\widetilde {C}, C)\) is a principally polarized abelian variety of dimension \(g\) and the above construction defines the Prym map
\[
P_{g+1}: {\mathcal R}_{g+1}=\{ \text{pairs \((\widetilde {C}, C)\) as above}\}\to {\mathcal A}_ g,
\]
where \({\mathcal A}_ g\) denotes the moduli space of principally polarized abelian varieties of dimension \(g\). The purpose of the paper under review is to give a lower bound for the genus of curves in \(P(\widetilde {C}, C)\). To be more precise, it is shown that for a generic Prym variety \(P\) of dimension \(g=2\) or \(\geq 4\) there is no nonconstant map \(D\to P\) from a smooth curve \(D\) of genus \(<2g -2\) into \(P\). The idea of the proof is to make use of a rigidity theorem [\textit{G. P. Pirola}, J. Reine Angew. Math. 431, 75-89 (1992; Zbl 0753.14040)] stating that under the above hypotheses for a generic \(P\) the only deformations of nonconstant maps \(D\to P\) are obtained by translations. étale double covering of a smooth curve; Prym map; moduli space of principally polarized abelian varieties; generic Prym variety Naranjo, J.C., Pirola, G.P.: On the genus of curves in the generic Prym variety. Indag. Math. (N.S.) \textbf{5}, 101-105 (1994) Algebraic moduli of abelian varieties, classification, Coverings of curves, fundamental group, Picard schemes, higher Jacobians On the genus of curves in the generic Prym variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(U(r,d)\) be the moduli space of semistable vector bundles of rank 2 and degree \(d\) on a smooth curve \(C\). In this paper the authors describe the groups of automorphisms and of polarized automorphisms of \(U(r,d)\) for \(r > 1\). Using the natural morphism \(\text{det} : U(r,d) \to J^d (C)\), sending a bundle \(E\) to its determinant \(\text{det} (E)\), one sees that the fiber \(\text{det}^{-1} (L_0)\) over a point \(L_0\) is just the moduli space \({\mathcal L} U(r,L_0)\) of semistable bundles of rank \(r\) and fixed determinant \(L_0\). Taking into consideration the isomorphism \(J^d(C) = U(1,d)\) and denoting by \(J^0 (C)[r]\) the group of \(r\)-torsion points one can formulate main result of paper:
If \(C\) is a curve without automorphisms, then the automorphism group of the moduli space \({\mathcal L} U(r, L_0)\) can be described as follows:
1. If \(r \nmid 2d\), then the natural map \(J^0 (C)[r] \to \Aut ({\mathcal L} U(r, L_0))\) is an isomorphism, and
2. If \(r \mid 2d\), then the natural map \(J^0 (C)[r] \rtimes \mathbb{Z}/2 \mathbb{Z} \to \Aut ({\mathcal L} U(r, L_0))\) is an isomorphism for \(r \geq 3\) and has kernel \(\mathbb{Z}/2 \mathbb{Z}\) for \(r = 2\). Here \(\rtimes\) is the semidirect product. Jacobian; determinantal variety; curve without automorphisms; automorphism group of the moduli space Kouvidakis, A., and Pantev, T., \textit{The automorphism group of the moduli space of semistable}\textit{vector bundles}, Math. Ann. 302 (1995), no. 2, 225--268. Vector bundles on curves and their moduli, Automorphisms of curves, Algebraic moduli problems, moduli of vector bundles The automorphism group of the moduli space of semi stable vector bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is an expository article containing notes for an Algebraic Geometry Summer School held at Bilkent University (Ankara, Turkey) during August, 1995. The author discusses the classical theorem of Abel on the fibers of the morphism \(C^{(n)}\to J(C)\) obtained by integrating a basis of regular differentials, where \(C^{(n)}\) is the \(n\)-th symmetric product of a complex, smooth, projective algebraic curve \(C\) and \(J(C)\) is its jacobian, as well as the theorem of Riemann (generalized by George Kempf) on the singularities of the image \({\mathcal W}_n\) of this morphism (when the genus of \(C\) is \(\geq 2)\).
Beginning with very basic material on holomorphic functions the author presents a rather complete treatment of Abel's theorem and explains the nature of that of Riemann-Kempf, including a sketch of a proof of the latter (due to Kempf), which uses deformation theory. To a considerable extent the discussion is self-contained. This is a good introduction to this subject. Riemann surface; periods; symmetric product of a complex smooth projective algebraic curve; Jacobian; Abel's theorem Riemann surfaces; Weierstrass points; gap sequences, Analytic theory of abelian varieties; abelian integrals and differentials, Jacobians, Prym varieties, Differentials on Riemann surfaces, Singularities of curves, local rings Integration of algebraic functions and the Riemann-Kempf singularity theorem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper dedicated to the memory of Victor Enolski, is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus \(2\). It contains new results, as well as a derivation from them of well-known results on these issues. The research is motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus \(2\). The authors consider a hyperelliptic curve \(V\) of genus \(2\) which admits a morphism of degree \(2\) to an elliptic curve. Then there exist two elliptic curves \(E_i\), \(i=1,2\), and morphisms of degree \(2\) from \(V\) to \(E_i\). They construct hyperelliptic functions associated with \(V\) from the Weierstrass elliptic functions associated with \(E_i\) and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. The authors show that the restrictions of hyperelliptic functions associated with \(V\) to the appropriate subspaces in \(\mathbb{C}^2\) are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with \(E_i\). Further, they express the hyperelliptic functions associated with \(V\) on \(\mathbb{C}^2\) in terms of the Weierstrass elliptic functions associated with \(E_i\). They derive these results by describing the homomorphisms between the Jacobian varieties of the curves \(V\) and \(E_i\) induced by the morphisms from \(V\) to \(E_i\) explicitly. This paper is organized as follows: Section 1 is an introduction to the subject. In Section 2 the authors recall the definitions of the Jacobi elliptic functions and the sigma functions for the curves of genus \(1\) and \(2\) and give facts about them. Sections 3 deals with the curve of genus \(2\) and the elliptic curves. Section 4 is devoted to the map from \(Jac(V)\) to \(Jac(E_i)\) and Section 5 to the map from \(Jac(E_i)\) to \(Jac(V)\). Section 6 deals with a comparison of the results obtained in this paper and the results of [\textit{E. D. Belokolos} and \textit{V. Z. Enolskiĭ}, J. Math. Sci., New York 106, No. 6, 3395--3486 (2001; Zbl 1059.14044); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 71 (2000); \textit{V. M. Buchstaber} et al., Multi-dimensional Sigma-functions'', Preprint, \url{arXiv:1208.0990}; \textit{V. Z. Enolskij} and \textit{M. Salerno}, J. Phys. A, Math. Gen. 29, No. 17, L\, 425-L\, 431 (1996; Zbl 0903.35070)]. hyperelliptic function; elliptic function; sigma function; reduction of hyperelliptic functions; Jacobian variety of an algebraic curve Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Theta functions and abelian varieties, Meromorphic functions of several complex variables, Elliptic functions and integrals Relationships between hyperelliptic functions of genus 2 and elliptic functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The general problem underlying the present paper is the determination of the Brauer group \(\text{Br}(X)\) of a smooth connected projective variety \(X\) defined over a field \(k\) of characteristic \(0\). This group can be identified in a natural way with the unramified Brauer group \(\text{Br}_{ur}(K/k)\), where \(K=k(X)\) denotes the function field of \(X\) over \(k\). The setting in this paper is the following : \(k\) is a finite extension of the \(p\)-adic numbers \({\mathbb Q}_p\), and \(X\) is a smooth projective model of an affine curve \(C\) defined by an equation \(y^2=f(x)\) with \(f(x)\) a polynomial of degree \(5\) without multiple roots. The aim is to obtain a presentation of generators of \(_2\text{Br}(X)\) (the exponent-2-part of \(\text{Br}(X)\)) in terms of quaternion algebras. In the present paper, this is achieved in the case where \(f\) factors completely over \(k\). The results are very explicit but rather technical.
The case of curves of genus zero is well known, and that of elliptic curves has been treated previously by the first and the last author. These results plus further ones in the case of hyperelliptic curves which are either quintics or possess good reduction can be found in a survey paper by the authors [\textit{V. Yanchevskij}, \textit{G. Margolin} and \textit{U. Rehmann}, ``Brauer groups of curves and unramified central simple algebras over their function fields'', J. Math. Sci., New York 102, 4071-4134 (2000; Zbl 0979.14010)]. Brauer group of a variety; unramified Brauer group; local field; hyperelliptic curve; quaternion algebras Margolin G. L., Quaternion Generation of the 2-torsion Part of the Brauer Group of a Local Quintic Brauer groups of schemes, Elliptic curves, Brauer groups (algebraic aspects), Curves over finite and local fields Generation of \(2\)-torsion part of Brauer group of local quintic by quaternion algebras, the totally splitting 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 Let \(X\) be a proper and integral curve over an algebraically closed field \(k\). The moduli space of line bundles of degree zero on \(X\) is then, in a natural way, a group scheme over \(k\), called the generalized Jacobian \(P(X)\) of \(X\).
If \(X\) is a smooth curve, then \(P(X)\) of \(X\) is just the usual Jacobian of \(X\), i.e., an abelian variety. However, if \(X\) is singular, then \(P(X)\) is only a non-proper group scheme, and the question of whether there is a natural ``compactification'' of \(P(X)\), that is a proper scheme \(\overline P(X)\) containing \(P(X)\) as an open subset and on which \(P(X)\) acts as a group scheme, arises quite inevitably. In fact, such a (so-called) compactified Jacobian can be constructed, namely by considering the functor of families of torsion-free sheaves of rank one on \(X\). This idea of construction goes back to \textit{A. Grothendieck} [cf. ``Fondements de la géométrie algébrique,'' Extraits du Séminaire Bourbaki 1957-1962 (Paris 1962; Zbl 0239.14002)]. Explicit constructions have been carried out, in the sequel, by A. Mayer and D. Mumford (1963), C. D'Souza (1973), T. Oda and C. S. Seshadri (1974), A. Altman, A. Iarrobino and S. Kleiman (1976), C. J. Rego (1980), and others. The fundamental paper of \textit{A. B. Altman} and \textit{S. L. Kleiman} ``Compactifying the Picard scheme'' [Adv. Math. 35, 50-112 (1980; Zbl 0427.14015)] gives the perhaps most general and complete account, with generalizations to higher dimensional varieties \(X\). Finally, the various attempts of constructing compactified Jacobians of curves have led to the now well-established theory of moduli spaces of semistable vector bundles over a curve.
The paper under review deals with the projectivity of the compactified Jacobian \(\overline P (X)\). Using the relative approach by A. Altman and S. Kleiman, the author constructs a Cartier divisor \(\Theta\) on \(\overline P^{g - 1} (X)\), the reduced \((g - 1)\)-component of the compactified Picard scheme \(\overline {\text{Pic}} (X/k)\) with respect to the arithmetic genus \(g\) of \(X\), and proves that \(\Theta\) is ample. -- The proof is given in such a way that it can be generalized to the relative case, i.e., to morphisms \(f : X \to S\) of finite type, flat and projective, whose fibers are integral curves of arithmetic genus \(g\). In this situation, the author's construction leads to a sheaf \({\mathcal D}\) on \(\overline {\text{Pic}}^{g - 1} (X/S)\) which is \(S\)-ample. ampleness; polarization; generalized Jacobian; projectivity of the compactified Jacobian; compactified Picard scheme 15. Soucaris, A.: The ampleness of the theta divisor on the compactified Jacobian of a proper and integral curve, Compositio Math., 93 (1994), 231--242 Jacobians, Prym varieties, Picard schemes, higher Jacobians, Algebraic moduli problems, moduli of vector bundles The ampleness of the theta divisor on the compactified Jacobian of proper and integral curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth, proper and irreducible curve of genus \(g \geq 2\) over an algebraically closed field of characteristic zero. A torsion packet on \(X\) is an equivalence class under the relation on \(X\) by defining \(P \sim Q\) if and only if the divisors \(mP\) and \(mQ\) are linearly equivalent for some positive integer \(m\). In other words, the equivalence class containing \(P \in X\) is the preimage of the torsion points of the Jacobian of \(X\), \(J(X) \), by the Abel map \(X\to J(X)\) given by the point \(P\). The Manin-Mumford conjecture, proved by \textit{M. Raynaud} [Invent. Math. 71, 207-233 (1983; Zbl 0564.14020)], states that the every torsion packet on \(X\) is finite. The main result of this article is concerned with the number and size of torsion packets on a curve. The authors prove that there are at most finitely many torsion packets of size greater than 2 on \(X\), and there are infinitely many non-trivial torsion packets on \(X\) if and only if either \(g=2\) or \(g=3\) and \(X\) is both hyperelliptic and bielliptic. Moreover, they deduce that there is a constant \(M\), depending on \(X\), such that every torsion packet on \(X\) has size at most \(M\). The authors ask the open questions: Does the above constant \(M\) depend only on the genus of \(X\)? Does there exist a constant \(M(g,s)\) for fixed numbers \(g\geq 2\) and \(s\geq 3\) such that for all curves \(X\) of genus \(g\), the number of torsion packets on \(X\) of size at least \(s\) is bounded by \(M(g,s)\)? In particular, they show that for every \(n\geq 1\), there exists a curve \(X\) of genus \(g\geq 2\) such that \(X\) has at least \(n\) torsion packets each of size at least \(n\). torsion points of the Jacobian; Manin-Mumford conjecture; torsion packets on a curve Baker, M; Poonen, B., Torsion packets on curves. Compositio Math. 127 (2001), 109-116. Zbl0987.14020 MR1832989 Jacobians, Prym varieties, Arithmetic ground fields for curves, Curves of arbitrary genus or genus \(\ne 1\) over global fields Torsion packets on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper is devoted to thorough analysis of algebraically integrable Hamiltonian systems (i.e., to complex integrable systems for which the compactified and desingularized joint level sets of the Hamiltonians are Abelian varieties with the linear structures given by the Hamiltonian [cf. \textit{P. van Moerbeke}, Proc. Symp. Pure Math. 49, Pt. 1, 107-131 (1989; Zbl 0688.70012)]), especially to the particular class of examples determined by some natural systems on the coadjoint orbits of a loop algebra of polynomials in one variable \(\lambda\) with values in a finite-dimensional semisimple Lie algebra [cf., e.g., \textit{M. Adler} and \textit{P. van Moerbeke}, Adv. Math. 38, 267-317 (1980; Zbl 0455.58017) and ibid., 318-379 (1980; Zbl 0455.58010)]. The reduced coadjoint orbit has a description as an open set of the union in the family of Jacobians, corresponding to a family of curves in the \((z,\lambda)\)-plane, and its symplectic geometry is related to that of the plane.
The author deals with the following problems: Which integrable systems fit into the loop algebra framework, if not, are there other algebraic surfaces which can be invariantly associated to them, and classification of these surfaces. We cannot refer the deep results here. In rough terms, the vanishing of a certain invariant of the integrable system ensures a surface \(Q\) generalising the \((z,\lambda)\)-plane, in the loop case this \(Q\) is rational, and the classification is made according to whether certain curves \(S_h\) in \(Q\) (determined by the Lagrangian fibration of the integrable system) are hyperelliptic or not. Several examples related to recent work by M. R. Adams, J. Harnad, the author (coadjoint orbits), N.J. Hitchin (moduli spaces of stable \({\mathfrak {gl}}(r)\) pairs), E. K. Sklyanin (unusual brackets), S. Makai (\(K-3\) surfaces), and the genus two case are discussed.
[The erratum concerns Proposition 3.5 of the paper]. Jacobian of a curve; algebraically integrable Hamiltonian systems; coadjoint orbit; loop algebra Hurtubise, J.: Integrable systems and algebraic surfaces. Duke Math. J. 83(1), 19--50 (1996) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Curves in algebraic geometry, Hamilton's equations Integrable systems and algebraic surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a projective smooth curve with a group \(G\) of automorphisms, \(J\) its Jacobian and \(\Theta\) the theta divisor. Here the author computes the trace of any \(h\in G\) on the \(G\)-representation \(H^0(J,\mathcal {O}(n\Theta ))\) when \(X\) is either the Klein quartic or the Macbeath curve of genus \(7\) or the Bring curve of genus \(4\). He also announces that he has a Maple program for doing similar computations. automorphism of a curve; Jacobian; theta divisor I. Moreno Mejía,The trace of an automorphism on \(H^0 \left( {J,\mathcal{O}\left( {n\Theta } \right)} \right)\) , Michigan Mathematical Journal53 (2005), 57--69. Automorphisms of curves, Jacobians, Prym varieties, Special algebraic curves and curves of low genus, Plane and space curves, Group actions on varieties or schemes (quotients), Geometric invariant theory, Index theory and related fixed-point theorems on manifolds The trace of an automorphism on \(H^0(J,\mathcal O(n \Theta))\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 theta constants of a Jacobian of a curve satisfy all of the Schottky- Jung relations. Let (A,\(\theta)\) be any principally polarized abelian variety of dimension \(g,\) with \(\theta\) symmetric, which satisfies all S- J relations. Let X be a quartic hypersurface which contains the Kummer variety of A, K(A)\(\subset | 2\theta |\), and which is invariant with respect to the action of the Heisenberg group. We show: K(A)\(\subset Sing(X)\) iff K(P)\(\subset X\) for every Prym P of A. Here the Prym's are the ppav's of dimension g-1 given by the Schottky-Jung relations and which are, intrinsically, subvarieties of \(| 2\theta |.\)
For every curve C one can map \(SU_ C(2)\), the moduli space of rank 2 bundles on C with trivial determinant, to \(| 2\theta |\). For C hyperelliptic we show that this map embeds \(SU_ C(2)/i\), where i is the hyperelliptic involution, and that the image is defined, at least as a set, by quartic polynomials. The intersection of the quadrics which contains K(Jac C) is shown to be isomorphic to the Grassmannian of all linear \({\mathbb{P}}^ g\subset Q\) of a fixed ruling of a smooth quadric \(Q\subset {\mathbb{P}}^{2g+1}\). theta constants of a Jacobian of a curve; Schottky-Jung relations; moduli space of rank 2 bundles van Geemen, B.: Schottky-Jung relations and vector bundles on hyperelliptic curves. Math. Ann.281, 431--449 (1988) Theta functions and abelian varieties, Algebraic moduli of abelian varieties, classification, Families, moduli of curves (algebraic), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Schottky-Jung relations and vectorbundles on hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian of a genus g hyperelliptic curve outside a theta divisor is explicitly isomorphic to an affine subvariety of \({\mathbb{C}}^{3g+1}\) [\textit{C. G. T. Jacobi}, J. Reine Angew. Math. 32, 220-226 (1846); cf. \textit{D. Mumford}, ``Tata lectures on theta. II: Jacobian theta functions and differential equations'', Prog. Math. 43 (1984; Zbl 0549.14014)]. This paper presents an analogous construction for general curves. The construction uses the interpretation of a generic line-bundle as a commutative ring of differential operators and is thus suited to a solution of the KP equation. KP flows; Jacobian of a genus g hyperelliptic curve; theta divisor; differential operators; KP equation Previato, E.: Generalized Weierstrass \wp-functions and KP flows in affine space. Comment. math. Helvetici 62, 292-310 (1987) Theta functions and abelian varieties, Partial differential equations of mathematical physics and other areas of application, Dynamics induced by flows and semiflows, Jacobians, Prym varieties, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Generalized Weierstrass \(\wp\)-functions and KP flows in affine 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 Let X be a nonsingular projective algebraic variety, D an ample subvariety and \({\mathcal F}(X,D)\) the sheaf on the Picard variety of X defined as a Fourier transform of the sheaf \({\mathcal O}_ X(_*D)\), with the universal family of flat line bundles on X as a kernel. \({\mathcal F}(X,D)\) is a Baker-Akhiezer module (i.e. a BA-module).
The author studies the BA-module of a principally polarized Abelian variety. Picard variety; Fourier transform; Baker-Akhiezer module; BA-module of a principally polarized Abelian variety Nakayashiki A.: Structure of Baker-Akhiezer modules of principally polarized abelian varieties, commuting partial differential operators and associated integrable systems. Duke Math. J. 62(2), 315--358 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules Structure of Baker-Akhiezer modules of principally polarized abelian varieties, commuting partial differential operators and associated integrable systems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author introduces a notion of relative deformation for a morphism \(f:Y \to X\) between smooth algebraic varieties, and he applies it to study the case where the fibers of \(f\) are Fano manifolds. -- Let \(k\) be a field, \(f : Y \to X\) and \(\pi : Z \to X\) \(k\)-morphisms between smooth \(k\)-varieties, and \((S,o)\) a connected punctured \(k\)-scheme. An \(S\)- morphism \(\widetilde f : Y \times_ kS \to X \times_ kS\), or a family \(\widetilde f = \{f_ s : Y \to X\}_{s \in S}\), is a ``relative deformation'' of \(f\) over \(Z\) with base subscheme \(B\subset Y\), if the following three conditions are satisfied:
\[
\widetilde f_{| Y \times \{0\}} = f,\quad \widetilde f_{| B \times S} = (f_{| B}, \text{pr}_ S) \quad \text{and} \quad (\pi, \text{pr}_ S) \circ \widetilde f = (\pi \circ f, \text{pr}_ S).
\]
\vskip3.mm We denote by \(\text{Deform}_ Z (S,o;f,B)\) the set of relative deformations of \(f\) over \(Z\) with base \(B\), and we define a contravariant functor from the category of punctured \(k\)-schemes to the category of sets: \(\text{Deform}_ Z(*;f,B) : (S,o) \mapsto \text{Deform}_ Z (S,o; f,B)\). We denote by \(\text{Sing} (\pi)\) the singular locus of \(\pi\), \(f^ \# T_{X/Z}\) the kernel of \(f^*d \pi : f^*T_ X \to f^* \pi^* T_ Z\).
Theorem 1. Assume that \(X\) and \(Y\) are both projective and that \(\pi\) is surjective.
(1) The functor \(\text{Deform}_ Z (*;f,B)\) is representable by a quasi-projective \(k\)-scheme \(D_ Z(f,B)\), the universal \(Z\)-deformation of \(f\) with base \(B\).
(2) If \(\dim_ k (Y) = 1\) and \(f(Y)\) is not contained in \(\text{Sing} (\pi)\), then the tangent space of \(D_ Z (f,B)\) at \([f] = 0\) is \(H^ 0(Y, {\mathcal I}_ B f^ \#T_{X/Z})\) and the obstruction lies in \(H^ 1 (Y, {\mathcal I}_ Bf^ \#T_{X/Z})\).
(3) When \(k\) is the fraction field of a ring finitely generated over \(\mathbb{Z}\) the statements (1) and (2) hold on almost every reduction of positive characteristic.
To prove the first part of the theorem, we use the theory of the Hilbert schemes and for the second part we study the sheaves of differential operators on \(X\) and \(Z\), and we look at infinitesimal deformations. Then we can use standard arguments on deformation. We have the following applications:
Theorem 2. Let \(X\) and \(Z\) be smooth projective \(k\)-varieties; let \(\pi: X \to Z\) be a surjective projective morphism with at least one smooth fiber. Then the relative anti-canonical sheaf \(\omega^{-1}_{X/Z}\) cannot be ample, unless \(Z\) is a single point.
Theorem 3. Let \(\pi : X \to Z\) be a morphism as above with \(X\) a Fano variety; let \(H\) be an ample divisor on \(Z\), \(\alpha>0\) such that \(-K_ X - \alpha \pi^*H\) is nef and let \(C\) be an irreducible curve on \(Z\). If \((C,-K_ Z - \alpha H) < 0\), then \(C\) is contained in the discriminant locus of \(\pi\).
If we assume that a morphism \(f:Y \to X\), where \(Y\) is a smooth projective curve, has a nontrivial deformation over \(Z\), by the theorem 1 we can show there exists a morphism \(f':Y\to X\) with \(\deg f'(Y)<\deg f(Y)\) and \(\pi\circ f'=\pi\circ f\). -- Then to prove theorems 2 and 3 we use Mori's arguments: we make reduction modulo \(p\) and we replace the map \(f\) by the composite with a suitable geometric Frobenius. relative deformation for a morphism; Fano manifold; Hilbert schemes; sheaves of differential operators; infinitesimal deformations; Fano variety Miyaoka Y.: Relative deformations of morphisms and applications to fibre spaces. Comment. Math. Univ. St. Paul. 42(1), 1--7 (1993) Local deformation theory, Artin approximation, etc., Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Fano varieties, Rational and birational maps, Parametrization (Chow and Hilbert schemes) Relative deformations of morphisms and applications to fibre spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0745.00052.]
For an integer \(N \geq 1\), let \(J_ 0(N)_ \mathbb{Q}\) be the Jacobian variety of the modular curve \(X_ 0(N)_ \mathbb{Q}\) which parametrizes elliptic curves together with a cyclic subgroup of order \(N\). \(J_ 0 (N)_ \mathbb{Q}\) is an abelian variety over \(\mathbb{Q}\).
The main purpose of this very well written paper is to compare two different ways of extending \(J_ 0(N)_ \mathbb{Q}\) to a group scheme over \(\mathbb{Z}\): First there is the Néron model \(J_ 0(N)\) which is smooth and separated, but whose mod \(p\) reduction has no obvious modular interpretation (in particular if \(p\) divides \(N)\). Secondly one can construct a model \(X_ 0(N)\) of \(X_ 0(N)_ \mathbb{Q}\) over \(\mathbb{Z}\) and then define \(\widetilde P\) as the kernel of the degree map \(\text{Pic}_{X_ 0(N)} \to \mathbb{Z}\). There is a natural homomorphism \(\widetilde P \to J_ 0(N)\) which on the connected component of 0 is an isomorphism for all primes \(p\) satisfying \(p \geq 5\) or \(p^ 2\not| N\). For such a prime \(p\) (i.e. \(N\) not divisible by \(p^ 2)\), the following results are obtained concerning the groups \(\Phi\) and \(\Phi'\) of connected components of \(J_ 0(N)\) and \(\widetilde P\), resp.: \(\Phi'\) is cyclic, \(\Phi/\Phi'\) is a torsion group of known exponent; \(\Phi\) is (as a Galois module) isomorphic with \(M^*/M\), where \(M\) is the character group of the maximal torus \(T\) in \(J_ 0(N)_ p\). Finally the Hecke correspondences on \(J_ 0(N)\) and their action on \(T\) are studied explicitly. Jacobian variety of the modular curve; Néron model; Hecke correspondences M. Raynaud, Jacobienne des courbes modulaires et opérateurs de Hecke, Courbes modulaires et courbes de Shimura, Astérisque 196-197, Société Mathématique de France, Paris (1991), 9-25. Jacobians, Prym varieties, Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties Jacobian of modular cures and Hecke operators | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review deals with the general question of how to compute the rank of an abelian variety over a number field K. The standard approach uses the injection \(A(K)/n\cdot A(K)\to H^ 1(0_ k,A[n])\) (flat cohomology): One tries to find an upper bound for the righthand side, and a lower bound for the left hand side which coincides with it. Here the authors deal with A \(= Eisenstein\)-quotient of \(J_ 0(N) = Jacobian\) of the modular curve \(X_ 0(N)\), \(N=p\) or \(N=p^ 2\) (p a prime. The case \(K={\mathbb{Q}}\), \(N=p\) has been treated extensively by B. Mazur).
Under certain hypotheses (on Bernoulli-numbers) the authors can show that either the right hand side vanishes, or that the theory of Heegner points suffices to account for all of it. The details of their arguments are a little bit involved, so we prefer not to comment on them. descent; Eisenstein-quotient of Jacobian of modular curve; rank of an abelian variety; Eisenstein-quotient of; Jacobian of the modular curve; Heegner points B. Gross & G. Lubin , The Eisenstein descent on J0(N). Invent . Math. 83 ( 1986 ), 303 - 319 . MR 818355 | Zbl 0594.14027 Jacobians, Prym varieties, Arithmetic ground fields for abelian varieties, Automorphic forms, one variable, Global ground fields in algebraic geometry, Picard schemes, higher Jacobians The Eisenstein descent on \(J_ 0(N)\) | 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 Let \(X\) be a smooth projective curve of genus \(g\geq 2\) over the complex numbers. Denote by \(W^ r_{n,d}\) the subset of the moduli variety of stable vector bundles \(E\) of rank \(n\) and degree \(d\) on \(X\) with \(\dim H^ 0(X,E)\geq r+1\). The author is concerned with the question: When is this subset nonempty? For \(n=1\), \textit{P. Griffiths} and \textit{J. Harris} [Duke Math. J. 47, 233-272 (1980; Zbl 0446.14011)] proved that there exist numbers \(\rho(1,r,d)\) such that for a general \(X\) one has \(W^ r_{1,d}\neq\emptyset\) if and only if \(\rho(1,r,d)\geq 0\). \textit{N. Sundaram} [Tôhoku Math. J., II. Ser. 39, 175-213 (1987; Zbl 0631.14021)] showed that there exist \(\rho(2,r,d)\) such that if \(W^ r_{2,d}\neq\emptyset\), then \(\dim W^ r_{2,d}\geq\rho(2,r,d)\) while \textit{M. Teixidor} [Europroj News Letter, 1991] pointed out that if \(W^ 2_{2,d}\) is nonempty then it has \(\dim=\rho(2,2,d)\) for a general \(X\). ---The author proves here the following results for a general \(X\):
(1) \(W^ 2_{2,d}\neq\emptyset\) if and only if \(\rho(2,2,d)\geq 0\).
(2) If \(\rho(2,r+1,d)\geq 0\) or if \(d\geq 3\), \(r \text{odd},(r+1)/2\) divides \(g\) and \(\rho(2,r,d)\geq 0\), then \(W^ r_{2,d}\neq\emptyset\).
The proofs are by direct construction and depend heavily on the above results of Griffiths and Harris plus the author's criterion of liftability of a section of a quotient line bundle of a vector bundle to a section of the vector bundle itself. projective curve; moduli variety of stable vector bundles; liftability of a section of a quotient line bundle Tan, X. J.: Some results on the existence of rank two special stable vector bundles. Manuscripta Math., 75, 365--373 (1992) Vector bundles on curves and their moduli, Families, moduli of curves (algebraic), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Some results on the existence of rank two special stable vector bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main results of the paper are the following theorems:
(1) Let \(C\) be a smooth projective curve of genus \(g \geq 3\) over the field of complex numbers and \(H\) an irreducible curve in the symmetric product \(C^{(2)}\). If the geometric genus of \(H\) is \(\leq{1\over 2} (g - 1)\) then \(C\) is a double covering of \(H\). -- A special case of this was obtained by Silverman and \textit{D. Abramovich} and \textit{J. Harris} in Compos. Math. 78, No. 2, 227-238 (1991; Zbl 0748.14010) by different methods.
(2) Let \(C\) be a non hyperelliptic, non bielliptic smooth curve of genus \(g \geq 3\) and \(D\) a smooth curve of genus \(\leq g\). There exist only finitely many families of irreducible curves \(\Gamma\) in \(D \times C\) of type \((m,2)\) for any \(m \geq 1\). -- The proof applies results of the authors' former paper [Math. Z. 205, No. 3, 333-342 (1990; Zbl 0685.14018)]. abelian subvariety of the Jacobian; symmetric product of curve; double covering of a curve; number of irreducible curves Jacobians, Prym varieties, Coverings of curves, fundamental group, Cycles and subschemes On curves in \(C^{(2)}\) generating proper abelian subvarieties of \(\text{J}(C)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians special linear series; general moduli; double cover of curve; Jacobian; Prym variety; special moduli A. Del Centina -S. Recillas,Some projective geometry associated with unramified double covers of curves of genus 4, Ann. Mat. Pura Appl.,33 (1983), pp. 125--140. Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Coverings of curves, fundamental group, Special algebraic curves and curves of low genus, Jacobians, Prym varieties Some projective geometry associated with unramified double covers of curves of 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 This paper gives a self-contained, detailed account of the construction and compactification of the moduli space of Higgs bundle on (families of) curves. It is divided into five parts: I. The theorem of the cube; II. \(G\)-bundles; III. Abelianisation; IV. Projective connections; V. Infinitesimal parabolic structure.
Part I starts with a theorem on the determinant of the cohomology of coherent sheaves on a curve. It is shown that the theorem of the cube follows from this result. A second application is at the basis of the construction, via theta-functions, of global sections of determinant bundles on the moduli space of Higgs bundles. More precisely, let \(S\) be a noetherian base scheme for the family of curves \(\pi:C\to S\), with \(\pi\) proper, all fibers of dimension \(\leq 1\), and such that \(\pi_ *({\mathcal O}_ C)={\mathcal O}_ S\). A Higgs bundle on \(C\) is a vector bundle \({\mathcal F}\) together with a section \(\theta\) of \(\Gamma(C,{\mathcal E}nd({\mathcal F})\otimes\omega_ C)\). The coefficients of the characteristic polynomial of \(\theta\) define global sections \(f_ i\in\Gamma(C,\omega^ i)\), and the affine space classifying such sections is called the characteristic variety \({\mathcal C}har\) (it depends on \(\text{rk}({\mathcal F}))\), and \({\mathcal F}\) defines a point \(\text{char}({\mathcal F})\) in \({\mathcal C}har\). Such a Higgs bundle will often be denoted \(({\mathcal F},\theta)\). One has the notion of (semi)-stability for Higgs bundles, and any semistable Higgs bundle admits a Jordan- Hölder (JH) filtration by subbundles with stable quotieents of constant ratio degree/rank. The isomorphism classes and multiplicities of these stable components are independent of the filtration. Two semi-stable Higgs bundles are called JH-equivalent if these coincide. A result on JH- equivalence is derived and used to show that the theta-functions separate points in the moduli space of Higgs bundles. The moduli-space of stable Higgs bundles of given rank and degree is constructed as an algebraic space \({\mathcal M}^ 0_ \theta\). Then \({\mathcal M}_ \theta^ 0\) embeds as an open subscheme into the onrmalization \({\mathcal M}_ \theta\) of \(\mathbb{P}^ N\times{\mathcal C}har\) (for suitable \(N)\) in \({\mathcal M}^ 0_ \theta\).
In part II one considers a reductive connected algebraic group \({\mathcal G}\) over a smooth projective connected curve \(C\) over a field \(k\). A \({\mathcal G}\)-torsor \(P\) on \(C\), together with an element
\[
\theta\in\Gamma(C,\text{Lie}({\mathcal G}_ P)\times\omega_ C)
\]
is called semistable if \((\text{Lie}({\mathcal G}_ P)\), ad\((\theta))\) is a semistable Higgs bundle of degree zero. One also has the notion of stable \(P\). The main result on semistable pairs \((P,\theta)\) is the following semistable reduction theorem: If \(V\) is a complete discrete valuation ring with fraction field \(K\), \(C\to V\) a smooth projective curve, \((P_ K,\theta_ K)\) a semistable pair (associated with a connected reductive group \({\mathcal G}\) over \(C)\) whose characteristic is integral over \(V\), then there exists a finite extension \(V'\) of \(V\) such that the base extension of \((P_ K,\theta_ K)\) extends to a semistable pair on \(C_{V'}\). Furthermore, if the special fiber of this extension is stable, then any other semistable extension is isomorphic to it. For stable \((P,\theta)\) one is led to construct an algebraic moduli stack \({\mathcal M}^ 0_ \theta({\mathcal G})\) and the coarse moduli space \(M^ 0_ \theta({\mathcal G})\) which is shown to be quasi-projective of explicitly calculated relative dimension over a suitable base. As before one defines a \(M_ \theta({\mathcal G})\) as the normalisation of a \(\mathbb{P}^ N\) in \(M^ 0_ \theta({\mathcal G})\). Then \(M_ \theta({\mathcal G})\) is projective over \({\mathcal C}har\) and contains \(M^ 0_ \theta({\mathcal G})\) as an open subscheme. Then, for example, if \(C\) has genus \(>2\), the boundary \(M_ \theta({\mathcal G})-M^ 0_ \theta({\mathcal G})\) has codimension \(\geq 4\). Many other results are derived.
In part III the theory is extended to exceptional groups. As a corollary of the theory one obtains, with the notations above, that the set of connected components of the moduli space \({\mathcal M}^ 0({\mathcal G})\) of stable (Higgs) \({\mathcal G}\)-bundles coincides with that of \({\mathcal M}^ 0_ \theta({\mathcal G})\), \(M_ \theta({\mathcal G})\) as well as that of a generic fiber of \({\mathcal M}^ 0_ \theta({\mathcal G})\to{\mathcal C}har\), under the natural mappings. Among many other results, one application of abelianisation is given by another corollary: On each connected component of \({\mathcal M}^ 0_ \theta({\mathcal G})\), all global functions are obtained by pullback from \({\mathcal C}har\).
In part IV the accent is on \({\mathcal M}^ 0({\mathcal G})\), where \({\mathcal G}\) is the twisted form of some semi-simple \(G\). The notion of \(\Omega_ C\)- connections \(\nabla\) on \({\mathcal G}\)-torsors \(P\) is introduced. \({\mathcal M}^ 0_ \nabla({\mathcal G})\) denotes the moduli stack of such pairs \((P,\nabla)\) with \(P\) stable. It is fibered over \({\mathcal M}^ 0({\mathcal G})\). Over \(\mathbb{C}\), \({\mathcal M}^ 0_ \nabla({\mathcal G})\) classifies bundles with integrable connections, i.e. representations of \(\pi_ 1(C)\). A locally faithful \({\mathcal G}\)-representation \({\mathcal F}\) defines a line bundle \({\mathcal L}={\mathcal L}({\mathcal F})\) on \({\mathcal M}^ 0({\mathcal G})\). Then the pullback of \({\mathcal L}\) to \({\mathcal M}^ 0_ \nabla({\mathcal G})\) has a connection \(\nabla\). Its curvature can be described explicitly.
The final part V discusses parabolic structures in the sense of C. Seshadri. The parabolic analogue of a Higgs bundle is introduced and a theory parallel to the one in the foregoing parts is sketched. torsor; moduli space of Higgs bundle; determinant of the cohomology of coherent sheaves on a curve; theorem of the cube; characteristic variety; theta-functions; semistable pairs; moduli stack; abelianisation; connection Faltings, Gerd, Stable {\(G\)}-bundles and projective connections, Journal of Algebraic Geometry, 2, 3, 507-568, (1993) Families, moduli of curves (algebraic), Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) Stable \(G\)-bundles and projective connections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(W\) be a smooth connected complete curve over the complex numbers \(\mathbb{C}\) admitting the dihedral group \(D_n\) of order \(2n\) as its group of automorphisms. Any pair of subgroups \(H\subset N\subset D_n\) defines a covering of curves \(f_{H,N}: W_H\to W_N\) where \(W_H= W/H\) and \(W_N =W/N\) denote the quotient curves. As such the subgroups \(H\) and \(N\) define three abelian varieties associated to \(W\), namely the Jacobians \(JW_H\) and \(JW_N\) of \(W_H\) respectively \(W_N\), and the Prym variety \(P(W_H/W_N)\) of the covering \(f_{H,N}\). In the paper it is shown that the Jacobian \(JW\) of \(W\) is isogenous to a product of abelian subvarieties where the group \(D_n\) acts on each factor by a suitable multiple of precisely one of the rational irreducible representations of \(D_n\). Some of the abelian subvarieties involved are Jacobians and Pryms of intermediate covers as described above, whereas some other kinds of abelian subvarieties also occur, namely orthogonal complements of Pryms inside Pryms. Jacobian; Prym variety; dihedral group; automorphism group of curve Carocca, A.; Recillas, S.; Rodríguez, R. E., Dihedral groups acting on Jacobians, Contemp. Math., 311, 41-77, (2011) Jacobians, Prym varieties, Automorphisms of curves Dihedral groups acting on Jacobians. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This thesis is dedicated to the connection between generalized polytopes and the Cartier divisors of a compact toric variety. Following \textit{G. Ewald} [``Combinatorial convexity and algebraic geometry'', Graduate Texts in Math. 168 (1996)], a generalized polytope is defined to be a finite set of halfspaces which is related in a natural way to a complete fan, such that the 1-skeleton of the fan constitutes the (outer) normals of the halfspaces. In the first two chapters the theory of generalized polytopes is developed in analogy to classical convex polytopes, including in particular the concept of mixed volumes. In the third chapter it is shown that the intersection number of \(\mathbb{Q}\)-Cartier-divisors in a compact toric variety corresponds to the mixed volume of the associated generalized polytopes. This leads to a simple and quick method for the computation of the intersection numbers. Finally some algebraic-geometric applications are derived. generalized mixed volume; Cartier-divisors; fan; generalized polytopes; convex polytopes; toric variety; intersection numbers Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Divisors, linear systems, invertible sheaves, Mixed volumes and related topics in convex geometry Combinatoric-geometric characterization and computation of the intersection numbers of Cartier divisors in compact 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 This article draws attention to results by \textit{A. Comessatti} [Mem. Soc. Ital. Sci. 21, 45-71 (1919)] whose significance for the construction of the Horrocks-Mumford bundle does not seem to be noticed. The aim of the article is to present new short proofs of Comessatti's results, the first one giving a simple criterion (theorem 1.1), when the Jacobian of a curve of genus \(2\) admits an embedding into \({\mathbb{P}}_ 4\). The author then explicitly constructs a 2-dimensional family of abelian surfaces, where this criterion applies, in the Hilbert moduli space of principally polarized abelian surfaces with real multiplication by the integers of \({\mathbb{Q}}(\sqrt{5})\). Comessatti's original result was only concerned with the embedding of abelian surfaces corresponding to a nonspecified open set of a certain subspace of the Siegel upper half plane \({\mathcal H}_ 2\). Finally a proof is given for Comessatti's theorem about the geometry of any such surface A in \({\mathbb{P}}_ 4\), saying that there are 25 planes in \({\mathbb{P}}_ 4\), whose intersection with A form Desargue-configurations. projective embedding; Horrocks-Mumford bundle; Jacobian of a curve; Hilbert moduli space of principally polarized abelian surfaces H. Lange, Jacobian surfaces in \(\mathbf P_ 4\) , J. Reine Angew. Math. 372 (1986), 71-86. Picard schemes, higher Jacobians, Embeddings in algebraic geometry, Jacobians, Prym varieties Jacobian surfaces in \({\mathbb{P}}_ 4\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A closed Riemann surface \(S\) is called a generalized Fermat curve of type \((k,n)\), where \(k,n \geq 2\) are integers, if it admits a group \(H \cong {\mathbb Z}_{k}^{n}\) of conformal automorphisms such that the quotient orbifold \(S/H\) has genus zero and exactly \((n+1)\) conical points, each one of order \(k\). An explicit equation for \(S\) is known as a fiber product of \(n-1\) Fermat curves of degree \(k\) [\textit{G. González-Diez} et al., J. Algebra 321, No. 6, 1643--1660 (2009; Zbl 1175.14023)] and it is also known that the group \(H\) is unique in \(\mathrm{Aut}(S)\) [\textit{R. A. Hidalgo} et al., J. Pure Appl. Algebra 221, No. 9, 2312--2337 (2017; Zbl 1400.14080)]. In the paper under review, the author constructs a set of generators for the first homology group of \(S\) and finds a set of generators for the period lattice of the associated Jacobian variety. Riemann surfaces; Jacobian variety; generalized Fermat curve Riemann surfaces; Weierstrass points; gap sequences, Compact Riemann surfaces and uniformization Periods of generalized Fermat curves | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.