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Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review is an excellent survey on the subject of counting curves, written by two leading experts in the field. A curve counting theory from the modern viewpoint, loosely speaking, involves a geometrically meaningful compactification of the families of curves in an algebraic variety \(X\) as well as (almost always) a deformation/obstruction theory (to define a virtual fundamental class). The authors summarize major ways of counting curves in the last two decades, including naive counting (classical enumerative geometry), Gromov-Written theory, Gopakumar-Vafa/BPS invariants, Donaldson-Thomas theory, stable pairs, stable unramified maps and stable quotients, with a focus on the case when \(X\) is a nonsingular projective \(3\)-fold. Moreover, the authors analyze in detail the advantages, drawbacks, and relationships amongst these approaches. This paper is well written and can serve as a great reference for students and general mathematicians looking for an elementary route into the subject.
As a remark, the fractional part \(1/2\) in the title reflects the authors' opinion that naive curve counting is not always well-defined and has many drawbacks, so it should be viewed as only \(1/2\) a method. This adds an amusing flavor to the paper, which coincides with the fact that sometimes counting invariants are rational numbers only. It also implies that many topics discussed in the paper are still developing and await further discovery. Gromov-Witten theory; stable maps; Hilbert schemes; stable pairs; stable quotients; BPS invariants; Donaldson-Thomas theory; virtual classes Pandharipande R. and Thomas R. P., 13/2 ways of counting curves, preprint 2011, . Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] 13/2 ways of counting 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 Gieseker-Petri theorem, asserted by \textit{K. Petri} [Math. Ann. 93, 182-209 (1925; F.d.M. 051.51001)] and proved by \textit{D. Gieseker} [Invent. Math. 66, 251-275 (1982; Zbl 0522.14015)], states that a general compact Riemann surface \(C\) of genus \(g\) satisfies the so-called Brill- Noether-Petri condition, i.e. for any line bundle \(A\) on \(C\) the multiplication map
\[
H^0 (A) \otimes H^0 (\omega_C \otimes A^\vee) \to H^0 (\omega_C)
\]
is injective. Next \textit{R. Lazarsfeld} [J. Differ. Geom. 23, 299-307 (1986; Zbl 0608.14026)] provided a new approach to the proof of the Gieseker-Petri theorem, based on the following theorem: Let \(S\) be a complex \(K3\) surface and \(L\) a line bundle on \(S\) such that the linear system \(|L |\) does not contain reducible or multiple curves. Then the general curve \(C \in |L |\), if smooth, satisfies the Brill-Noether-Petri condition.
This proves the Gieseker-Petri theorem since, as it is known, for any \(g \geq 2\), there are \(K3\) surfaces \(S\) such that \(\text{Pic} (S) = \mathbb{Z} \circ [C]\), with \(C\) a smooth irreducible curve of genus \(g\).
In the present paper the author, following some of Lazarsfeld's ideas, gives a simpler and quick proof of Lazarsfeld's theorem. Gieseker-Petri theorem; compact Riemann surface; genus; linear system; K3 surfaces Pareschi, Giuseppe, A proof of Lazarsfeld's theorem on curves on \(K3\) surfaces, J. Algebraic Geom., 4, 1, 195-200, (1995) Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic), \(K3\) surfaces and Enriques surfaces, Infinitesimal methods in algebraic geometry A proof of Lazarsfeld's theorem on curves on \(K3\) 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 H be the complement in the semi-group of natural numbers of the set of gaps at a point of an algebraic curve. H is a semigroup. The author is concerned with realizing certain 4-generator semigroups in this way, and he uses a construction due to \textit{H. C. Pinkham} [Ásterisque 20, 1-131 (1974; Zbl 0304.14006)] combined with curves which are of torus embedding type (pull-backs of fibres of torus embeddings). This construction realizes 3-generator semigroups and extensive classes of 4-generator semigroups. Weierstrass points; semigroups of natural numbers Komeda, J., On the existence of Weierstrass points with a certain semigroup, \textit{Tsukuba J. Math.}, 6, 2, 237-270, (1982) Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic) On the existence of Weierstrass points with a certain semigroup generated by 4 elements | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal M}(n,d)\) denote the moduli space of stable holomorphic vector bundles of coprime rank \(n\) and degree \(d\) over a fixed Riemann surface of genus at least two. Let \(\Lambda\) denote a fixed line bundle of degree \(d\) and let \({\mathcal M}_\Lambda(n,d)\) denote the space consisting of those bundles with determinant \(\Lambda\). The authors determine the Hodge numbers of these two moduli spaces. The method involves viewing these spaces as finite-dimensional quotients in the sense of Mumford's geometric invariant theory, and then applying results of \textit{F. Kirwan} [``Cohomology of quotients in symplectic and algebraic geometry'', Mathematical Notes 31 (Princeton 1984; Zbl 0553.14020), Ark. Mat. 24, 221-275 (1986; Zbl 0625.14026)] to give an inductive approach for determining the Hodge numbers. The authors go on to show that the \(\chi(t)\)-characteristic of \({\mathcal M}_\Lambda(n,d)\) has an especially simple form, while the \(\chi(t)\)-characteristic of \({\mathcal M}(n,d)\) is identically zero. moduli space of vector bundles; Hodge-Poincaré polynomial; geometric invariant theory Earl, R., Kirwan, F.: The Hodge numbers of the moduli spaces of vector bundles over a Riemann surface. Q. J. Math. \textbf{51}, 465-484 (2000). arXiv:math/0012260 Transcendental methods, Hodge theory (algebro-geometric aspects), Vector bundles on curves and their moduli, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Geometric invariant theory The Hodge numbers of the moduli spaces of vector bundles over a Riemann surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(R\) be a commutative ring. There is a natural decomposition of the Picard group
\[
\text{Pic}(R[x,x^{-1}])\cong\text{Pic}(R)\oplus \text{NPic}(R)\oplus\text{NPic}(R)\oplus\text{LPic}(R)
\]
due to the reviewer [Invent. Math. 103, No. 2, 351-377 (1991; Zbl 0794.13008)]. Here \(\text{NPic}(R)= \text{Pic}(R[x])/\text{Pic}(R)\) and \(\text{LPic}(R)= H^ 1_{\text{et}}(\text{spec }R,\mathbb{Z})\). It is well known that \(\text{NPic}(R)= 0\) if and only if \(R_{\text{red}}\) is seminormal. The paper under review describes the structure of \(\text{LPic}(R)\) when \(R\) is a 1-dimensional reduced Noetherian ring with finite normalization \(\overline R\). From the \(H^ 0\)-LPic sequence of the reviewer [op. cit.], one obtains a combinational formula for the integer \(d\) such that \(\text{LPic}(R)= \mathbb{Z}^ d\); the author uses this to verify a conjecture of S. Greco: If \(R\) is 1-dimensional Noetherian then \(\text{LPic}(R)= 0\) if and only if \(X= \text{Spec}(R)\) contains no ``polygons'', distinct components of \(X\) meet in at most one point, and each irreducible component of \(X\) is unibranch. Adding seminormality gives a necessary and sufficient geometric condition for \(\text{Pic}(R)=\text{Pic}(R[x, x^{- 1}])\).
\{Reviewer's remarks: The finite normalization condition is not needed. The vanishing of \(\text{LPic}(R)\) is much harder to describe for 2- dimensional rings\}. natural decomposition of Picard group; Noetherian ring; finite normalization; seminormality Asanuma, T, Picr[X,X\(-\)1] for R a one-dimensional reduced Noetherian ring, J. Pure Appl. Algebra, 71, 111-128, (1991) Negative \(K\)-theory, NK and Nil, Picard groups, Grothendieck groups, \(K\)-theory and commutative rings, Grothendieck groups and \(K_0\) Pic\(R[X, X^{-1}]\) for \(R\) a one-dimensional reduced Noetherian ring | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G := \text{SL}_3({\mathbb C})\), \(\overline{G} := \text{PGL}_3({\mathbb C})\), \(V = {\mathbb C}^3\) be the standard representation of \(G\), \(e_1,e_2,e_3\) the canonical \(\mathbb C\)-basis of \(V\) and \(x_1,x_2,x_3\) the dual basis of \(V^{\vee}\). For any positive integer \(d\), let \(V(d)\) denote the irreducible representation \(\mathrm{Sym}^d(V^{\vee})\) of \(G\). In the paper under review, the authors show that the quotient space \(\text{Hyp}(d,2) := {\mathbb P}(V(d))^{\text{ss}}/\overline{G}\) is rational for \(d \equiv 1\) (mod 3), \(d \geq 37\) and also for \(d \equiv 2\) (mod 3), \(d \geq 65\). Combining this result with a result of \textit{P. Katsylo} [Math. USSR, Sb. 64, No. 2, 375--381 (1989; Zbl 0679.14028)] asserting that \(\text{Hyp}(d,2)\) is rational for \(d \equiv 0\) (mod 3), \(d \geq 1821\), one gets that \(\text{Hyp}(d,2)\) is rational for \(d \gg 0\).
The structure of the proof is similar to that used by \textit{N.I. Shepherd-Barron} [Compos. Math. 67, No. 1, 51--88 (1988; Zbl 0661.14022)] to settle the case \(d \equiv 1\) (mod 9), \(d \geq 19\), but the computational aspects of the proof are quite different. One starts with the fact, proved by Shepherd-Barron, that the quotient map \({\mathbb P}(V(4))^{\text{ss}} \rightarrow {\mathbb P}(V(4))^{\text{ss}}/\overline{G}\) (resp., \({\mathbb P}(V(8))^{\text{ss}} \rightarrow {\mathbb P}(V(8))^{\text{ss}}/ \overline{G}\)) is a principal \(\overline{G}\)-bundle in the Zarisky topology over a non-empty open subset of the quotient variety. Then one constructs, using the \textit{symbolic method}, a \(G\)-equivariant map \(S_d : V(d) \rightarrow V(4)\) for \(d = 3n+1\) (resp., \(T_d : V(d) \rightarrow V(8)\) for \(d = 3n+2\)) defined by homogeneous polynomials of degree 4. \(S_d\) appears already in the paper of Shepherd-Barron. The authors need, actually, from the symbolic method only the following observation: for every homogeneous polynomial \(P\) of degree \(d\) in three indeterminates, there exists a unique \(\mathbb C\)-linear function \({\ell}_P : V(d) \rightarrow {\mathbb C}\) such that \({\ell}_P({\alpha}_x^d) = P(\alpha)\), \(\forall \alpha = ({\alpha}_1,{\alpha}_2,{\alpha _3}) \in {\mathbb C}^3\), where \({\alpha}_x := {\alpha}_1x_1 + {\alpha}_2x_2 + {\alpha}_3x_3 \in V^{\vee}\). More precisely, one has the formula:
\[
{\ell}_P = \frac{1}{d!}\, P\left( \frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \frac{\partial}{\partial x_3} \right)\;.
\]
For \(\alpha ,\beta ,\gamma ,\delta \in {\mathbb C}^3\), the authors consider the expression:
\[
I = I(\alpha ,\beta ,\gamma ,\delta) := (\alpha \beta \gamma)(\alpha \beta \delta)(\alpha \gamma \delta) (\beta \gamma \delta)
\]
where \((\alpha \beta \gamma)\) is the determinant of the \(3 \times 3\) matrix with rows \(\alpha ,\beta, \gamma\). According to the above observation, there exists, for \(d = 3n+1\), a unique \(\mathbb C\)-linear map \({\widetilde S}_d : V(d)^{\otimes 4} \rightarrow V(4)\) such that:
\[
{\widetilde S}_d({\alpha}_x^d \otimes {\beta}_x^d \otimes {\gamma}_x^d \otimes {\delta}_x^d) = I^n{\alpha}_x{\beta}_x{\gamma}_x{\delta}_x\, .
\]
\(S_d\) is, by definition, \({\widetilde S}_d\) composed with the map \(V(d) \rightarrow V(d)^{\otimes 4}\), \(f \mapsto f^{\otimes 4}\). For the definition of \(T_d\), \(d = 3n+2\), one uses the symbolic expression \(I^n{\alpha}_x^2{\beta}_x^2{\gamma}_x^2{\delta}_x^2\).
Now, the authors consider the linear subspaces \(L_S := x_1^{2n+3}{\mathbb C}[x_1,x_2,x_3]_{n-2} \subset V(d)\) for \(d = 3n+1\), and \(L_T := x_1^{2n+5}{\mathbb C}[x_1,x_2,x_3]_{n-3} \subset V(d)\) for \(d = 3n+2\), and notice that \({\mathcal I}_{{\mathbb P}(L_S)}^3\) (resp., \({\mathcal I}_{{\mathbb P}(L_T)}^3\)) contains the ideal sheaf of the base locus of the rational map \(S_d : {\mathbb P}(V(d)) \dashrightarrow {\mathbb P}(V(4))\) (resp., \(T_d : {\mathbb P}(V(d)) \dashrightarrow {\mathbb P}(V(8))\)). They deduce that for \(g \in V(d)\) not in \(L_S\) (resp., \(L_T\)) the restricted map \(S_d\, | \, {\mathbb P}(L_S+{\mathbb C}g) \dashrightarrow {\mathbb P}(V(4))\) (resp., \(T_d \, | \, {\mathbb P}(L_T+{\mathbb C}g) \dashrightarrow {\mathbb P}(V(8))\)) is linear.
The key point of the proof consists in showing that, under the above assumptions on \(d\), there exists \(g\) such that the above linear map is \textit{dominant}. The authors reduce the proof of this fact, after a number of elementary but ingenious tricks, to some straightforward computations with a computer algebra program. The proof of the rationality of \(\text{Hyp}(d,2)\) can be now completed as in the paper of Shepherd-Barron. plane curve; moduli space; rational variety; symbolic method; covariant Böhning, Chr., Graf v. Bothmer, H.-Chr.: The rationality of the moduli spaces of plane curves of sufficiently large degree. Invent. Math. 179(1). doi: 10.1007/s00222-009-0214-6 (2010). preprint available at arXiv:0804.1503 Rationality questions in algebraic geometry, Families, moduli of curves (algebraic), Rational and unirational varieties, Geometric invariant theory, Representation theory for linear algebraic groups Rationality of the moduli spaces of plane curves of sufficiently large degree | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study the compactification of the space of twisted cubics of three-degree \((1,1,1)\) in the Segre variety \(X = \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1\) in the stable sheaves space \(\mathbf{S}\). They show that it is a smooth, irreducible, rational variety of dimension \(6\) with Poincaré polynomial \(q^6+3q^5+7q^4+10q^3+7q^2+3q+1\). Moreover, \(\mathbf{S}\) is isomorphic to the compactification in the stable maps space \(\mathbf{M}\) or the Hilbert scheme \(\mathbf{H}\). rational curves; stable maps; stable sheaves Chung, K; Lee, W, Twisted cubic curves in the Segre variety, C. R. Math., 353, 1123-1127, (2015) Families, moduli of curves (algebraic), Determinantal varieties Twisted cubic curves in the Segre 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 k be a field of characteristic zero, V(d) the space of binary forms of degree \(d\) and \({\mathbb{P}}(d):={\mathbb{P}}(V(d))\) its projective space. The author proves: there is a SL(2)-equivariant map from \({\mathbb{P}}(4m-2)\) to \({\mathbb{P}}(2m)\), which is after blow-up a projective space bundle. This theorem is used to give a new proof for the fact that \(H_{2g}\), the moduli space of hyperelliptic curves of genus \(2g\), is rational. Then explicit rational bases (i.e. an algebraically independent set of generators) for the function fields \(k(H_ 4)\) and \(k(H_ 3)\) are given.
Furthermore, the author proves that the moduli space \(M_ n(2,3)\) for quadro-cubic complete intersections in \({\mathbb{P}}^ n\) is rational if \(n+1\) is prime to 3. Finally, a rational basis of \(k(M_ 3(2,3))\) is computed. rationality of moduli space of hyperelliptic curves; quadro-cubic complete intersections Shepherd-Barron, Apolarity and its applications, Invent Math 97 (2) pp 433-- (1989) Rational and unirational varieties, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles Apolarity and its 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 notion of paracanonical curves is an analog of canonical curves, with a twist by a non-trivial torsion line bundle. The Prym-Green Conjecture, which was first formulated in [\textit{A. Chiodo} et al., Invent. Math. 194, No. 1, 73--118 (2013; Zbl 1284.14006)], predicts that the minimal free resolution of the paracanonical curve corresponding to a general level curve of genus \(g \geq 5\) has at most one non-zero entry in each diagonal of its Betti table. This conjecture was verified for odd genus with level \(2\) or sufficiently high level in the authors' previous work [Invent. Math. 203, No. 1, 265--301 (2016; Zbl 1335.14009)] by using certain \(K3\) surfaces. The main result of this impressive paper proves the conjecture fully for odd genus, i.e. for any level, by using elliptic ruled surfaces instead, to provide explicit examples of pointed Brill-Noether general curves. syzygy; paracanonical curve; ruled elliptic surface Families, moduli of curves (algebraic), Syzygies, resolutions, complexes and commutative rings The resolution of paracanonical curves of odd 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 The present article is a long and very detailed survey on hyperelliptic and superelliptic curves. The point in the aim of the authors is the fact that the study of hyperelliptic curves has been crucial in algebraic geometry. Then, a natural wish is to generalize results on hyperelliptic curves to general curves or, at least, to a wider class of curves. A hyperelliptic curve admits a cyclic Galois cover to the projective line, which is of degree \(2\). Thus, a natural generalization is to consider cyclic Galois covers of degree \(n \geq 2\). In that case, for a curve \(\mathcal{C}\) with automorphism group Aut(\(\mathcal{C}\)), there is a cyclic subgroup \(H = \langle \tau \rangle\) normal in Aut(\(\mathcal{C}\)) such that \(\mathcal{C}/H\) is isomorphic to \(\mathbb{P}^1\). Those curves are called superelliptic, and \(\tau\) is called the superelliptic automorphism of \(\mathcal{C}\). The paper is devoted to the generalization of the theory of hyperelliptic curves to that of superelliptic curves, focusing on the theories that can be extended and on the pending problems that arise in the generalization.
The paper is organized as follows. A part 1 considers general algebraic curves and hyperelliptic curves. Its three Sections study generalities of algebraic curves (Sect. 2), Weierstrass points (Sect. 3), and the full automorphism groups of surfaces (Sect. 4). This last point is intrinsically important in this context, since hyperelliptic and superelliptic curves exist just because the hyperelliptic and the superelliptic automorphism exist. As an exhibition of that importance, the automorphism groups of all hyperelliptic curves over any characteristic are shown in Theorem 15.
Part 2 is devoted to superelliptic curves, and is the core of the work. It splits in as much as eleven Sections. In Sect. 5 superelliptic curves are introduced, and their Weierstrass points and automorphism groups are studied. Sect. 6 is devoted to the loci of superelliptic curves in the moduli space. The inclusions between these loci are studied, and the authors give a complete stratification of the moduli space for genera \(3\) and \(4\). In Sect. 7 the open problem of determining the equation of a curve \(\mathcal{C}\) having a given group \(G = \) Aut(\(\mathcal{C})\) is considered. The question is solved for particular cases, and for \(g \leq 3\), but it remains open for \(g \geq 4\). However, Theorem 21 gives its solution for superelliptic curves. Sects. 8 and 9 consider the invariant theory of binary forms and weighted moduli spaces. In Sect. 10 the authors study minimal models of superelliptic curves. Conditions on the set of invariants of the curve are given in order that it has such a minimal model, as well as an algorithm to obtain it. Sect. 11 discusses when the field of moduli is a minimal field of definition. In particular, superelliptic curves of genus \(1 \leq g \leq 4\) which are definable over their field of moduli are described in 11.4. In Sect. 12 theta functions and Thomae's formulas are considered for hyperelliptic curves, and how to generalize for superelliptic curves. Sect. 13 deals with Jacobian varieties, and Sect. 14 is devoted to those having complex multiplication, with a particular consideration of curves with many automorphisms. Finally, Sect. 15 explores how to extend this study to even more general curves, for instance, with dihedral groups or abelian covers.
All in all, this is an amazing piece of work which puts together a lot of results obtained by the authors, an important set of collaborators of them, and other researchers. There are also scattered in the text a number of new results, as well as the latest state-of-the-art. hyperelliptic curves; superelliptic curves Research exposition (monographs, survey articles) pertaining to algebraic geometry, Special algebraic curves and curves of low genus, Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Automorphisms of curves, Jacobians, Prym varieties From hyperelliptic to superelliptic 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 short appendix to [\textit{T. Ashikaga} and \textit{K.-i. Yoshikawa}, ibid. 1--34 (2009; Zbl 1196.14024)], a Horikawa index is constructed [Algebraic geometry 2000, Azumino. Proceedings of the symposium, Nagano, Japan, July 20--30, 2000. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 36, 1-49 (2002; Zbl 1088.14010)] for genus \(4\) fibred surfaces \(f: S\longrightarrow B\) whose general fibre is EH-general (i.e. has two dinstinct \(g^1_3\)'s). It has to be remarked that the fibred surfaces need not be semistable. This result is achieved by cleverly studying the multiplication map of relative canonical algebras
\[
\text{Sym}^2f_*\omega_f\longrightarrow f_*\omega_f
\]
and in particular by estimating its cokernel. >From this, using a formula in [loc. cit.], the author obtains a localization for the signature of \(S\) slope; fibred surface Fibrations, degenerations in algebraic geometry, Families, moduli, classification: algebraic theory, Families, moduli of curves (algebraic) Appendix to ``A divisor on the moduli space of curves associated to the signature of fibered surfaces'' by T. Ashikaga and K.-I. Yoshikawa | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\subset\mathbb{A}^n\) be a smooth hypersurface defined by a polynomial \(f\) of degree \(k\ge 3\) over a field \(K\), and assume that the leading terms of \(f\) define a smooth projective hypersurface, \(Z\) say. For a fixed \(K\)-point \(P=[(a_1,\ldots,a_n)]\) of \(Z\) one is interested in the space Mor\(_{d,P}(\mathbb{A}^1,X)\) of \(n\)-tuples of polynomials \((g_1,\ldots,g_n)\) of degree \(d\), with leading terms \(a_1,\ldots,a_n\), satisfying \(f(g_1,\ldots,g_n)=0\). Then it is shown that Mor\(_{d,P}(\mathbb{A}^1,X)\) is irreducible and has the expected dimension \(d(n-k)\) for any field \(K\) whose characteristic is either zero or \(>k\), provided that \(d\ge k-1\ge 2\) and
\[
\left\lfloor\frac{d}{k-1}\right\rfloor \left(\frac{n}{2^k}-k+1\right)\ge 1.
\]
This is proved as a corollary to the main theorem, which describes the compactly supported cohomology of the space of rational curves on a smooth hypersurface.
To establish the principal result the authors use ``spreading out'', so that it suffices to examine the situation over the algebraic closure of a finite field. The key new feature of the proof is a geometric analogue of the circle method. The major arcs are handled geometrically, although the treatment is guided by calculations familiar from the traditional setting. For the minor arcs the problem reduces to a point counting problem for function fields over finite fields, where existing circle method techniques apply. circle method; étale cohomology; mapping space; rational curves; hypersurface Families, moduli of curves (algebraic), Applications of the Hardy-Littlewood method, Étale and other Grothendieck topologies and (co)homologies, Rational points A geometric version of the circle method | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Stable curves were introduced by \textit{P. Deligne} and \textit{D. Mumford} in Publ. Math., Inst. Haut. Étud. Sci. 36 (1969), 75-109 (1970; Zbl 0181.488). Their moduli space \(\bar {\mathcal M}_ g\) is a compactification of the classical moduli space \({\mathcal M}_ g\) of Riemann surfaces. Both spaces have dimension 3g-3 and the locus \({\mathcal D}=\bar {\mathcal M}_ g- {\mathcal M}_ g\) is the sum of \(1+[g/2]\) divisors of \(\bar {\mathcal M}_ g\). These define homology classes in \(H_{6g-8}(\bar {\mathcal M}_ g;{\mathbb{Q}})\); the Weil-Peterson Kähler form \(\omega\) on \({\mathcal M}_ g\) extends to a closed form on \(\bar {\mathcal M}_ g\) and so defines, by Poincaré duality, another class in \(H_{6g-8}(\bar {\mathcal M}_ g;{\mathbb{Q}})\). It is proved in this paper that these \(2+[g/2]\) cycles essentially form a basis of \(H_{6g-8}(\bar {\mathcal M}_ g;{\mathbb{Q}})\), if \(g>2\) (theorem 5.1). From this result, the author deduces interesting consequences about the form \(\omega\). The main technique is the construction of \(2+[g/2]\) analytic 2-cycles in \(\bar {\mathcal M}_ g\) and the computation of their intersection pairing with the above \(2+[g/2]\) classes in \(H_{6g-8}({\mathcal M}_ g;{\mathbb{R}})\), which turns out to be a non-singular pairing. This technique for constructing analytic cycles can also be used to construct higher dimensional homology classes. moduli space of stable curves; Weil-Peterson Kähler form; analytic cycles S. Wolpert. On the homology of the moduli of stable curves. \textit{Annals of Mathematics}, \textbf{118:2} (1983), 491-523 Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Cycles and subschemes On the homology of the moduli space 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 Let \(C\) be a smooth algebraic curve. The Mukai bundle \(E_C\) is a special stable rank-\(2\) bundle whose global sections map general canonical curves \(C\) of low genus into the Grassmannians as sections in the Plücker embedding. Given a prime number \(\ell\), the modular variety \(\mathcal R_{g,\ell}\) parameterizes pairs \([C,\eta]\) for nontrivial torsion line bundles \(\eta\) of order \(\ell\) on smooth curves \(C\) of genus \(g\). Inside \(\mathcal R_{g,\ell}\) the locus \(\mathcal B_{g,\ell}\) parameterizing \([C,\eta]\) with \(H^0(C, E_C\otimes \eta) \neq 0\) is a virtual divisor, i.e. it has expected codimension equal to one.
The main result of the paper shows that for both \(g=6\) and \(g=8\) and for every prime \(\ell\), the locus \(\mathcal B_{g,\ell}\) is an actual divisor, and in these cases the author also computes their divisor classes in certain partial compactification of \(\mathcal R_{g,\ell}\). As an application, the author shows that \(\mathcal R_{8,3}\) is of general type, improving previous results about the birational type of \(\mathcal R_{g,\ell}\) in [\textit{G. Bruns}, Algebra Number Theory 10, No. 9, 1949--1964 (2016; Zbl 1351.14018); \textit{A. Chiodo} et al., Invent. Math. 194, No. 1, 73--118 (2013; Zbl 1284.14006); \textit{G. Farkas} and \textit{K. Ludwig}, J. Eur. Math. Soc. (JEMS) 12, No. 3, 755--795 (2010; Zbl 1193.14043)]. Mukai bundle; level structures; moduli spaces of curves; effective divisor classes; birational type Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Rationality questions in algebraic geometry, Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification Twists of Mukai bundles and the geometry of the level 3 modular variety over \(\overline{\mathcal {M}}_{8}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Recent progress in the structure theory of higher-dimensional varieties has developed into different directions. One of them concentrates on the study of the geometry of curves on varieties: Based on classical geometric ideas, the book under review investigates rational curves on varieties of arbitrary dimension. Main techniques are the use of Hilbert and Chow schemes. This is worked out in chapter I in the necessary generality, taking into account especially the case of positive characteristic where the Chow-functor does not behave as well as in characteristic 0. Also, there are hints and references to the cases of algebraic and analytic spaces, respectively, as well as an appendix with some background from commutative algebra.
Chapter II applies the preceding techniques to study curves on varieties and morphisms from curves to varieties: rational curves yield a closed subvariety of the Chow variety. Néron-Severi's theorem is presented as a result of the investigation of the cone of curves. The chapter culminates in Mori's ``bend-and-break'' technique which is applied to show that for a projective variety \(X\) containing a curve having negative intersection with the canonical class, there exists such rational curve as well. Other applications are vanishing results in positive characteristic obtained using a construction of Ekedahl. The final section of this chapter is devoted to smoothing of morphisms of curves and contains results for reduced curves which are ``combs'' with rational ``teeth''.
The next chapter III of the book aims to give a short illustration of the ideas of Mori's minimal model program in some special cases. It contains a proof of the cone theorem following Mori's original arguments. There is a section devoted to the case of surfaces which presents the classical theory from the viewpoint of the minimal model theory, accompanied by an introduction to del Pezzo surfaces \(X\) with \((K_X^2)\leq 4\) over an arbitrary field. The following chapters constitute the main part of the book: Chapter IV on ``Rationally connected varieties'' studies birational properties of varieties covered by rational curves. There are important parallels between the class of uniruled and rationally connected varieties, some of their numerical properties being stable under smooth deformations (at least if we exclude the case of characteristic \(>0\); this is necessary for some parts of this chapter). The following topics are covered in different sections:
Ruled and uniruled varieties;
Minimizing families of rational curves;
Rationally connected varieties;
Growing chains of rational curves;
Maximal rationally connected fibrations;
Rationally connected varieties over nonclosed fields.
Chapter V on Fano varieties considers the higher-dimensional generalizations of del Pezzo surfaces which are of increasing interest for Mori's program. Though the singular case is not considered, this should be important for a theory of Fano varieties with terminal singularities as well. There are various assertions to support the following principle: ``The geometry of a Fano variety is governed by rational curves of low degree''. It is explained how (in principle) a list of all Fano varieties in any given dimension could be obtained. Further, there is a proof of Mori's theorem characterizing the projective space as the only algebraic variety with ample tangent bundle. The final sections study concrete cases; especially section 5 is devoted to nonrational Fano varieties.
The appendix (chapter VI) gives a proof of Abhyankar's theorem which says that any birational morphism \(X\to Y\) with \(Y\) smooth has only ruled exceptional divisors (without using resolution of indeterminacies). Further, there is an introduction to the intersection theory of divisors (following an unpublished paper of \textit{S. Kleiman}) which is applied to obtain the asymptotic Riemann-Roch theorem for nef line bundles.
This book will be of continuing interest as well as a source of reference as for its presentation of the material which is encouraging the reader to get through even with the technical demanding parts. It is (in a reasonable sense) self-contained and accompanied by many exercises. Mori theory; minimal models; Fano variety; Hilbert scheme; rational curves; Chow schemes; vanishing; positive characteristic; Mori's minimal model program; cone theorem; del Pezzo surfaces; Fano varieties Kollár, J., Rational Curves on Algebraic Varieties, (1995), Springer: Springer Berlin Minimal model program (Mori theory, extremal rays), Rational and ruled surfaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Parametrization (Chow and Hilbert schemes), Fano varieties, Rational and unirational varieties, Riemann-Roch theorems, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory Rational curves on algebraic varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a complex projective variety, and let \(Y\subset X\) be a smooth very ample hypersurface. Let \(\beta \in H_2(X)\) be a homology class, and let \(\alpha_1,\ldots, \alpha_n\) be a collection of nonnegative integers. Consider the moduli space of relative stable maps \(\overline{M}_{(\alpha_1,\ldots,\alpha_n)}^Y(X,\beta)\) consisting of all stable genus zero curves \(f:C\rightarrow X\) such that \(f\) cuts \(Y\) at the point \(x_i\) with multiplicity \(\alpha_i\). The relative Gromov-Witten invariants are defined using (the virtual fundamental classes of) these moduli spaces.
This paper proves the following equality of cycles in the Chow groups of the moduli spaces of relative stable maps
\[
\begin{multlined}(\alpha_k \psi_k + ev_{x_k}^* [Y]) \cdot [\overline{M}_{(\alpha_1,\ldots,\alpha_n)}^Y(X,\beta)]^{\text{virt}}= \\ =[\overline{M}_{(\alpha_1,\ldots,\alpha_k+1,\ldots, \alpha_n)}^ Y(X,\beta)]^{\text{virt}} +[D_{(\alpha_1,\ldots,\alpha_n),k}(X,\beta)]^{\text{virt}},\end{multlined}
\]
where \(D_{(\alpha_1,\ldots,\alpha_n),k}\) is the moduli space (of virtual codimension 1) of reducible stable curves in which the component containing the point \(x_k\) is included in \(Y\), and \(\psi_k\) is the cotangent class (the Euler class of the bundle which puts over each stable \((C,f)\) the cotangent space of \(C\) at \(x_k\)). Geometrically, this formula follows by requiring the curve \((C,f)\) to cut \(Y\) with multiplicity \(\alpha_k+1\) at \(x_k\).
The method of proof consists in embedding \(X\) in a projective space \(\mathbb{P}^N\) in such a way that the hyperplane section is \(Y\subset X\). The case of a hyperplane \(H\subset \mathbb{P}^N\) is studied in detail, and then the information is transferred to \(Y\subset X\) via pull-back. This is an extension of results found by \textit{R. Vakil} [J. Reine Angew. Math. 529, 101--153 (2000; Zbl 0970.14029)].
This result is used to prove that the Gromov-Witten invariants of \(X\) give all the relative invariants as well as all genus zero Gromov-Witten invariants of \(Y\) whose homology and cohomology classes are induced by \(X\). Gromov-Witten invariants; hypersurfaces, Chow groups Gathmann, A.: Gromov--Witten invariants of hypersurfaces. Habilitation thesis, University of Kaiserslautern Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Enumerative problems (combinatorial problems) in algebraic geometry, Hypersurfaces and algebraic geometry, (Equivariant) Chow groups and rings; motives Absolute and relative Gromov-Witten invariants of very ample 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 Let \(R_e(X)\) be the family of smooth rational curves of degree \(e\) on a hypersurface \(X\) of degree \(d\). The expected dimension, denoted by \(\mu\), of \(R_e(X)\) is equal to the Euler characteristic of the normal bundle on a smooth rational curve. In the paper under review, the author studies \(R_e(X)\) without restriction on the characteristic of the ground field. Under some conditions on \(d,e\) and \(\mu\), it is shown that \(R_e(X)\) is empty, smooth, or connected for general \(X\). Hilbert schemes; rational curves; hypersurfaces Katsuhisa Furukawa, Rational curves on hypersurfaces, J. Reine Angew. Math. 665 (2012), 157 -- 188. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Hypersurfaces and algebraic geometry Rational curves on 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 [For the entire collection see Zbl 0619.00007.]
This paper is related with classification of torsion-free modules over a reduced local ring R of a curve and generalizes in some sense the paper by \textit{G.-M. Greuel} and the author in Math. Ann. 270, 417-425 (1985; Zbl 0553.14011). The starting point is that any torsion-free R-module of rank n is isomorphic to a R-module M such that \(R^ n\subset M\subset \tilde R^ n\), and isomorphisms between two such modules are introduced by automorphisms of \(\tilde R^ n,\) where \(\tilde R\) is the integral closure of R in its total quotients ring. Following this way the author defines the ``number of parameters of torsion-free R-modules of rank n'': \(par_ n(R)\). Now consider a flat family of reduced curves \(f: X\to S,\) over a smooth curve S, and put for any \(s\in S\), \(Z_ n(s)=\sum_{x\in f^{-1}(s)}par_ n({\mathcal O}_{f^{-1}(s),x}).\) The main result is that \(s\mapsto Z_ n(s)\) is an upper semicontinuous function. singularities of curves; classification of torsion-free modules; number of parameters Knörrer, H. : Torsionfreie Moduln bei Deformation von Kurvensingularitäten , In: Singularities, Representation of Algebras and Vector Bundles, Lambrecht 1985 (Eds.: Greuel, G.-M.; Trautmann, G.). Lecture Notes in Math., Vol. 1273, Springer, Berlin-Heidelberg - New York (1987) pp. 150-155. Singularities of curves, local rings, Families, moduli of curves (algebraic), Deformations of singularities Torsionsfreie Moduln bei Deformation von Kurvensingularitäten. (Torsion free modules at deformation of singularities of curves) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes \(\operatorname{GU} (2, 1, \mathbb{Q} (\sqrt{ - 3}))\). The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula.
Along the way we prove new results on characteristic polynomials of Frobenius acting on the first cohomology group of cyclic triple covers of any genus, dimension formulas for spaces of Picard modular forms and formulas for the numerical Euler characteristics of the local systems. Picard modular form; Picard modular surface; Hecke eigenvalue; curves over finite fields; Harder conjecture Modular and automorphic functions, Arithmetic aspects of modular and Shimura varieties, Curves over finite and local fields, Families, moduli of curves (algebraic), Modular and Shimura varieties, Moduli, classification: analytic theory; relations with modular forms Picard modular forms and the cohomology of local systems on a Picard modular 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 \(\mathcal I_{d,g,r}\) denote the Hilbert scheme parametrizing smooth curves of degree \(d\) and genus \(g\) in complex projective space \(\mathbb P^{r}_{\mathbb C}\).
\textit{L. Ein} proved \(\mathcal I_{d,g,r}\) is irreducible if \(d \geq \frac{(2r-2)g+(2r+3)}{r+2}\) [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 469--478 (1986; Zbl 0606.14003); Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], which proves Severi's claim of irreducibility for \(d \geq g+r\). He also gave examples showing that Severi's claim fails for \(r \geq 6\).
In the paper under review, the author expands the known irreducibility range in case \(r=5\), showing that \(\mathcal I_{d,g,r}\) is irreducible for \(d \geq \max\{\frac{11}{10}g+2,g+5\}\). The outline of the proof runs as follows. If \(V \subset \mathcal I_{d,g,r}\) is any irreducible component, then \(V\) is generically a fibre bundle over a closed subset of a component \(\mathcal G \subset \mathcal G^{r,d}\) with fibre \(\text{Aut}(\mathbb P^{r})\) and there is \(\alpha \geq r\) such that \(\mathcal G\) is generically a fibre bundle over a closed subset of an irreducible component \(\mathcal W \subset \mathcal W^{\alpha,d}\) with fibre \({\roman Gr}(r,\alpha)\), thus we arrive at the inequality
\[
(r+1)d-(r-3)(g-1) \leq \mathcal W + r^{2}+2r+(r+1)(\alpha-r)
\]
(the spaces \(\mathcal G^{r,d}\) and \(\mathcal W^{\alpha,d}\) are the global versions of \(G^{r,d}(C)\) and \(W^{\alpha,d}(C)\) from the book of \textit{E. Arbarello, M. Cornalba, P. A. Griffiths} and \textit{J. Harris} [Geometry of Algebraic Curves, Grundlehren der Mathematischen Wissenschaften, 267. New York etc.: Springer-Verlag (1985; Zbl 0559.14017)]).
Now if the expected dimension \(\rho(d,g,r)=g-(r+1)(g-d+r)\) is positive, one constructs a unique irreducible component \(V \subset \mathcal I_{d,g,r}\) which dominates the moduli space \(\mathcal M_{g}\). If \(V_{1}\) were a different irreducible component, we would produce \(\mathcal W\) as above which does not dominate \(\mathcal M_{g}\), but then a series of estimates contradicts the inequality above. Hilbert scheme; Brill-Noether theory; linear series H. Iliev, On the irreducibility of the Hilbert scheme of curves in \(\mb{P}^5\), Comm. Algebra 36 (2008), no. 4, 1550--1564. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) On the irreducibility of the Hilbert scheme of curves in \(\mathbb P^{5}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies the cohomology of the Artin stack \(M_1\) parametrizing genus one curves \textit{without} a marked point. There is an obvious forgetful functor \(M_{1,1} \to M_1\), which has a left inverse \(J : M_1 \to M_{1,1}\) assigning to a genus one curve its Jacobian. With coefficients in \(\mathbb Q\) or even \(\mathbb Z[1/6]\) it is very easy to compute the cohomology of \(M_1\) using the Leray spectral sequence for \(J\) and the fact that the fiber of \(J\) over an elliptic curve \(E\) is the classifying stack \(BE\). The author therefore focuses on torsion, and computes the cohomology with \(\mathbb Z[1/2]\) degrees completely and with \(\mathbb Z\) coefficients in low degrees.
The cohomology of \(M_{1,1}\) is a direct summand of the cohomology of \(M_1\) and the author refers to the complement as ``dagger classes''. The dagger classes are cohomological obstructions to the existence of rational points on genus one curves, and the author investigates questions concerning vanishing and nonvanishing of these characteristic classes over various base schemes. Rationally, every class in positive degree is of course a dagger class. The author observes that for the natural mixed Hodge structure on \(H^k(M_1,\mathbb Q)\) each class has weight \(<k\), so for a family of genus one curves over a smooth base these characteristic classes will always vanish by Deligne's weight bounds. However, the author constructs explicitly singular schemes over which they are nonzero. The author also studies the \(\pmod 2\) cohomology, and shows that in this case the dagger classes can be nonzero also over smooth bases. moduli of curves; characteristic classes; torsors for elliptic curves; cohomology Taelman, L., Characteristic classes for curves of genus one, Michigan math. J., 64, 3, 633-654, (2015) Families, moduli of curves (algebraic), Elliptic curves Characteristic classes for curves of genus one | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The note follows a lecture given by the author during the school ``Liaison and related topics'' (Torino 2001). It refers the results already proved by the author in [Le Matematiche, LV, 517--531 (2000)] about the connectedness of Hilbert scheme of space curves of degree \(d\) and genus \((d-3)(d-4)/2 - 1\). With respect to that paper, a complete list of degrees for which smooth irreducible curves appear together with the corresponding Hartshorne-Rao modules. Plane and space curves, Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Families, moduli of curves (algebraic) A note on the Hilbert scheme of curves of degree \(d\) and genus \({d-3\choose 2}-1\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Picard group \(\mathrm{Pic}(R)\) of an \(E_\infty\)-ring spectrum \(R\) is the group of invertible objects in the homotopy category of \(R\)-modules.
\textit{A. Mathew} and \textit{V. Stojanoska}, [Geom. Topol. 20, No. 6, 3133--3217 (2016; Zbl 1373.14008)] introduced the Picard spectrum \(\mathfrak{pic}(R)\), a connective spectrum whose zeroeth homotopy group recovers \(\mathrm{Pic}(R)\). For a Galois extension of ring spectra \(A \to B\) with finite Galois group, they derive a descent spectral sequence relating the homotopy groups of \(\mathfrak{pic}(A)\) and \(\mathfrak{pic}(B)\), and use it to compute Picard groups related to topological modular forms.
In this paper, the authors work at the prime 2, and consider the homotopy fixed points of the Lubin-Tate spectrum \(E_n\) under a certain \(C_2\)-action, which is induced from the formal inverse of its formal group law via the Goerss-Hopkins-Miller theorem. They show that for all \(n \geq 1\), \(\mathrm{Pic}(E_n^{hC_2}) \cong \mathbb{Z}/2^{n+2}\), generated by \(\Sigma E_n^{hC_2}\) (Theorem 3.12). This is the first systematic computation at all chromatic heights. The height 1 case is a well-known unpublished result of Hopkins, which says that \(\mathrm{Pic}(\mathrm{KO}) \cong \mathbb{Z}/8\) - in this case, taking \(C_2\)-fixed points corresponds to the passage from complex to real \(K\)-theory.
Using Theorem 3.12, the authors show that the Gross-Hopkins dual \(IE_n\) of \(E_n\) is \(C_2\)-equivariantly equivalent to \(\Sigma^{4+n}E_n\) (Theorem 4.15). This time, the height 1 case is an equivariant refinement of the well-known result that \(IE_1^{hC_2} \simeq \Sigma^5 E_1^{hC_2}\). chromatic homotopy; Picard group; Brown-Comenetz; Morava; Gross-Hopkins; real \(K\)-theory Topological \(K\)-theory, Equivariant homotopy theory in algebraic topology, Picard groups, Duality in applied homological algebra and category theory (aspects of algebraic topology), Generalized (extraordinary) homology and cohomology theories in algebraic topology, Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) Picard groups and duality for real Morava \(E\)-theories | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Here we study the existence of codimension one subvarieties of \(\overline {\mathcal M}_g\) parametrizing curves with very low gonality with respect to balanced line bundles (i.e. the line bundles with multidegrees satisfying certain inequalities introduced by D. Gieseker and L. Caporaso). We also study the gonality sequence of certain hyperelliptic stable curves. stable curve; gonality; Brill-Noether theory; balanced line bundle Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic) On the gonality sequence of a stable 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 \(\bar {\mathcal M}_ g\) denote the Deligne-Mumford compactification of the moduli space \(\bar {\mathcal M}_ g\) of curves of genus g. Mumford has determined the Chow ring of \(\bar {\mathcal M}_ 2\) in terms of generators and relations. Let \(\lambda_ 2, \kappa_ 2\) denote the tautological classes in \(\Lambda^ 1(\bar {\mathcal M}_ 3)\) and \(\delta_ 0, \delta_ 1\) denote the \({\mathbb{Q}}\)-class of the boundary components \(\Delta_ 0\) and \(\Delta_ 1\) of \(\bar {\mathcal M}_ 3\). Then, the Chow ring of \(\bar {\mathcal M}_ 3\) (as the author shows) is generated by \(\lambda_ 2, \delta_ 0, \delta_ 1\) and \(\kappa_ 2\) modulo an ideal I generated by three relations in codimension 3 and six relations in codimension 4. The dimensions of the Chow groups are 1, 3, 7, 10, 7, 3, 1 and the pairing of the Chow groups in complementary dimensions is perfect. The proof is by an exhaustive case by case analysis. The ample divisor classes of \(\bar {\mathcal M}_ 3\) are also found.
[See also part II of this paper, ibid. 132, No.3, 421-449 (1990).] Deligne-Mumford compactification of the moduli space; curves of genus g; dimensions of the Chow groups Faber, C, Chow rings of moduli spaces of curves. I. the Chow ring of \(\overline{\mathcal M}_3\), Ann. Math. (2), 132, 331-419, (1990) Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles Chow rings of moduli spaces of curves. I: The Chow ring of \(\bar {\mathcal M}_ 3\) | 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, the structure of the Cartan prolongation for the family of plane curves \(x_{1}x_{2} = t\) (with nodal central member) is investigated. In this direction, the monster/Semple tower construction is used. As well, in order to illustrate the theoretical results, some examples are proposed. curve families; nodal singularity; vector distributions; prolongation Vector distributions (subbundles of the tangent bundles), Differential invariants (local theory), geometric objects, Normal forms on manifolds, Families, moduli of curves (algebraic), Plane and space curves Cartan prolongation of a family of curves acquiring a node | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 irreducible algebraic variety over the field \(k\). Then one can define its Picard group \(\text{Pic} (X)\). \(\text{Pic} (X)\) is a quotient of the divisor group \(\text{Div} (X)\), i.e. it is defined by the codimension one subvarieties of \(X\). For absolutely irreducible and complete \(X\) and the field extension \(k \subset F\) the natural map \(\text{Pic} (X) \to \text{Pic} (X \times_ kF)\) is injective. Also, in this case there exists an abelian variety \(J = \underline {\text{Pic}}^ 0_{X/k}\) defined over \(k\) which `represents' \(\text{Pic} (X)\) in the sense that there exists a \(\text{Gal} (\overline k/k)\)- equivariant short exact sequence
\[
0 \to J (\overline k) \to \text{Pic} (\overline X) \to NS(\overline X\to 0,
\]
where \(\overline X = X \times_ k \overline k\) and \(NS (\overline X)\) is an abelian group of finite type, the Néron-Severi group of \(X\). One also has various finiteness properties of the torsion part \(\text{Pic} (X)_{\text{tors}}\) of \(\text{Pic} (X)\), eventually depending on the nature of \(k\).
In higher codimension one tries to generalize the notion of the Picard group and one is led to define the Chow groups \(CH_ i (X)\), or, in case \(X\) is equidimensional of dimension \(d\), \(CH^ j(X) = CH_{d-j} (X)\). One has \(CH^ 1(X) = \text{Pic} (X)\). However, in codimension \(\geq 2\) one has no injectivity or representability results as for the Picard group. The main theme of the paper is to study torsion phenomena for \(CH^ 2\) and to give an overview of the author's results, his joint work with W. Raskind, J.-J. Sansuc and C. Soulé, work of P. Salberger, S. Saito and of N. Suwa. As one might expect, important ingredients such as the Merkurev-Suslin theorem and ideas of S. Bloch on algebraic cycles and algebraic \(K\)-theory are at the basis of these results. A basic theorem, following from the work of these people (and A. Ogus), says that for smooth \(X/k\) and an integer \(n\) invertible in \(k\), the subgroup \(_ nCH^ 2(X)\) of elements in \(CH^ 2(X)\) killed by multiplication by \(n\) is a subquotient of the étale cohomology group \(H^ 3_{\acute e t} (X, \mu^{\otimes 2}_ n)\), and for \(\ell\) prime to \(\text{char} (k)\), the group \(CH^ 2(X)_{\ell \text{ - tors}}\) is a subquotient of \(H^ 3_{\acute e t} (X, \mathbb{Q}_ \ell/ \mathbb{Z}_ \ell (2))\). This result (and the ideas behind it) is used at various places in the sequel where the torsion of \(CH^ i(X)\) \((i=2\) most of the time) is discussed for (i) \(k\) a separately closed field: (ii) \(k\) a finite field; (iii) \(k\) a number field; and (iv) \(k\) a local field, respectively.
In case (i) it is shown, among other things, that for smooth connected \(X\) of dimension \(d\) and \(n\) invertible in \(k\), \(_ nCH^ d(X)\) is finite. Also, Roitman's theorem on torsion zero-cycles is shown without reducing to the case of surfaces. The main result in case (ii) for smooth projective and geometrically connected \(X\) says that \(CH^ 2(X)_{\text{tors}}\) is finite. Here one uses a Hochschild-Serre spectral sequence and Deligne's theorem on the Weil conjecture with twisted coefficients. Also, a sketch of the proof of the theorem of Kato and Saito which says that for such \(X\), the group of degree zero 0-cycles modulo rational equivalence is finite and isomorphic to the geometric abelian fundamental group \(\pi_ 1^{ab, \text{geom}} (X)\) of \(X\), is given. In case (iii) the most general known result is proved in various steps. It is due to the author and Raskind, and Salberger. It says that for smooth projective geometrically connected \(X\) over the number field \(k\) with \(H^ 2(X, {\mathcal O}_ X) = 0\), \(CH^ 2 (X)_{\text{tors}}\) is finite. Saito has a different approach, also discussed in the underlying paper, to obtain this result. In case (iv), \(k\) a finite extension of \(\mathbb{Q}_ p\), one has the general result, due to the author, Sansuc and Soulé, establishing the finiteness of \(_ nCH^ 2(X)\) for any integer \(n>0\), where \(X\) is a smooth \(k\)-variety. Several other finiteness results, assuming that \(H^ 2 (X, {\mathcal O}_ X) = 0\), are discussed and proved. In the final section, again on varieties over number fields, Salberger's ideas on the finiteness of the exponent of \(CH^ 2 (X)_{\text{tors}}\) for \(X\) with \(H^ 2 (X, {\mathcal O}_ X) = 0\), used in the proof of the main result in case (iii), are explained. algebraic \(K\)-theory; Picard group; Néron-Severi group; Chow groups; Merkurev-Suslin theorem; algebraic cycles Jean-Louis Colliot-Thélène, Cycles algébriques de torsion et \?-théorie algébrique, Arithmetic algebraic geometry (Trento, 1991) Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, pp. 1 -- 49 (French). Algebraic cycles, Picard groups, Applications of methods of algebraic \(K\)-theory in algebraic geometry Torsion algebraic cycles and algebraic \(K\)-theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The study of intersection numbers of \(\psi\)-classes (called correlators) on the moduli space of stable curves with marked points has been a fundamental topic in the past several decades, which has produced remarkable applications in many fields, most notably through the Kontsevich-Witten theorem. Generating functions of these correlators often satisfy fascinating relations and can be studied recursively. However, their large genus asymptotic properties are not fully understood, despite of some special cases. In particular, Delecroix-Goujard-Zograf-Zorich conjectured about a limit behavior for large genus asymptotic of correlators in certain range of the genus and the number of marked points. This conjecture was motivated from computing dynamical invariants on moduli spaces of quadratic differentials (i.e., half-translation surfaces) such as their Masur-Veech volumes and (area) Siegel-Veech constants. The main result of this paper analyzes large genus asymptotics of correlators and proves the conjectural prediction on Masur-Veech volumes and Siegel-Veech constants for moduli spaces of quadratic differentials with simple zeros. Besides various combinatorial techniques, a novel idea is to compare with the jump probabilities of a certain asymmetric simple random walk. \(\psi\)-classes; intersection numbers; quadratic differentials; Masur-Veech volumes; large genus asymptotic Families, moduli of curves (algebraic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Differentials on Riemann surfaces Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We discuss the flatness property of some fiber type contractions of complex smooth projective varieties of arbitrary dimensions. We relate the flatness of some morphisms having one-dimensional fibers with their conic bundles structures, also in the general case in which some mild singularities of the varieties are admitted. flatness; fiber type contractions; conic bundles Rational and birational maps, Algebraic cycles, Families, moduli of curves (algebraic) A note on flatness of some fiber type contractions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathcal{M}_{g,n}\) denote the moduli stack of smooth \(n\)-marked curves of genus \(g\). Besides the well-known Deligne-Mumford compactification \(\overline{\mathcal{M}}_{g,n}\), there is the larger class of so-called \textit{Hassett-spaces}. These are modular compactifications \(\mathcal{M}_{g,n} \subset \overline{\mathcal{M}}_{g,w}\) depending on a weight vector \(w \in \mathbb{Q}^n \cap (0,1]^n\). The limiting objects are \textit{\(w\)-stable curves}, i.e. curves \(C\) for which \(K_C + \sum w_i p_i\) is ample and markings are allowed to coincide if their combined weight is at most 1. Just like we have a tropical analogue of \(\overline{\mathcal{M}}_{g,n}\) (the moduli space \(M_{g,n}^{\mathrm{trop}}\) of stable tropical \(n\)-marked curves of genus \(g\)), there are also tropical Hassett spaces \(M_{g,w}^{\mathrm{trop}}\). Denote the link of the cone point of \(M_{g,w}^{\mathrm{trop}}\) by \(\Delta_{g,w}\).
In the present paper, the authors study the automorphism group of \(\Delta_{g,w}\). For \(g \geq 1\) they identify \(\operatorname{Aut}(\Delta_{g,n})\) with the automorphism group of the symmetric \(\Delta\)-complex \(K_w\) with vertices \(1, \ldots, n\) which has a simplex joining \(S \subseteq \{1, \ldots, n\}\) if and only if \(\sum_{i \in S} w_i \leq 1\). This does not coincide with the automorphism group of the algebraic space \(\overline{\mathcal{M}}_{g,w}\), but rather the authors show that \(\operatorname{Aut}(\Delta_{g,w})\) is precisely the subgroup of \(\operatorname{Aut}(\overline{\mathcal{M}}_{g,w})\) preserving the locus of singular curves. Furthermore, in the rational case (i.e. \(g = 0\)) the authors give a partial result: \(\operatorname{Aut}(\Delta_{0,w}) = \operatorname{Aut}(K_w)\) is still true if \(w\) is heavy/light.
The technical ideas build on [\textit{S. Kannan}, Trans. Am. Math. Soc. 374, No. 8, 5805--5847 (2021; Zbl 1476.14105)], where the automorphism group of \(M_{g,n}^{\mathrm{trop}}\) was computed. tropical geometry; moduli space; weighted stable curves Foundations of tropical geometry and relations with algebra, Families, moduli of curves (algebraic), Combinatorial aspects of algebraic geometry Automorphisms of tropical Hassett 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 The author proves the existence of infinitely many components of the Noether-Lefschetz locus in degree \(d=4s\), \(s\gg0\). This contradicts a conjecture by J. Harris. exceptional component; Noether-Lefschetz locus Voisin, C.: Conterexample a une conjecture de J. Harris. C.R. Acad. Sci. Paris S?r. I Mat.313, 685-687 (1991) Picard groups Contrexemple à une conjecture de J. Harris. (A counterexample to a conjecture of J. Harris) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(k\) denote an algebraically closed field of characteristic \(p>0\). Let \(f(x)\in k[x]\) be a polynomial of degree \(m\), where \((m,p)=1\). Let \(C\) denote the Artin-Schreier curve \(y^p-y=f(x)\). Then \(C\rightarrow {\mathbb P}^1_k\) is a \(p\)-cyclic cover of the projective line (totally) ramified in exactly one point, the point over \(\infty\). Let \(G_\infty(f)\subset \text{Aut}_k(C)\) denote the inertia group at this point and let \(G_{\infty,1}(f)\) denote the wild inertia group at this point.
The authors review the results of \textit{H.~Stichtenoth} [Arch. Math. 24, 615--631 (1973; Zbl 0282.14007)] concerning these \(G_{\infty,1}(f)\), and then prove new results. They give an algorithm to compute \(G_{\infty,1}(f)\), and they give an explicit example using Maple. They extend Stichtenoth's bounds on the order of \(G_{\infty,1}(f)\). They show that if the genus of \(C\) is at least 2, then the class of groups that occur as a \(G_{\infty,1}(f)\) is exactly the class of finite groups that are extensions of a \(p\)-cyclic group by a finite elementary abelian \(p\)-group, and they also show that these groups are certain subgroups of extraspecial groups. Finally, they characterize those polynomials \(f(x)\) that give a maximum for \(| G_{\infty,1}(f)|/g(f)^2\), where \(g(f)=(m-1)(p-1)/2\). The authors remark that they plan to apply the results in this paper to a study of special fibers of semi-stable models for \(p\)-cyclic covers of the projective line over a \(p\)-adic field \(K\) with residue field \(k\). automorphism; Artin-Schreier cover; extraspecial group Lehr, C.; Matignon, M., Automorphism groups for \textit{p}-cyclic covers of the affine line, Compos. Math., 141, 1213-1237, (2005) Automorphisms of curves, Families, moduli of curves (algebraic), Computational aspects of algebraic curves, Finite nilpotent groups, \(p\)-groups Automorphism groups for \(p\)-cyclic covers of the affine line | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper serves essentially a three-fold purpose:
(1) it provides a very readable and gentle introduction to the key objects and definitions arising in the moduli spaces of tropical curves and abelian varieties, largely a summary of Mikhalkin et al.'s work [\textit{G. Mikhalkin}, Not. Am. Math. Soc. 54, No. 4, 511--513 (2007; Zbl 1142.14300); \textit{G. Mikhalkin} and \textit{I. Zharkov}, Contemp. Math. 465, 203--230 (2008; Zbl 1152.14028)] and the subsequent development and refinements by Caporaso, Viviani et al. [\textit{S. Brannetti} et al., Adv. Math. 226, No. 3, 2546--2586 (2011; Zbl 1218.14056); \textit{L. Caporaso} and \textit{F. Viviani}, Duke Math. J. 153, No. 1, 129--171 (2010; Zbl 1200.14025)], etc.;
(2) it puts forth a minor modification to a definition appearing in these latter works, namely that of a ``stacky fan,'' in order to fix a gap and render some basic claims in those papers correct;
(3) it uses computational methods, elementary combinatorics, and some deeper pre-existing combinatorial results (some intriguingly classical ones, in particular, pre-dating tropical geometry!) to prove some basic results on these tropical moduli spaces and provide explicit descriptions of their combinatorial structure in low genus.
Let us now elaborate slightly on these three aspects. The perspective taken on these tropical objects (curves and abelian varieties) and their moduli spaces is a now widely-accepted one that is both intrinsic yet ad hoc. In other words, tropicalization per se does not play a role, rather it is used simply to motive the definition of tropical curves simply as decorated graphs, and similarly for tropical abelian varieties. In this way, one has entirely combinatorial/polyhedral objects that assume a geometric significance through the tropical analogy with algebraic geometry. For instance, these objects have automorphisms so when parameterizing them one must introduce a type of combinatorial/polyhedral orbifold which is essentially something that is locally a balanced polyhedral complex (``tropical variety'') modulo a finite group action. This leads to an ad hoc definition of a stack in the tropical word (quite primitive in many ways, as descent and gluing play no role, one simply wants to remember automorphisms pointwise, families are not considered) that nonetheless seems adequate for many purposes. This ad hoc notion was introduced and termed a ``stacky fan'' (related to, though distinct from, the situation of toric varieties where a stacky structure can be encoded in the defining fan) by Caporaso and Viviani et al. in the above-cited papers, though in this paper Chan astutely points out a deficiency in their definition: it disallows the moduli spaces of tropical curves and tropical abelian varieties, what are supposed to be the principal examples of tropical stacks! Chan's modification is minor and still ad hoc, but suffices to correct this inadequacy. One hopes a more rigorous notion of stack/moduli space will eventually be introduced to the tropical literature, but for now the present definition still permits a nice combinatorial study of these objects. Along the way to proving that the standard constructions of the moduli spaces of tropical curves and abelian varieties are indeed stacky fans in this new sense, Chan provides some nice elementary examples of the objects/constructions.
In a more extensive example, Chan computes the poset structure of the tropical moduli spaces of curves for genus \(g \leq 5\). This is accomplished by a simple algorithm described in the paper, implemented in Mathematica, and with accompanying numerical data posted on the author's website. The paper also includes a combinatorial study of a locus in the moduli space of tropical abelian varieties containing the image of the tropical Torelli map (i.e., this locus includes in particular all tropical Jacobians of graphs). Computational methods return to describe the poset structure of the tropical Schottky locus, namely, the locus of tropical Jacobians. This is possible due to a convenient fact that these have a matroidal interpretation which ties them into pre-existing literature, and also due to a clever algorithmic approach described by Chan in this paper. Finally, the paper concludes with some preliminary investigation into rigidifying these tropical stacks by considering level structures. tropical geometry; tropical curves; metric graphs; Torelli map; moduli of curves; abelian varieties 10 M. Chan, 'Combinatorics of the tropical Torelli map', \textit{Algebra Number Theory}6 (2012) 1133-1169. , Families, moduli of curves (algebraic), Enumeration in graph theory Combinatorics of the tropical Torelli map | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper studies modular properties of nodal curves on general \(K3\) surfaces.
Let \((S,L)\) be a smooth, primitively polarized \(K3\) surface of genus \(p\geq 2\), with \(L\) a globally generated, indivisible line bundle with \(L^2=2p-2\). Let \({\mathcal{K}}_p\) denote the moduli space (or stack) of smooth primitively polarized \(K3\) surfaces of genus \(p\) which is of dimension \(19\).
For \(m\geq 1\), the arithmetic genus \(p(m)\) of the curves in \(|mL|\) is given by \(p(m)=m^2(p-1)+1\). Let \(\delta\) be an integer such that \(0\leq \delta\leq p(m)\). Consider the quasi-projective scheme \({\mathcal{V}}_{p,m,\delta}\) called the \((m,\delta)\)-universal Severi variety. There is the projection \({\mathcal{V}}_{p,m,\delta} \to {\mathcal{K}}_p\) whose fiber over \((S,L)\) is the variety \(V_{m,\delta}(S)\) called the Severi variety of \(\delta\)-nodal irreducible curves in \(|mL|\). The variety \(V_{m,\delta}(S)\) is smooth of dimension \(g:=p(m)-\delta\) (the geometric genus of any curve in \({\mathcal{V}}_{m,\delta}\)). There is the moduli map \(\psi_{m,\delta}: {\mathcal{V}}_{m,\delta}\to M_g\) where \(M_g\) is moduli space of genus \(g\)-curves. The purpose of this paper is to find conditions on \(p, m, \delta\) for the existence of an irreducible component \({\mathcal{V}}\) of \({\mathcal{V}}_{p,m,\delta}\) on which the moduli map \(\psi: {\mathcal{V}}\to M_g\) (with \(g=p(m)-\delta)\) has generically maximal rank differential.
The results are summarized in the following theorem.
{Theorem}. (A) For the following values of \(p\geq 3, m\) and \(g\), there is an irreducible component \({\mathcal{V}}\) of \({\mathcal{V}}_{m,\delta}\) such that the moduli map \({\mathcal{V}}\to M_g\) is dominant: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] \(m=1\) and \(0\leq g\leq 7\); \item[{\(\bullet\)}] \(m=2\), \(p\geq g-1\) and \(0\leq g\leq 8\); \item[{\(\bullet\)}] \(m=3\), \(p\geq g-2\) and \(0\leq g\leq 9\); \item[{\(\bullet\)}] \(m=4\), \(p\geq g-3\) and \(0\leq g\leq 10\); \item[{\(\bullet\)}] \(m\geq 5\), \(g\geq g-4\) and \(0\leq g\leq 11\).
\end{itemize}} (B) For the following values of \(p, m\) and \(\delta\), there is an irreducible component \({\mathcal{V}}\) of \({\mathcal{V}}_{m,\delta}\) such that he moduli map \({\mathcal{V}}\to M_g\) is generically finite into its image: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] \(m=1\) and \(p\geq g\geq 15\); \item[{\(\bullet\)}] \(2\leq m\leq 4\), \(p\geq 15\) and \(g\geq 16\); \item[{\(\bullet\)}] \(m\geq 5\), \(p\geq 7\) and \(g\geq 11\).
\end{itemize}} To prove this, it suffices to exhibit some specific curves in the universal Severi variety such that a component of the fiber of the moduli map at that curve has the right dimension, i.e., \(\min\{0,22-2g\}\). To do this, consider partial compactifications \(\overline{\mathcal{K}}_p\) and \(\overline{\mathcal{V}}_{m,\delta}\) and prove the assertion for curves in the boundary.
The partial compactification \(\overline{\mathcal{K}}_p\) is obtained by adding to \({\mathcal{K}}_p\) a divisor parametrizing pairs \((S,T)\) where \(S\) is a reducible \(K3\) surface of genus \(p\) that can be realized in \({\mathbb{P}}^p\) as the union of two rational normal scrolls intersecting along an elliptic normal curve \(E\), and \(T\) is the zero scheme of a section of the first cotangent sheaf \(T_S^1\), consisting of \(16\) points on \(E\). These \(16\) points together with subtle deformation argument of nodal curves establishes the assertion for \(m=1\), and for \(m>1\), specialization argument of curves in the Severi variety to suitable unions of curves used for \(m=1\) plus other types of limit curves establishes the assertions. moduli of curves; moduli maps; moduli spaces of \(K3\) surfaces; Severi varieties; deformations theory; degenerations Ciliberto, Ciro; Flamini, Flaminio; Galati, Concettina; Knutsen, Andreas Leopold, Moduli of nodal curves on \(K3\) surfaces, Adv. Math., 309, 624-654, (2017) Families, moduli of curves (algebraic), \(K3\) surfaces and Enriques surfaces, Structure of families (Picard-Lefschetz, monodromy, etc.), Projective techniques in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Moduli of nodal curves on \(K3\) 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 first part of this paper (pp. 305-323) surveys the fundamental concepts in topological field theory and related areas: Gerstenhaber algebras (\(G\)-algebras) (section 1). Topological vertex operator algebras (TVOA) in section 2, where the Lian-Zuckermann conjecture is stated in an updated form: does a TVOA carry a natural \(G_\infty\)-algebra structure? \(A_\infty\) and \(L_\infty\)-algebras (section 3). Homotopy \(G\)-algebras (section 4). The authors are interested in the homotopy \(G\)-algebra structure provided by the decorated moduli spaces \(\underline{\mathcal M}(n)\) [\textit{Murray Gerstenhaber} and \textit{Alexander A. Voronov}, Int. Math. Res. Not. 1995, No. 3, 141-153 (1995; Zbl 0827.18004)]: they choose to define \(G_\infty\)-algebras as the algebras over the associated Getzler-Jones' cellular operad \(K_\bullet\underline{\mathcal M}\).
Section 5 (pp. 323-327) contains the main result of the article. A reduced topological conformal field theory (reduced TCFT) can be viewed as an algebra over the smooth singular chain operad of the operad \(\underline{\mathcal M}\). The authors prefer to define a reduced TCFT on a complex \(V\) of vector spaces as a collection of operator-valued differential forms in \(\Omega^\bullet (\underline{\mathcal M}(n), \Hom(V^{\otimes n},V))\) (satisfying the expected conditions: equivariance, behaviour with respect to differentials and to the operad laws\dots). Integration over the cells provides a reduced TCFT with a structure of a \(G_\infty\)-algebra. Let now \(\mathcal P\) be the operad of moduli spaces of Riemann spheres with holomorphically embedded unit disks. A morphism of operads (string vertices) from \(\underline{\mathcal M}\) to \(\mathcal P\) can be constructed explicitely using a result of \textit{M. Wolf} and \textit{B. Zwiebach} [J. Geom. Phys. 15, No. 1, 23-56 (1994; Zbl 0860.53005)]. This construction is sketched pp. 326 [see also \textit{Takashi Kimura, Jim Stasheff} and \textit{Alexander A. Voronov}, Commun. Math. Phys. 171, No. 1, 1-25 (1995; Zbl 0844.57039)]. Since a TCFT may be defined as a complex \(V\) provided with a collection of forms in \(\Omega^\bullet ({\mathcal P}(n), \Hom (V^{\otimes n}, V))\) (satisfying the expected conditions: equivariance, behaviour with respect to differentials and to the operad laws...), the authors conclude from the definition of a reduced TCFT and from the string vertices construction that any TCFT is a reduced TCFT, hence admits a \(G_\infty\)-algebra structure.
The last section (pp. 327-331) contains observations and indications which relate the string vertices construction to double loop spaces and to Kontsevich's graph complex and weight systems. homotopy Gerstenhaber algebras; topological field theory; operad Kimura, T., Voronov, A.A., Zuckerman, G.J.: Homotopy Gerstenhaber algebras and topological field theory. In: Operads: Proceedings of Renaissance Conferences, J.-L. Loday, J. Stasheff, A.A. Voronov, (eds.), Contemporary Math. 202, Providence, RI: Am. Math. Soc., 1997, pp. 305--333 Secondary and higher cohomology operations in algebraic topology, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Knots and links in the 3-sphere, Vertex operators; vertex operator algebras and related structures, Algebraic topology on manifolds and differential topology, Applications of differential geometry to physics, Families, moduli of curves (algebraic) Homotopy Gerstenhaber algebras and topological field theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\to S\) be an analytic fiber bundle with fiber \(F\) of genus \(g\geq 2\) over a curve of genus \(g_b\). Let \(K^2=8(g_b-1) (g-1)\) and \(\chi= (g_b-1) (g-1)\). The aim of this paper is to study the subvariety parametrizing these fiber bundles in the moduli space \({\mathcal M}_{K^2,\chi}\) of surfaces of general type. The author proves the following theorem:
If \(g_b> (g+1)/2\), then the connected component \({\mathcal M}\) of \({\mathcal M}_{K^2,\chi}\) containing the modulus of \(X\), parametrizes precisely the surfaces admitting analytic fiber bundle structure with the same \(g,g_b\) and \({\mathcal G}\) where \({\mathcal G}=\text{Image} (\pi_1(S) \to\Aut (F))\). fiber bundles; surfaces of general type; moduli space H. Önsiper, On the moduli spaces of fiber bundles of curves of genusg. Arch. Math.75, 346--348 (2000). Families, moduli, classification: algebraic theory, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic) On the moduli spaces of fiber bundles of curves of genus \(\geqq 2\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For an affinoid algebra A over a non-archimedean valued complete and stable field k, this paper gives a method for the calculation of the Picard group Pic(A) of A. The main result is: ''If the reduction \(\bar A\) of A has no zero-divisors then the canonical map \(\beta:Pic(\bar A)\to Pic(A)\) is injective. If moreover \(\bar A\) is non-singular then \(\beta\) is an isomorphism.'' This result was already known for fields with a discrete valuation. There is a list of examples, with \(\bar A\) singular, for which the cokernel of \(\beta\) is calculated. In one example one uses the structure of the formal group of an elliptic curve over a valuation ring. affinoid space; rigid analytic geometry; non-archimedean valued ground field; affinoid algebra; Picard group; formal group of an elliptic curve E. Heinrich , M. van der PUT . '' Uber die Picardgruppen affinoider Algebren ''. Math Z. 186 , 9 - 28 ( 1984 ). Article | MR 735047 | Zbl 0543.14011 Local ground fields in algebraic geometry, Picard groups, Non-Archimedean valued fields, Formal groups, \(p\)-divisible groups, Special algebraic curves and curves of low genus Über die Picardgruppen affinoider Algebren | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 shows that cotangent bundles of moduli spaces of vector bundles over a Riemann surface are algebraically completely integrable Hamiltonian systems. More precisely, let G be a complex semisimple Lie group, let N be the moduli space of stable G-bundles with prescribed topological invariants on a compact Riemann surface and let n be the dimension of N. The cotangent space to N at the point represented by a G- bundle P is \(H^ 0(M;ad(P\otimes K))\) where ad(P) is the bundle associated to P via the adjoint representation of G on its Lie algebra g. Thus a choice of basis \(p_ 1,...,p_ k\) for the ring of invariant polynomials on g induces a holomorphic map \(\phi: T*N\to \oplus H^ 0(M;K^{d_ i})\) where \(d_ i\) is the degree of \(p_ i\). The components of \(\phi\) are n functionally independent Poisson-commuting functions on T*N, and when G is a classical group the generic fibre of \(\phi\) is an open set in an abelian variety on which the Hamiltonian vector fields defined by the components of \(\phi\) are linear. This is what it means to say that T*N is an algebraically completely integrable Hamiltonian system. The abelian varieties occurring are either Jacobian or Prym varieties of curves covering M. cotangent bundles of moduli spaces of vector bundles over a Riemann surface; completely integrable Hamiltonian systems; adjoint representation; Jacobian; Prym varieties N. Hitchin, \textit{Stable bundles and integrable systems}. Duke Math. J. 54 (1987), no. 1, 91--114. Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification Stable bundles and 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 main result of the paper is a complete classification of real smooth trigonal, maximally inflected curves of type I (i.e., real curves, whose complex point set is divided into two connected components by the real part) on rational ruled surfaces. The answer is given in terms of specific graphs, and the main tool is the dessins d'enfants techniques. This paper heavily relies on earlier works by \textit{A. Degtyarev} et al. [Am. J. Math. 130, No. 6, 1561--1627 (2008; Zbl 1184.14065)] and \textit{V. Zvonilov} [Vestnik Syktyvkarskogo Universiteta Ser. 1 Mat. Mekh. Inform. No. 6, 45--66 (2006)]. As application, the authors characterize real Jacobian elliptic surfaces of type I (i.e., with the real point set realizing the second Stiefel-Whitney class of the complex surface) that cover real rational ruled surfaces with ramification along the union of a real trigonal curve and the base divisor. real trigonal curves; real algebraic curves of type I; inflexion point; dessins d'enfants; real elliptic surfaces Degtyarev, A.; Itenberg, I.; Zvonilov, V., Real trigonal curves and real elliptic surfaces of type I, \textit{J. Reine Angew. Math}.,, \textbf{686}, 221-246, (2014) Topology of real algebraic varieties, Families, moduli of curves (algebraic), Dessins d'enfants theory, Rational and ruled surfaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations Real trigonal curves and real elliptic surfaces of type I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal H}_{g,n}\) be the Hurwitz space of meromorphic functions of degree \(n\) over a curve of genus \(g\), with the properties: the pole of each function are 1) simple, 2) numbered, and 3) the sum of the critical values is equal to zero. \({\mathcal H}_{g,n}\) is fibered over \({\mathcal M}_{g,n}\), the moduli space of curves of genus \(g\) with \(n\) marked points. \({\mathcal H}_{g,n}\) has a compactification \({\mathcal H}_{g,n}\) by stably meromorphic functions [see \textit{T. Ekedahl}, \textit{S. Lando}, \textit{M. Shapiro} and \textit{A. Vainshtein}, Invent. Math. 146, No. 2, 297--327 (2001; Zbl 1073.14041)].
In the present paper the authors study the cohomology ring of \(\overline{{\mathcal H}_{g,n}}\) modulo \(\mathbb{C}^*\)-action. They give explicit expressions for the classes determined by the subvarieties of functions with fixed ramification type, by means of a very small number of certain tautological classes: computations are based on \textit{R. Thom's} theory of universal polynomials for singularities [Comment. Math. Helv. 28, 17--86 (1954; Zbl 0057.15502)] recently extended to the case of multisingularities by the first author. They also compute the ``Hurwitz numbers''. M. E. Kazaryan and S. K. Lando, ''Towards an Intersection Theory on Hurwitz Spaces,'' Izv. Ross. Akad. Nauk, Ser. Mat. 68(5), 91--122 (2004) [Izv. Math. 68, 935--964 (2004)]. Coverings of curves, fundamental group, Families, moduli of curves (algebraic) On intersection theory on Hurwitz 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 We give an inductive proof that the generalized Severi varieties -- the varieties which parametrize (irreducible) plane curves of given degree and genus, with a fixed tangency profile to a given line at several general fixed points and several mobile points -- are irreducible. Plane and space curves, Families, moduli of curves (algebraic), Fibrations, degenerations in algebraic geometry The irreducibility of the generalized Severi varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this article is to report contents of the author's lecture delivered in the workshop ``Algebraic Number Theory and Related Topics 2016'' in Kyoto University. We survey some known facts about the Grothendieck conjecture for algebraic varieties of dimension greater than one, and then present the author's recent works on hyperbolic polycurves -- algebraic varieties in the form of successive fiberations by hyperbolic curves. In particular, we focus on a certain hyperbolic polycurve obtained as a finite étale cover of \(\mathcal{M}_{2,r}\), the moduli space of curves of genus two with ordered \(r\) marked points. anabelian geometry; hyperbolic polycurve; moduli of curves Coverings of curves, fundamental group, Families, moduli of curves (algebraic) Recent progress in higher-dimensional anabelian geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0628.00007.]
Let \(f: X\to S\) be a (degenerating) family of curves over a base scheme \(S=\text{Spec}(B)\), where B is a complete discrete valuation ring with algebraically closed residue field. Denote by s (resp. \(\eta\), resp. \({\bar \eta}\)) the closed (resp. generic, resp. geometric generic) point of S, assume that X is a regular scheme and the generic fibre \(X_{\eta}\) is smooth, and denote by \(\chi(X_ z):= \sum_{i}(-1)^ i\cdot \dim (H^ i_{et}(X_ z,{\mathbb{Q}}_{\ell}))\) the étale Euler characteristic of the fibre \(X_ z\) over \(z\in S\). The present paper is devoted to the problem of calculating the number \(\chi(X_ s)- \chi(X_{{\bar\eta}})\). In the case of characteristic zero (i.e., \(B=C[[t]])\), one can interprete this number as the local contribution to the cycle-theoretic self-intersection of the diagonal \(\Delta_ X\), i.e., \((\Delta_ x\cdot \Delta_ X)_ s= \chi(X_ s)- \chi(X_{{\bar\eta}})\) [cf. \textit{W. Fulton}, ``Intersection theory'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 2 (1984; Zbl 0541.14005)]. In the case of pure positive characteristic p, the global Grothendieck-Ogg-Šafarevič formula implies the modified formula \((\Delta_ X\cdot \Delta_ X)_ s= \chi(X_ s)- \chi(X_{{\bar\eta}})-sw(X/S)\), where the correction term sw(X/S) is just the so-called Swan conductor [cf. \textit{J.-P. Serre}, ``Corps locaux'' (Paris 1962; Zbl 0137.02601; see also the third edition, Paris 1980)]. Now the present paper provides a proof of the fact that the latter formula is also valid in the remaining case of mixed characteristic. A more general and detailed treatment of this topic is given in the author's subsequent article published in: Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, 421-450 (1987; see the review 14004). Chern classes; Chow groups; family of curves; Euler characteristic; cycle-theoretic self-intersection; Swan conductor Families, moduli of curves (algebraic), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Topological properties in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology Euler characteristics and Swan conductors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 primitive curves (of order \(n\)) were defined by Bănică and Forster in 1981 (manuscript). They are characterized [cf. \textit{C. Bănică, O. Forster}, in: Algebraic geometry, Proc. Lefschetz Centen. Conf., Mexico City/Mex. 1984, Part I, Contemp. Math. 58, 47--64 (1986; Zbl 0605.14026)] as nilpotent schemes \(Y\) embedded in a smooth threefold \(Z\), such that \(Y_{\mathrm{red}}=X\) is a smooth curve, \(Y\) having the property that the graded \({\mathcal O}_X\) -algebra associated to the Bănică -- Forster filtration has the form \({\mathcal B}(Y)=\bigoplus _{i=0}^{n-1} L^i\), with \(L\) a line bundle on \(X\). When \(n=2\) one has the so-called ``double curves''. The ``doubling'' of a curve was studied first by \textit{D. Ferrand} [C. R. Acad. Sci., Paris, Sér. A 281, 345--347 (1975; Zbl 0315.14019)]. The purely algebraic case was studied by several authors, but the final setting was achieved by \textit{R. Fossum} [Proc. Am. Math. Soc. 40, 395--400 (1973; Zbl 0271.13013)]. The double curves were studied as ``abstract'' schemes by \textit{D. Bayer} and \textit{D. Eisenbud} [Trans. Am. Math. Soc. 347, No. 3, 719--756 (1995; Zbl 0853.14016)].
In the paper under review one studies ``abstract'' primitive curves in general. The main idea is the following : take \(Z_n = \text{Spec}({\mathbb C}[t]/(t^n))\); the primitive curves of multiplicity \(n\) are then obtained by taking an open cover \(U_i\) of \(X\) and by glueing \(U_i \times Z_n\) conveniently (i.e. using automorphisms of \((U_i\cap U_j)\times Z_n\) leaving \(U_i\cap U_j\) invariant). The main tool is the study of the sheaf of nonabelian groups \({\mathcal G}_n\) of automorphisms of \(X\times Z_n\). The paper, although technical, is very well written and almost self-content. multiple structures; primitive multiple curves Drézet, J. M., Paramétrisation des courbes multiples primitives, Adv. Geom., 7, 559-612, (2007) Families, moduli of curves (algebraic) Parametrizations of primitive multiple 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 \(Y\) be a smooth, connected, projective complex curve of genus \(\geq 0\). \textit{R. Biggers} and \textit{M. Fried} [J. Reine Angew. Math. 335, 87--121 (1982; Zbl 0484.14002), Trans. Am. Math. Soc. 295, No. 1, 59--70 (1986; Zbl 0601.14022)] proved the irreducibility of the Hurwitz spaces which parametrize coverings of \(\mathbb{P}^1\) whose monodromy group is a Weyl of type \(W(D_{d})\). Here we prove the irreducibility of Hurwitz spaces that parametrize coverings of \(Y\) with monodromy group a Weyl group of type \(W(B_{d})\). F. Vetro, Irreducibility of Hurwitz spaces of coverings with monodromy groups Weyl groups of type W(Bd), Boll. Unione Mat. Ital., in press, preprint No. 279, April, 2005, Dipartimento di Matematica e Applicazioni, Università di Palermo Coverings of curves, fundamental group, Families, moduli of curves (algebraic) Irreducibility of Hurwitz spaces of coverings with monodromy groups Weyl groups of type \(W(B_d)\) | 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 The intermediate Jacobians of Fano threefolds have turned out to be highly interesting objects of study in algebraic geometry. Especially the geometry of their theta divisors is of greatest significance, above all in view of various concrete applications and constructions and the detailed study of manageable examples has been a popular topic of research in complex algebraic geometry over the past thirty years. One of the most beautiful examples, in this regard, is the so-called quartic double solid, i.e., a double cover of \(\mathbb{P}^3\) ramified in a quartic surface \(W\subset \mathbb{P}^3\). Since 1981, when \textit{G. Welters} [Abel-Jacobi isogenies for certain types of Fano threefolds, Math. Centre Tracts, 141, Mathematisch Centrum, Amsterdam (1981; Zbl 0474.14028)] exhibited a family of curves parametrizing the intermediate Jacqbian \(J(X)\) of a quartic double solid \(X\), many authors have contributed to a refined study of this example. In the paper under review, the authors give an improvement of previous results concerning the parametrization of the theta divisor \(\Theta\) of \(J(X)\), together with important applications to the study of certain moduli spaces of stable vector bundles on Fano threefolds. In 2003, \textit{A. S. Tikhomirov} [Acta Appl. Math. 75, 271--279 (2003; Zbl 1075.14044)] found a simple parametrization of that theta divisor \(\Theta\) by the family \(C^1_5(X)\) of elliptic quintics in \(X\). The advantage of this parametrization is that the elliptic quintics define stable vector bundles on \(X\) (via the Serre construction), by which the associated Abel-Jacobi map \(C_5^1(X)\to J(X)\) admits a factorization through the moduli space \(M(2;0,3)\) of the quartic double solid \(X\). The paper under review provides a detailed account of this previous result of Tikhomirov, whose proof was only sketched in the foregoing paper, and it even gives a substantial refinement of the precise statement. Namely, the main result of the present paper establishes the fact that the Abel-Jacobi map sends the family \(C^1_5(X)\) of elliptic quintics in the quartic double solid \(X\) onto an open subset of a translate of the theta divisor \(\Theta\) in \(J(X)\), and that the Serre construction defines a factorization
\[
C_5^1(X) @>\text{Serre}>> M@>g>> \Theta + \text{const}
\]
through a component \(M\) of the moduli space \(M(2;0,3)\) such that the map \(g\) is generically finite of degree 84. Moreover, it is shown that the family \(C_5^1(X)\) is indeed irreducible (within the corresponding Hilbert scheme). The fine analysis carried out in this beautiful, extremely comprehensive and lucid paper also demonstrates the enormous power of G. Welters's methods developed more than twenty years ago, and the fascinating appeal of concrete algebraic geometry likewise. intermediate Jacobians; Fano varieties; threefolds; theta divisors; families of curves; quartic double solids; moduli spaces of stable vector bundles; Abel-Jacobi map Markushevich, D.G., Tikhomirov, A.S.: A parametrization of the theta divisor of the quartic double solid. Int. Math. Res. Not. \textbf{2003}(51), 2747-2778 (2003) Picard schemes, higher Jacobians, \(3\)-folds, Fano varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves A parametrization of the theta divisor of the quartic 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 [For the entire collection see Zbl 0725.00006.]
A cubic graph is a graph whose vertices have valence three; if \(G\) is a cubic graph, one can define a unique stable curve \(X_ G\) from \(G\) as follows: \(X_ G\) is the union of \(\nu\) (= number of vertices) rational curves which meet in ordinary nodes according to the edges of \(G\). \(X_ G\) is a stable curve of genus \(g=\nu/2+1\). On a smooth curve \(X\), we have the Gaussian map \(\varphi\) from \(\bigwedge^ 2H^ 0(X,\omega_ X)\) to \(H^ 0(X,\Omega^ 1_ X\otimes\omega^ 2_ X)\), which can be defined also on stable curves, hence on curves such as \(X_ G\). The main result of the paper is to show that, when \(G\) is planar (i.e. embeddable in a 2- sphere) plus some extra hypotheses, the Gaussian map \(\varphi\) of \(X_ G\) has corank 1. --- The interest of such result lies in the fact that for the general curve of genus \(g\geq 12\) or \(g=10\), \(\varphi\) is surjective. On the other hand, this is not true for all curves, e.g. for the hyperelliptic ones having \(\text{coker} \varphi\) of dimension \(3g-2\). It would then be interesting to stratify \({\mathcal M}_ g\) into strata \({\mathcal M}_{g,k}\), where by \({\mathcal M}_{g,k}\) is the locus with \(\dim(\text{coker} \varphi)=k\).
Since curves like the \(X_ G\)'s appear as hyperplane sections of unions of planes, if one could show that such union can be smoothed to a \(K3\) surface \(S\), then the general curve on \(S\) would have a corank 1 Gaussian map. This would in turn imply that the generic element in \({\mathcal M}_{g,1}\) is a curve on a \(K3\) surface. moduli of curves; stable curve according to a cubic graph; Gaussian map; curve on a \(K3\) surface Rick Miranda, The Gaussian map for certain planar graph curves, Algebraic geometry: Sundance 1988, Contemp. Math., vol. 116, Amer. Math. Soc., Providence, RI, 1991, pp. 115 -- 124. Families, moduli of curves (algebraic), Graph theory, Special algebraic curves and curves of low genus, \(K3\) surfaces and Enriques surfaces The Gaussian map for certain planar graph curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present paper is closely based on the lecture given by the second author at The Unity of Mathematics symposium and is based on our joint work in progress on Gromov-Witten invariants with values in complex cobordisms. We will mostly consider here only the simplest example, elucidating one of the key aspects of the theory. We refer the reader to [\textit{A. Givental}, in: Frobenius Manifolds: Quantum Cohomology and Singularities, Aspects Math. E 36, 91--112 (2004; Zbl 1075.53091)] for a more comprehensive survey of the subject and to [\textit{T. Coates}, Riemann-Roch Theorems in Gromov-Witten Theory, Ph.D. thesis, Berkeley, 2003; \texttt{http://abel.math.harvard.edu/\(\sim\)tomc}] for all further details.
Consider \(\overline{\mathcal M}_{0,n}\), \(n\geq 3\), the Deligne-Mumford compactification of the moduli space of configurations of \(n\) distinct ordered points on the Riemann sphere \(\mathbb{C}\mathbb{P}^1\). Obviously, \(\overline{\mathcal M}_{0,3}=PT\), \(\overline{\mathcal M}_{0,4}=\mathbb{C} \mathbb{P}^1\), while \(\overline{\mathcal M}_{0,5}\) is known to be isomorphic to \(\mathbb{C}\mathbb{P}^2\) blown up at four points. In general, \(\overline {\mathcal M}_{0,n}\) is a compact complex manifold of dimension \(n-3\), and it makes sense to ask what is the complex cobordism class of this manifold. The Thom complex cobordism ring, after tensoring with \(\mathbb{Q}\), is known to be isomorphic to \(U^*=\mathbb{Q}[\mathbb{C} \mathbb{P}^1,\mathbb{C}, \mathbb{P}^2,\dots]\), the polynomial algebra with generators \(\mathbb{C} \mathbb{P}^k\) of degree \(-2k\). Thus our question is to express \(\overline {\mathcal M}_{0,n}\), modulo the relation of complex cobordism, as a polynomial in complex projective spaces.
This problem can be generalized in the following three directions. First, one can develop intersection theory for complex cobordism classes from the complex cobordism ring \(U^*(\overline{\mathcal M}_{0,n})\). Such intersection numbers take values in the coefficient algebra \(U^*=U^* (\text{pt})\) of complex cobordism theory. Second, one can consider the Deligne-Mumford moduli spaces \(\overline{\mathcal M}_{g,n}\) of stable \(n\)-pointed genus-\(g\) complex curves. They are known to be compact complex orbifolds and for an orbifold, one can mimic (as explained below) cobordism-valued intersection theory using cohomology intersection theory over \(\mathbb{Q}\) against a certain characteristic class of the tangent orbibundle. Third, one can introduce more general moduli spaces \(\overline{\mathcal M}_{g,n} (X,d)\) of degree-\(d\) stable maps from \(n\)-pointed genus-\(g\) complex curves to a compact Kähler (or almost-Kähler) target manifold \(X\). One defines Gromov-Witten invariants of more, using virtual tangent bundles of the moduli spaces of stable maps and their characteristic classes, one can extend Gromov-Witten invariants to take values in the cobordism ring \(U^*\). The Quantum Hirzebruch-Riemann-Roch theorem expresses cobordism-valued Gromov-Witten invariant of \(X\) in terms of cohomological ones. Cobordism-valued intersection theory in Deligne-Mumford spaces is included as the special case \(X= \text{pt}\). In the notes, we will mostly be concerned with the special case and with curves of genus zero, i.e., with cobordism-valued intersection theory in the manifolds \(\overline{\mathcal M}_{0,n}\). The cobordism classes of \(\overline{\mathcal M}_{0,n}\) than we seek are then interpreted as the self-intersections of the fundamental classes. Coates, T., Givental, A.: Quantum cobordisms and formal group laws, The unity of mathematics, Progr. Math., vol. 244, pp. 155-171. Birkhäuser, Boston (2006) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Bordism and cobordism theories and formal group laws in algebraic topology, Riemann-Roch theorems, Formal groups, \(p\)-divisible groups Quantum cobordisms and formal group laws | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Analogous to the polarized \(K3\) surfaces, in this paper, the Severi varieties of nodal curves on polarized abelian surfaces are studied. As the first main result, it has been shown that for a general abelian surface \(S\) with a polarization \(L\) of type \( (1, n) \), the Severi variety parametrizing \(\delta\)-nodal curves in \(|L|\) is nonempty and smooth of expected codimension \(\delta\). The second result computes the dimension of the locus \(\mathfrak{U}_{g,n}\subset \mathcal{M}_g\) of smooth curves of genus \(g\) possessing a \(\delta\)-nodal model as a hyperplane section of a \( (1, n) \)-polarized abelian surface. The major part of this paper deals with the Brill-Noether theory of curves in the linear series \(|L|\). In particular, the expected dimension of the Brill-Nother loci \(|L|^r_{\delta,d}\), parametrizing \(\delta\)-nodal curves whose stable model admits a linear series \(g^r_d\), and a necessary condition for their nonemptiness is provided. The proof of most results relies on degeneration to a semi-abelian surface constructed by identifying two sections of a \(\mathbb{P}^1\)-bundle over an elliptic curve. degeneration; Severi varieties; polarization; abelian surfaces; nodal curves; Brill-Noether theory Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves, Fibrations, degenerations in algebraic geometry, Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory), Projective techniques in algebraic geometry, Abelian varieties of dimension \(> 1\) Severi varieties and Brill-Noether theory of curves on abelian 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 \(B\) be an indefinite division quaternion algebra over \(\mathbb{Q}\), and let \(\Gamma\) be some principal congruence subgroup in the norm 1 elements of a maximal order in \(B\). The Shimura curve \(X/\mathbb{Q}\), which over \(\mathbb{C}\) is isomorphic to \({\mathfrak H}/\Gamma\), is the base space for a canonical family \({\mathcal A}\to X\) of abelian surfaces with quaternionic multiplication by \(B\). The author constructs complex multiplication cycles in the fibers of \({\mathcal A}\). For any \(r> 0\), he shows that the \(r\)th symmetric powers of these cycles span an infinite-dimensional subspace of the \(r\)th Griffiths group of the \(r\)th fibered power \({\mathcal A}^r\), and that therefore this group has infinite rank. -- This is an extension of work of \textit{C. Schoen} [Duke Math. J. 53, 771-794 (1986; Zbl 0623.14018)], and a first example of an infinite rank Griffiths group in codimension \(>2\). It is also mentioned that work of \textit{M. Nori} [Invent. Math. 111, 349-373 (1993; Zbl 0822.14008)] might be used to produce other examples. CM cycles; Shimura curves; Abel-Jacobi map in Hodge numbers; abelian surfaces with quaternionic multiplication; complex multiplication cycles; Griffiths group of infinite rank A. Besser, CM cycles over Shimura curves, J. Algebraic Geom. 4 (1995), no. 4, 659-691. Picard groups, Modular and Shimura varieties, Complex multiplication and abelian varieties, Arithmetic aspects of modular and Shimura varieties, Picard schemes, higher Jacobians CM cycles over Shimura curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We define the Weierstrass filtration for Teichmüller curves and construct the Harder-Narasimhan filtration of the Hodge bundle of a Teichmüller curve in hyperelliptic loci and low-genus nonvarying strata. As a result we obtain the sum of Lyapunov exponents of Teichmüller curves in these strata. Teichmüller curves; Lyapunov exponents; Harder-Narasimhan filtration F. Yu; K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents, J. Mod. Dyn., 7, 209-237, (2013) Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Teichmüller theory for Riemann surfaces, Families, moduli of curves (algebraic) Weierstrass filtration on Teichmüller curves and Lyapunov exponents | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 monograph (based on the author's Harvard dissertation) is concerned with the problem of compactification of the moduli space of principally polarised (\(g\)-dimensional) abelian varieties over \(\mathbb{Z}\). For elliptic curves \(E\), with \(j\) = the \(j\)-invariant of \(E\), the ''\(j\)-line'' \(\text{Spec}\mathbb{Z}[j]\) gives the moduli; the natural completion of the affine \(j\)-line \(\mathbb{A}^1\) is just \(\mathbb{P}^1\) and we have a 'canonical' solution for the compactification problem. When \(g>1\), the isomorphism classes of principally polarized abelian varieties correspond to points of the (coarse) moduli space \(A_ g\) of such abelian varieties; Mumford solved the classification problem, constructing a coarse moduli scheme \(A_ g\) over Spec \(\mathbb{Z}\), via his geometric invariant theory. Associated with the moduli scheme \(A_ g\), there exists a fundamental geometric problem underlying the need for the compactification: namely, for a given prime number \(p\),
\[
\begin{cases} \vtop{=.85 noindent is the geometric fibre \(A_ g \underset{\text{Spec}\mathbb{Z}}\times \text{Spec}\mathbb{F}_ p\) irreducible or equivalently, \smallskip\noindent is the moduli space of principally polarized abelian varieties irreducible, in characteristic \(p\)?} \end{cases} \tag{\(*\)}
\]
An affirmative answer to (\(*\)) for \(p>2\) is a consequence of the author's construction of toroidal completions of Siegel moduli schemes over \(\mathbb{Z}[\frac12]\) and extension (to positive characteristics) of a theorem of Tai on the ''projectivity of toroidal compactifications'' (substituting local holomorphic functions in Tai's proof with the algebraic machinery of theta functions). Let \(M(\mathbb{Z},k)\) denote the ring of Siegel modular forms of degree \(g\) and weight \(k\) with Fourier coefficients in \(\mathbb{Z}\), and \(R\) the graded ring \(\oplus_{k\geq 0} M(\mathbb{Z},k)\). For \(g=1\), it is well-known that \(R\) is finitely generated. The corresponding question for general \(g\) was raised by \textit{J. Iqusa} who also provided in a nice paper [Am. J. Math. 101, 149-183 (1979; Zbl 0415.14026)] an explicit set of generators for \(g=2\). The author's affirmative answer to (\(*\)) above (for \(p>2\)) implies that the graded ring of Siegel modular forms with Fourier coefficients in \(\mathbb{Z}[]\) is finitely generated over \(\mathbb{Z}[]\). --- A footnote on page xi refers to the question (\(*\)) of irreducibility for the case \(p=2\) having since been settled by Faltings.
Chapter I reviews the major results on Siegel moduli schemes used in subsequent chapters. The next chapter contains a treatment of semi-abelian varieties and with a vital definition of ''polarization'', the canonical construction of the quotient of a semi-abelian scheme by a discrete subgroup. These results are applied in chapter III to construct polarized semi-abelian schemes providing local coordinates, at the boundary, of toroidal completions of Siegel moduli schemes. Chapter IV contains the main result (theorem 4.2) of the author extending Tai's theorem to positive characteristics, and chapter V contains nice applications to Siegel modular forms.
There are three excellent appendices (dealing with theta functions); the results in the appendix on 2-adic theta functions with values in complete local fields \(k\) with a uniformization theorem for abelian varieties over k (with residue characteristic \(\neq2)\) are unpublished results due to Mumford. compactification of the moduli space of principally polarised; abelian varieties; coarse moduli scheme; toroidal completions of Siegel moduli schemes; semi-abelian varieties; 2-adic theta functions Chai, C.-L.: Compactification of Siegel moduli schemes. London Math. Soc. Lecture Note Series, vol. 107. Cambridge University Press (1985) Algebraic moduli problems, moduli of vector bundles, Algebraic moduli of abelian varieties, classification, Geometric invariant theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Structure of modular groups and generalizations; arithmetic groups, Families, moduli of curves (algebraic) Compactification of Siegel moduli schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is common to describe the deformation theory of algebraic objects by cohomological means; it is very interesting to note that this description is limited. Equivalence classes of infinitesimal deformations of a Lie algebra \({\mathfrak g}\) are in bijection to the Chevalley-Eilenberg cohomology space \(H^2({\mathfrak g},{\mathfrak g})\), second cohomology with adjoint coefficients. But Lie algebras with trivial \(H^2({\mathfrak g},{\mathfrak g})\) still may have non-trivial deformations. Examples for this phenomenon were given for the Lie algebra of meromorphic vector fields on the Riemann sphere which are holomorphic outside \(\{0,\infty\}\) in [\textit{A. Fialowski} and \textit{M. Schlichenmaier}, ``Global deformations of the Witt algebra of Krichever-Novikov type'', Commun. Contemp. Math. 5, No. 6, 921--945 (2003; Zbl 1052.17011)] and for current algebras of the form \({\mathbb C}[z,z^{-1}]\otimes {\mathfrak g}\) for a complex simple finite-dimensional Lie algebra \({\mathfrak g}\) in [\textit{A. Fialowski} and \textit{M. Schlichenmaier}, ``Global geometric deformations of current algebras as Krichever-Novikov type algebras'', Commun. Math. Phys. 260, No. 3, 579--612 (2005; Zbl 1136.17307)].
These examples illustrate the fact that the condition \(H^2({\mathfrak g},{\mathfrak g})=0\) implies infinitesimal and formal rigidity, but in general not rigidity (while for finite dimensional Lie algebra, one may conclude towards general rigidity).
The present article reviews several of these examples and discusses their geometric origin. Indeed, the deformations come from deforming the underlying algebraic curve with marked points. The point is that the Lie algebras of vector fields and the algebra of functions can be written down in terms of generators and relations for the explicit families of (marked) elliptic curves which degenerate to nodal or cuspidal cubics. An abstract way of capturing this transition from families of curves to deformations of Lie algebras of vector fields has been proposed in [\textit{F. Wagemann}, ``Deformations of Lie algebras of vector fields arising from families of schemes'', J. Geom. Phys. 58, No. 2, 165--178 (2008; Zbl 1175.17007)]. global deformations of Lie algebras; Lie algebras of meromorphic vector fields on marked Riemann surfaces; cuspidal cubic; nodal cubic; current Lie algebra; Krichever-Novikov Lie algebra Homological methods in Lie (super)algebras, Cohomology of Lie (super)algebras, Infinite-dimensional Lie (super)algebras, Lie algebras of vector fields and related (super) algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Virasoro and related algebras, Families, moduli of curves (algebraic), Elliptic curves Deformations of the Witt, Virasoro, and current algebra | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this article it is shown that any family of curves can be altered into a semi-stable family. This implies that if \(S\) is an excellent scheme of dimension at most 2 and \(X\) is a separated integral scheme of finite type over \(S\), then \(X\) can be altered into a regular scheme. This result is stronger than the author's previous results [ ``Smoothness, semi-stability and alterations'', Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996)]. In addition we deal with situations where a finite group acts. family of curves; alterations; group actions de Jong, A. Johan, Families of curves and alterations, Université de Grenoble. Annales de l'Institut Fourier, 47, 599-621, (1997) Families, moduli of curves (algebraic) Families of curves and alterations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 conjecture by \textit{R. F. Coleman} [Torsion points on curves. Adv. Stud. Pure Math. 12, 235--247 (1987; Zbl 0653.14015)], still unsolved, can be studied by geometric means by considering the moduli space of Jacobians (called the Torelli locus) inside the moduli space of principally polarized abelian varieties. Using the André-Oort conjecture, now,a theorem in this case, see Chapter 3, this leads to the study of special subvarieties generically contained in the open Torelli locus, see Expectation 2.3 in this chapter. Both this expectation and the Coleman conjecture are challenging open problems. Families, moduli of curves (algebraic), Torelli problem Special subvarieties in the Torelli locus | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0509.00008.]
Let C be the rational normal curve of degree n in \(P^ n\), and consider the osculating flag \(F(C,p)=\{F^ 0(C,p)\subset...\subset F^{n- 1}(C,p)\}\) to C at the point \(p\in C\), where \(F^ k(C,p)\) is the intersection of all hyperplanes of \(P^ n\) meeting \(C\geq k+1\) times (it turns out that \(F^ k(C,p)\) is a k-plane). If \(n-k\geq a_ 0\geq...\geq a_ k\geq 0\), then define the Schubert cycle \(\sigma_{a_ 0,...,a_ k}(p)\) by \(\{all\quad k-planes\quad V\subset P^ n| \dim(V\cap F^{n-k-a_ i+i}(C,p)\geq i \quad for\quad i=0,1,...,k\}.\) In this note the author proves the following result obtained jointly with J. Harris [and used by them in their investigation of the Brill-Noether problem via rational curves with ordinary cusps (instead of the former approach via rational curves with ordinary nodes of Griffiths and Harris); see the author and \textit{J. Harris} [Invent. Math. 74, 371-418 (1983; Zbl 0527.14022)]]. - Theorem. Let k be an integer such that 0\(\leq k\leq n-1\), let \(p_ 1,...,p_ g\) be g points of C and for \(i=1,...,g\) let \(\Sigma_ i\) be any Schubert cycle \(\sigma_{a_ 0^{(i)},...,a_ k^{(i)}}(p_ i)\) defined with the reference to the osculating flag \(F(C,p_ i)\) as above. If the points \(p_ 1,...,p_ g\) are distinct, then either \(co\dim(\cap^{g}_{i=1}\Sigma_ i)=\sum^{g}_{i=1}co\dim(\Sigma_ i)\), or \(\cap^{g}_{i=1}\Sigma_ i=\emptyset,\) the latter case being precisely when the product of the corresponding cohomology classes is 0. A dual form of this result was obtained in 1973 by \textit{A. Iarrobino} (but never published). As an immediate consequence of the above theorem one gets that the dimension of the space of nondegenerate rational curves of degree n in \(P^ r\) having \(\geq d\) ordinary cusps is \((r+1)(n+1)-4- (r-1)d\) if \(d\leq(n-r)(r+1)/r,\) and the space is empty otherwise. moduli of curves; rational curves with cusps; osculating flag; Schubert cycle; Brill-Noether problem Singularities of curves, local rings, Families, moduli of curves (algebraic), Enumerative problems (combinatorial problems) in algebraic geometry, Rational and unirational varieties, Singularities in algebraic geometry Rational curves with cusps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Soient \(V\) une variété de Fano et \(H\) un système de hauteurs d'Arakelov définissant un accouplement entre le groupe de Picard \(\text{Pic} V\) et les points rationnels de \(V\) à valeur dans \(\mathbb{R}\). Soit \(\zeta_H\) la fonction zêta associée sur \(\text{Pic} V \otimes\mathbb{C}\). Batyrev, Manin et Tschinkel [woir \textit{J. Franke}, \textit{Yu. I. Manin} et \textit{Yu. Tschinkel}, Invent. Math. 95, No. 2, 421-435 (1989; Zbl 0674.14012); \textit{V. V. Batyrev} et \textit{Yu. I. Manin}, Math. Ann. 286, No. 1-3, 27-43 (1990; Zbl 0679.14008) and \textit{Yu. I. Maninini}, Proc. Conf. Geometry Topology, Cambridge 1993, Surv. Differ. Geom. Suppl. J. Differ. Geom. 2, 214-245 (1995; Zbl 0847.14015)] ont conjecturé que cette fonction est holomorphe sur un cône de sommet le faisceau anticanonique \(\omega^{-1}_V\). Il est en outre possible de donner une expression conjecturale du terme principal de cette fonction \(\zeta_H\) au voisinage de ce sommet. Le but de ce texte est de montrer comment cette expression conjecturale peut s'écrire naturellement en passant aux torseurs universels au-dessus de \(V\). Fano variety; Picard group; zeta function; height; universal torror; Zariski dense rational points E. Peyre, ''Terme principal de la fonction zêta des hauteurs et torseurs universels,'' in Nombre et Répartition de Points de Hauteur Bornée, , 1998, vol. 251, pp. 259-298. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Picard groups, Arithmetic varieties and schemes; Arakelov theory; heights, Heights, Rational points Principal term of the zeta function of heights and universal torsors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(S_ n\) be the set of isomorphism classes of complex curves of genus \(2\) having a ``maximal'' map of degree \(n\) onto an elliptic curve (maximal means that the map does not factorize through an unramified covering of the elliptic curve). The purpose of this article is to prove that the class of a curve belongs to \(S_ n\) if and only if the class of its Jacobian belongs to the Humbert surface \(H_{n^ 2}\) in the moduli space of principally polarized abelian surfaces. This amounts to proving that it has a period matrix of the form \((\begin{smallmatrix} z_ 1 & 1/n & 1 & 0 \\ 1/n & z_ 2 & 0 & 1 \end{smallmatrix} )\) with \(\text{Im} z_ 1\), \(\text{Im} z_ 2 > 0\). moduli space of curves; covering of the elliptic curve; Jacobian; Humbert surface; moduli space of principally polarized abelian surfaces; period matrix Murabayashi, N.: The moduli space of curves of genus two covering elliptic curves. Man. Math. 84, 125--133 (1994) Elliptic curves, Families, moduli of curves (algebraic), Algebraic moduli of abelian varieties, classification, Coverings of curves, fundamental group The moduli space of curves of genus two covering 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 If \({\mathcal F}\) is a coherent sheaf on \(\mathbb{P}_2(\mathbb{C})\), pure of dimension 1, let \({\mathcal F}^\vee= \underline{\text{Ext}}^1({\mathcal F}, \omega_{\mathbb{P}_2})\). A theta-characteristic on \(\mathbb{P}_2\) is a pair \(({\mathcal F}, \sigma)\), where \({\mathcal F}\) is a coherent sheaf on \({\mathbb{P}}_2\), pure of dimension 1, and \(\sigma:{\mathcal F}\to {\mathcal F}^\vee\) is a symmetric isomorphism. The moduli space \(\Theta(d)\) of semistable theta-characteristics on \(\mathbb{P}_2\) of multiplicity \(d\) has been constructed by the author [\textit{C. Sorger}, Reine Angew. Math. 435, 83-118 (1993; Zbl 0757.14024)]. If \(d\) is odd, \(\Theta(d)\) has a trivial irreducible component \(\Theta_{\text{can}}(d)\) containing the canonical theta-characteristics, i.e. of the form \({\mathcal O}_C((d-3)/2)\), where \(C\) is a curve in \(\mathbb{P}_2\) of degree \(d\). If \(d\geq 3\) there are two other irreducible components, \(\Theta_0(d)\) and \(\Theta_1(d)\). The first [respectively, second] one corresponds to semistable noncanonical pairs \(({\mathcal F},\sigma)\) such that \(h^0( {\mathcal F})\) is even [respectively, odd].
In this paper the author computes the Picard groups of \(\Theta_0(d)\) and \(\Theta_1(d)\): They are free and except for small values of \(d\) they are of rank 2. He proves that for \(d\neq 3,5\) the moduli space \(\Theta_0(d)\) is not locally factorial, by finding a suitable open subset \(U\) of \(\Theta_0(d)\) whose complement has codimension at least 2, and a generator \(L\) of \(\text{Pic} (\Theta_0(d))\) such that \(L_{|U}\) has a square root \(L'\): The line bundle \(L'\) on \(U\) cannot be extended to \(\Theta_0(d)\). He has similar results for \(\Theta_1(d)\).
Suppose that \(d\geq 3\). Let \(\Theta^{ \text{reg}}(d)\subset \Theta(d)\) be the smooth open subset consisting of pairs \(({\mathcal F},\sigma)\) such that \({\mathcal F}\) is stable. The author proves that if \(d\) is even then there is no universal family of theta-characteristics on \(U \times\mathbb{P}_2\). If \(d\) is odd it is only possible to define locally such a universal family. coherent sheaf; theta-characteristic; Picard groups Picard groups, Theta functions and curves; Schottky problem, Algebraic moduli problems, moduli of vector bundles, Theta functions and abelian varieties The Picard group of the variety of theta-characteristics 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 The author shows how one can use elementary combinatorial arguments to obtain explicit formulae and relations for the universal cotangent classes \(\psi_p\) and the Mumford classes \(\kappa_m\) on the moduli spaces of stable curves of genus 0 and 1.
More precisely, in genus 0 it is known [\textit{S. Keel}, Trans. Am. Math. Soc. 330, No.2, 545--574 (1992; Zbl 0768.14002)] that any cohomology class in \(H^*(\overline{\mathcal M}_{0,n};{\mathbb Q})\) can be written as a linear combination of monomials in Keel's boundary classes \(\delta_{0,S}\). Using the formula \(\psi_p=\sum_{q_1,q_2\notin S\ni p}\delta_{0,S}\), and the recursive relations \(\kappa_m=\pi_p^*(\kappa_m)+\psi_p^m\) and \(\pi_p^*(\delta_{0,A})=\delta_{0,A}+\delta_{0,A\cup \{p\}}\) from \textit{E. Arbarello} and \textit{M. Cornalba} [Publ. Math., Inst. Hautes Étud. Sci. 88, 97--127 (1998; Zbl 0991.14012)], the author is able to determine explicit formulae expressing \(\psi_p^m\) and \(\kappa_m\) as linear combinations of monomials in the Keel classes. These formulae are then described in terms of the cohomology of the De Concini-Procesi model for the hyperplane arrangement of type \(A_{n-1}\).
In genus 1, explicit formulae for \(\psi_p^{m}\) and \(\kappa_m\) are found by means of the gluing morphisms \(\xi_{\text{irr.}}: \overline{\mathcal M}_{0,P\cup\{q_1,q_2\}}\to \overline{\mathcal M}_{1,P}\) and \(\xi_S: \overline{\mathcal M}_{0,S\cup\{r_1\}}\times\overline{\mathcal M}_{1,(P\setminus S)\cup\{r_2\}}\to \overline{\mathcal M}_{1,P}\), using the formula \(\psi_p={1\over {24}}\xi_{\text{irr.},*}(1)+ \sum_{S\ni p,| S| \geq 2}\xi_{S,*}(1)\) [\textit{E. Arbarello} and \textit{M. Cornalba}, loc. cit.] and the recursive relation \(\kappa_m={1\over {24}}\xi_{\text{irr.},*}(\kappa_{m-1})+ \sum_{| S| \geq 2}\xi_{S,*}(\kappa_{m-1}\otimes 1)\) from \textit{A. Kabanov} and \textit{T. Kimura} [Commun. Math. Phys. 194, No. 3, 651--674 (1998; Zbl 0974.14018)]. moduli of curves; cohomology; stable graphs Families, moduli of curves (algebraic), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Combinatorics of Mumford-Morita-Miller classes in low 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 It is known that the moduli space of plane quartic curves is birational to an arithmetic quotient of a six-dimensional complex ball [\textit{S. Kondo}, J. Reine Angew. Math. 525, 219--232 (2000; Zbl 0990.14007)]. In this paper, we shall show that there exists a 15-dimensional space of meromorphic automorphic forms on the complex ball, which gives a birational embedding of the moduli space of plane quartics with level-2 structure into \(\mathbb P^{14}\). This map coincides with the one given by \textit{A. Coble} [Algebraic geometry and theta functions. American Mathematical Society, New York (1929; JFM 55.0808.02)] by using Göpel invariants. Kondo, S, Moduli of plane quartics, Göpel invariants and borcherds products, Int. Math. Res. Not., 2011, 2825-2860, (2011) Moduli, classification: analytic theory; relations with modular forms, Relations with algebraic geometry and topology, Families, moduli of curves (algebraic), \(K3\) surfaces and Enriques surfaces, Plane and space curves Moduli of plane quartics, Göpel invariants and Borcherds products | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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, two projective structures canonically associated to a compact Riemann surface are compared, one provided by the uniformization theorem and the other from a meromorphic \(2\) form previously constructed by \textit{A. Ghigi} and two of the current authors [Int. J. Math. 26, No. 1, Article ID 1550005, 21 p. (2015; Zbl 1312.14076)]. The main result is that these two projective structures differ and it relies on the study of the \((0,1)\)-component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves. This had been computed previously for the projective structure given by the uniformization theorem by \textit{P. G. Zograf} and \textit{L. A. Takhtadzhyan} [Math. USSR, Sb. 60, No. 2, 297--313 (1988; Zbl 0663.32017); translation from Mat. Sb., Nov. Ser. 132(174), No. 3, 304--321 (1987)], and it is now computed in the paper for the other involved canonical structure. projective structure; moduli space; Weil-Petersson form; Siegel form Families, moduli of curves (algebraic), Projective connections, Analytic theory of abelian varieties; abelian integrals and differentials, Jacobians, Prym varieties A Hodge theoretic projective structure on compact Riemann surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Fix integers \(r\ge 3\), \(g\ge 0\) and \(d\) such that \(\rho (g,r,d):= (r+1)d- rg -r(r+1)\ge 0\). It is well-know that there is a unique irreducible component \(\Gamma\) of the Hilbert scheme of smooth curves \(C\subset \mathbb {P}^r\) parametrizing curves with general moduli and that a general \(C\in \Gamma\) has maximal rank in the range of quadrics, i.e. \(h^0(\mathcal{I} _C(2)) =\max \{0,\binom{r+2}{2} -2d-1+g\}\). In this paper the author study the stratification by ranks of the set of quadric hypersurfaces containing \(C\), giving a conjectural number of the dimension with fixed rank and proving it when \(d\ge g+r-1\). He use this theorem to construct new algebraic cohomology classes in \(\overline{\mathcal{M}}_{g,n}\) proving that \(\overline{\mathcal{M}}_{15,9}\) is of general type. moduli space; singular quadrics; Hilbert scheme of curves; Brill-Noether theory; Kodaira's dimension Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory) Variety of singular quadrics containing a projective curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, we initiate a systematic study of entanglements of division fields from a group theoretic perspective. For a positive integer \(n\) and a subgroup \(G\subseteq \operatorname{GL}_2( \mathbb{Z}/n\mathbb{Z})\) with surjective determinant, we provide a definition for \(G\) to represent an \((a,b)\)-entanglement and give additional criteria for \(G\) to represent an explained or unexplained \((a,b)\)-entanglement. Using these new definitions, we determine the tuples \(((p,q),T)\), with \(p<q\in \mathbb{Z}\) distinct primes and \(T\) a finite group, such that there are infinitely many non-\( \overline{\mathbb{Q}} \)-isomorphic elliptic curves over \(\mathbb{Q}\) with an unexplained \((p,q)\)-entanglement of type \(T\). Furthermore, for each possible combination of entanglement level \((p,q)\) and type \(T\), we completely classify the elliptic curves defined over \(\mathbb{Q}\) with that combination by constructing the corresponding modular curve and \(j\)-map. elliptic curves; division fields; entanglement; modular curves Elliptic curves over global fields, Galois representations, Families, moduli of curves (algebraic) A group theoretic perspective on entanglements of division 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 \(X\) be a smooth projective variety over the complex numbers and \(N_1(X)\) be the real vector space
\[
N_1(X)=_{\text{def}}\{1\text{-cycles on \(X\) modulo numerical equivalence}\}\otimes \mathbb{R}.
\]
Denote by \(NE(X)\) the cone of curves on \(X,\) i.e. the convex cone in \(N_1(X)\) generated by the effective 1-cycles. One knows by the cone theorem that it is rational polyhedral whenever \(c_1(X)\) is ample. The purpose of this paper is to study the cone of curves of abelian varieties. Specifically, the author focusses on the problem of determining the abelian varieties \(X\) such that the closed cone \(\overline{NE}(X)\) is rational polyhedral. Let \(\text{Nef}(X)\) be the nef cone and \(NS^{+}(X)\) is the semi-group of homology classes of effective line bundles, i.e. the subset
\[
NS^{+}(X)=_{\text{def}} \{\lambda\in NS(X)\mid \lambda =c_1(L)\text{ for some }L\in \text{Pic}(X)\text{ with }h^0(X,L)>0\}
\]
of the Néron-Severi group of \(X.\)
Theorem. Let \(X\) be an abelian variety over the field of complex numbers. Then the following conditions are equivalent:
(ia) The closed cone of curves \(\overline{NE}(X)\) is rational polyhedral.
(ib) The nef cone \(\text{Nef}(X)\) is rational polyhedral.
(ic) The semi-group \(NS^{+}(X)\) is finitely generated.
(ii) \(X\) is isogenous to a product \(X_1\times\cdots\times X_r\) of mutually non-isogenous abelian varieties \(X_i\) with \(NS(X_i)\cong {\mathbb Z} \) for \(1\leq i\leq r.\) abelian variety; cone of curves; Néron-Severi group; cone theorem \beginbarticle \bauthor\binitsT. \bsnmBauer, \batitleOn the cone of curves of an Abelian variety, \bjtitleAmer. J. Math. \bvolume120 (\byear1998), no. \bissue5, page 997-\blpage1006. \endbarticle \OrigBibText T. Bauer. On the cone of curves of an abelian variety. American Journal of Mathematics , 120(5), 1998. \endOrigBibText \bptokstructpyb \endbibitem Isogeny, Algebraic cycles, Plane and space curves, Picard groups On the cone of curves of an abelian variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author gives sufficient conditions for the existence of the torsion component of the Picard functor of an algebraic stack, and for the finite generation of the Néron-Severi groups or of the Picard group itself. For this he proves a stacky version of the relative representability theorem from [\textit{P. Berthelot, A. Grothendieck} and \textit{L. Illusie} (eds.), Séminaire de géométrie algébrique du Bois Marie 1966/67. Lect. Notes Math. 225 (1971; Zbl 0218.14001), exp.XII]. A special care was taken to avoid superfluous tameness assumptions. Because of the fact that an algebraic stack might have infinite cohomological dimension, it is often convenient to assume that the algebraic stacks under consideration are tame. In Appendix A, the author shows that the cohomology of an arbitrary algebraic stack is tractable as soon as the base scheme is regular and noetherian. As a byproduct he get the semicontinuity theorem for algebraic stacks. algebraic stacks; Picard functor; Néron-Severi groups; torsion component; semicontinuity theorem Brochard, S, Finiteness theorems for the Picard objects of an algebraic stack, Adv. Math., 229, 1555-1585, (2012) Generalizations (algebraic spaces, stacks), Picard groups, (Equivariant) Chow groups and rings; motives Finiteness theorems for the Picard objects of an algebraic stack | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies the cohomology of various moduli spaces of curves of genus \(3\) with level \(2\) structures, including the moduli space \(\mathcal M_3[2]\) of genus \(3\) curves with level \(2\) structure, the moduli space \(\mathcal M_{3,1}[2]\) of genus \(3\) curves with level \(2\) structure and one marked point, and the moduli space \(\mathcal{H}ol_3[2]\) of genus \(3\) curves with level \(2\) structure with a holomorphic differential (or canonical divisor). The core of the study is to compute the cohomology of the moduli space \(\mathcal Q[2]\) of plane quartics with level \(2\) structure as a representation of \(\mathrm{Sp}(6, \mathbb F_2)\). moduli spaces; curves of low genus; plane quartics; Del Pezzo surfaces; configurations of point sets; equivariant cohomology Families, moduli of curves (algebraic), Classical real and complex (co)homology in algebraic geometry, Plane and space curves, Families, moduli, classification: algebraic theory, Configurations and arrangements of linear subspaces Equivariant cohomology of moduli spaces of genus three curves with level two structure | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a continuation of the author's joint paper with \textit{D. Mumford} [Invent. Math. 67, 23-86 (1982; Zbl 0506.14016)]. Here the author proves the following very important theorem: for g even and \(\geq 40\) the moduli space \(M_ g\) of curves of genus g is of general type. The main point is the hard calculation of a few intersection numbers, hence the proof of a few enumerative formulas about \(g^ 1_ d\) on a general curve. Recently \textit{D. Eisenbud} and the author [Bull. Am. Math. Soc. 10, 277-280 (1984; Zbl 0533.14013)] announced that \(M_ g\) is of general type if \(g\geq 24\). canonical divisor; Hurwitz scheme; Chern classes; Kodaira dimension; moduli space of curves of general type \textsc{D. Eisenbud and B. Ulrich}, The regularity of the conductor, In: A Celebration of Algebraic Geometry, 267-280 Clay Math. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 2013. Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Algebraic moduli problems, moduli of vector bundles, Coverings of curves, fundamental group On the Kodaira dimension of the moduli space of curves. II: The even-genus 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 In [Invent. Math. 99, No. 2, 321--355 (1990; Zbl 0705.14026)] \textit{J. Harris} and \textit{I. Morrison} studied simply branched covers of \(\mathbb P^1\). By varying a branch point, they constructed a sweeping family of curves for bounding slopes of effective divisors on the moduli space of stable genus \(g\) curves \(\overline{\mathcal M}_g\). In this paper, the authors elaborate on Harris-Morrison's construction by considering the base curve of the family with arbitrary genus and obtain asymptotic results about the slope and index of the family. moduli of curves; fibration; gonality Fine and coarse moduli spaces, Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic) A note on Harris Morrison sweeping families | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given any \(s\) points \(P_ 1, \dots, P_ s\) in the projective plane and \(s\) positive integers \(m_ 1, \dots, m_ s\), let \(\Sigma_ n\) be the linear system of plane curves of degree \(n\) through the \(P_ i\)'s with multiplicity at least \(m_ i\). When the \(P_ i\)'s are in general position, i.e. never three of them on a line, we give numerical criteria to the effect that \(\Sigma_ n\) be regular and/or irreducible. linear system of plane curves Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves Regularity and irreducibility of linear systems of plane curves through points of \(\mathbb{P}^ 2\) in general position with prescribed multiplicity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Erratum to the author's paper [ibid. 270, No. 3--4, 871--887 (2012; Zbl 1270.14011)]. Halic, Mihai, Erratum to: Modular properties of nodal curves on \(K3\) surfaces [ MR2892928], Math. Z., 280, 3-4, 1203-1211, (2015) Families, moduli of curves (algebraic), \(K3\) surfaces and Enriques surfaces, Formal methods and deformations in algebraic geometry Erratum to: ``Modular properties of nodal curves on \(K3\) 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 This paper deals with the following question: Can one find a linear pencil of plane curves of a given degree \(m\), defined over the rational number field \(\mathbb{Q}\), with \(m^2\) distinct \(\mathbb{Q}\)-rational base points, such that every curve belonging to the pencil is irreducible?
The answer for \(m=3\) is well known, which gives an elliptic curve defined over \(\mathbb{Q} (t)\) with Mordell-Weil rank \(r=8\). For general \(m\), an affirmative answer will give an algebraic curve over \(\mathbb{Q}(t)\) with rank \(r=m^2-1\) [cf. \textit{T. Shioda}, Proc. Japan Acad, Ser. A 69, No. 1, 10-12 (1993; Zbl 0784.14013)].
The case \(m=4\) is solved in the affirmative in this paper. The question is open for \(m>4\). rational base points; linear pencil of plane curves Pencils, nets, webs in algebraic geometry, Plane and space curves, Families, moduli of curves (algebraic), Rational points, Algebraic functions and function fields in algebraic geometry On \(\mathbb{Q}\)-split Bézout intersection | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\pi: X\to T\) be a proper flat morphism of complex algebraic varieties and \(L\) a line bundle on \(X\) with \(X\) and \(T\) separated, \(T\) irreducible and \(X\) pure dimensional. Suppose that \(k=\dim(T)\) and \(d\) is the relative dimension of \(\pi\). Consider the divisor class \({\mathcal E}(L,F)=\pi_*((rc_ 1(L)-\pi^*c_ 1(F))^{d+1}\cap [X]\in A_{k+1}(T)\) for a coherent subsheaf \(F\) of \(\pi_*(L)\). Suppose that the following conditions hold:
(i) If \(t\) is a general point of \(T\), then \(F_ t\otimes C\subset H^ 0(\pi^{-1}(t)\), \(L_{(\pi^{-1}(t))})\) is base point free, very ample and yields a semi-stable embedding of \(\pi^{-1}(t)\).
(ii) \(L\) is relatively ample.
Then, the main result shows
(1) that \({\mathcal E}(L,F)\) lies in the closure of the cone in the \(A_{k- 1}(T)\otimes \mathbb Q\) generated by the effective Weil divisors.
(2) if \(F\) is locally free, \({\mathcal E}(L,F)\) lies in the closure of the cone generated by the effective Cartier divisors.
A counter example due to Ian Morrison shows that the semi-stability condition is essential.
As a significant application of the above result the authors show: Let \(\overline M_ g\) be the moduli space of stable genus \(g\) curves with \(g\geq 2\) and \(\lambda,\delta \in \text{Pic}(\overline M_ g)\otimes\mathbb Q\) be the Hodge and boundary class. Then the class \(a\lambda-b\delta\) has non-negative degree on every curve in \(\overline M_ g\) not contained in \(\Delta =\overline M_ g- M_ g\) if and only if \(a\geq (8+4/g)b\). Furthermore, \(a\lambda-b\delta\) is ample if and only if \(a>11.b>0\). These results are known in part. Hodge class; divisor class; effective Weil divisors; effective Cartier divisors; moduli space of stable genus g curves; boundary class Cornalba, M; Harris, J, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. École Norm. Sup. (4), 21, 455-475, (1988) Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves Divisor classes associated to families of stable varieties with applications to the moduli space of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(B\) be a projective algebraic curve over an algebraically closed field, and let \(S\subset B\) a finite set of points. The so-called Shafarevich conjecture for families of curves, proved in the 1970s by \textit{A. Parshin} [Sov. Math. Dokl. 9, 1419--1422 (1968; Zbl 0176.50903)] and \textit{A. Arakelov} [Math. USSR, Izv. 5, 1277--1302 (1971; Zbl 0248.14004)], states the following:
(i) There are only finitely many isomorphism classes of smooth and non-isotrivial families of curves over \(B- S\).
(ii) If \(2\cdot\text{genus}(B)- 2+ \text{card}(S)\leq 0\), then there are no such families at all.
The paper under review is devoted to analogous questions for smooth families of higher-dimensional algebraic manifolds over \(B- S\) which are currently subject to intensive study by various authors.
Due to \textit{G. Faltings's} work on Arakelov's theorem for abelian varieties [Invent. Math. 73, 337--347 (1983; Zbl 0588.14025)], statement (i) seems to be too much to hope for in the higher-dimensional case, and so the authors propose to split it up into two separate questions regarding boundedness and rigidity. Then, in this note, they prove their formulated boundedness conjecture for surfaces of general type, and also for canonically polarized manifolds in the special case of \(S=\emptyset\).
The method of proof is based on particular ampleness conditions and cohomological vanishing theorems for suitable sheaves on the manifolds under consideration. Some earlier partial results in this context, obtained by several authors, are reproved within the authors' general framework. families of surfaces; famlies of polarized algebraic varieties; vanishing theorem E. Bedulev and E. Viehweg, ''On the Shafarevich conjecture for surfaces of general type over function fields,'' Invent. Math., vol. 139, iss. 3, pp. 603-615, 2000. Families, moduli, classification: algebraic theory, Surfaces of general type, Families, moduli of curves (algebraic), Vanishing theorems in algebraic geometry, Fibrations, degenerations in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials On the Shafarevich conjecture for surfaces of general type 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 Let \(\pi:X\to D\) be a proper surjective holomorphic map of a complex manifold \(X\) of dimension 2 to a small open disk \(D= \{t\in\mathbb{C} \mid|t|< \varepsilon\}\). We assume that \(\pi\) is smooth over a punctured disc \(D'=D-\{0\}\). Moreover we assume that for every \(t\in D'\) the fiber \(X_t=\pi^{-1} (t)\) is a nonsingular curve of genus \(g\) and that \(X\) contains no exceptional curves of the first kind. By \(L_t\), we denote the effective divisor in \(X\) defined by the equation \(\pi=t\) \((t\in D)\). We call the divisor \(L_0\) the singular fiber of \(\pi\). For every \(t\in D'\) we call the divisor \(L_t\) a generic fiber.
In the study of elliptic surfaces, Kodaira showed that there exist only ten types of singular fibers of pencils of curves of genus one. Iitaka and Ogg gave a numerical classification of singular fibers of curves of genus 2. Namikawa and Ueno classified their numerical types completely, constructed all their singular fibers and calculated the monodromies around them. In a previous paper [\textit{K. Uematsu}, Jap. J. Math., New Ser. 23, No. 2, 281-301 (1997; Zbl 0905.14006)], the author studied the numerical properties of singular fibers in pencils of curves of genus \(g\) \((\geq 2)\) and gave a method to classify all the numerical types of the singular fibers. In this article, by using this method, we give the complete numerical classification of singular fibers in pencils of curves of genus three. If the number of irreducible components is more then one, we have \(\Gamma^2 <0\) and \(\Gamma\cdot K_X\geq 0\), where \(K_X\) is a canonical divisor. If \(\Gamma\cdot K_X>0\), we call this component \(\Gamma\) a trunk. If \(\Gamma\cdot K_X=0\), then we have \(\Gamma^2 =-2\). Thus we call this component \(\Gamma\) a \((-2)\)-curve. Further we call a connected component consisting of \((-2)\)-curves in a singular fiber a branch. The method of numerical classification of singular fibers in the paper under review is as follows.
First, the author determines all the combinations of trunks, the number of which is finite. Secondly, he classifies the possible branches in the singular fiber in pencils of curves of genus 3. Finally by combining trunks with branches, he classifies all the singular fibers. In his calculation there exist 4343 types of singular fibers in pencils of curves of genus 3, while on the other hand, in the case of genus 2 there exist 140 types. numerical classification; elliptic surfaces; singular fibers of pencils of curves; curves of genus three Kazuhiro Uematsu, Numerical classification of singular fibers in genus 3 pencils, J. Math. Kyoto Univ. 39 (1999), no. 4, 763 -- 782. Pencils, nets, webs in algebraic geometry, Special algebraic curves and curves of low genus, Fibrations, degenerations in algebraic geometry, Families, moduli of curves (algebraic), Elliptic surfaces, elliptic or Calabi-Yau fibrations, Special divisors on curves (gonality, Brill-Noether theory) Numerical classification of singular fibers in genus 3 pencils | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal M}\) be the fine moduli space of the smooth curves of genus g (defined over \({\mathbb{C}})\) with a level-\(\nu\)-structure for some fixed \(\nu \in {\mathbb{Z}}_{\geq 3}\). Let \(p:\quad {\mathcal C}\to {\mathcal M}\) be the associated smooth family of curves and let \(\mu:\quad {\mathcal M}\to {\mathcal M}_ g\) be the natural morphism to the coarse moduli space \({\mathcal M}_ g\). For \(k,r\in {\mathbb{Z}}_{\geq 1}\), let
\[
W^ r_ k=\{x\in {\mathcal C}:\quad \dim (H^ 0(p^{-1}(p(x)),{\mathcal O}(kx))\geq r+1\}.
\]
One knows that \(\dim (W^ r_ k)\geq 3g-2-r(g+r-k)\) [\textit{R. F. Lax}, Math. Ann. 216, 35-42 (1975; Zbl 0291.32028)].
In the case \(r=1\), one has that: \((i)\quad W^ 1_ k\) is equidimensional of dimension \(2g-3+k\); \((ii)\quad if\) x is a general point of \(W^ 1_ k\), then its Weierstrass gap sequence is the so- called hyperordinary gap sequence; \((iii)\quad p| W^ 1_ k:\quad W^ 1_ k\to {\mathcal M}\) is generically injective; \((iv)\quad \mu (p(W^ 1_ k))\) is irreducible [\textit{E. Arbarello}, Compos. Math. 29, 325-342 (1974; Zbl 0355.14013); \textit{S. Diaz}, Duke Math. J. 51, 905-922 (1984; Zbl 0581.14019); \textit{D. S. Rim} and \textit{M. A. Vitulli}, J. Algebra 48, 454-476 (1977; Zbl 0412.14002); the author, ''The number of Weierstrass points on some special curves. I'' in Arch. Math. 46, 453-465 (1986)].
This paper is an attempt to prove similar results for the Weierstrass points with two prescribed non-gaps. Let \(n,s\in {\mathbb{Z}}_{\geq 1}\) with \(n<s\) and let \(g=a(n-1)+b\) with \(0\leq b<n-1\) and such that \(s<g+a\) and s is not a multiple of n. Let \(\overset\circ W_{n,s}=\{x\in {\mathcal C}:\quad n\quad is\) the first non-gap of x and \(\dim (H^ 0(p^{- 1}(p(x)),{\mathcal O}(sx))\geq e+2\}.\) We prove that, if Z is an irreducible component of \(\overset\circ W_{n,s}\) and \(| sx|\) is a simple linear system on \(p^{-1}(p(x))\) if x is a general point on Z, then \(\dim (Z)=n+s+g-4-e\) and \(\dim (H^ 0(p^{-1}(p(x)),{\mathcal O}(sx))=e+2.\) This is closely related to some repeatedly criticized formula of \textit{K. Hensel} and \textit{G. Landsberg} [''Theorie der algebraischen Funktionen einer Variablen und ihre Anwendung auf algebraische Kurven und abelsche Integrale'' (Leipzig 1902)]. It is also shown that the assumption ''\(| sx|\) is a simple linear system'' can not be ommited in general.
Let \({\mathcal W}_{n,s}=\{x\in {\mathcal C}:\) n is the first non-gap of x, s is the first non-gap of x which is not a multiple of n and \(| sx|\) is a simple linear system\(\}\). - It is proved that \({\mathcal W}_{n,s}\) is not empty whenever it make sense. Also some existence results of Weierstrass points with prescribed gap sequence are obtained. Those are closely related to the Weierstrass points obtained by \textit{H. Knebl} [Manuscr. Math. 49, 165-175 (1984; Zbl 0575.14022)]. Also, some open questions related to this paper are stated. Weierstraß points with two prescribed non-gaps; prescribed gap sequence Coppens, M., Weierstrass points with two prescribed nongaps,Pacific J. Math. 131 (1988), 71--104. Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (analytic), Families, moduli of curves (algebraic), Compact Riemann surfaces and uniformization Weierstrass points with two prescribed non-gaps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 develops a method to establish the projectivity of certain moduli spaces. The general framework uses some results of \textit{E. Viehweg} [``Weak positivity and the stability of certain Hilbert points'', Invent. Math. 96, No.3, 639-667 (1989)] but develops in a different direction. The general criteria are applied to show the projectivity of the moduli of stable curves in every characteristic. Partial results are obtained for the moduli of surfaces. The compactification of the Picard schemes constructed by \textit{A. B. Altman} and \textit{S. L. Kleiman} [Adv. Math. 35, 50-112 (1980), and Am. J. Math. 101, 10-41 (1979; Zbl 0427.14015 and 14016)] is also shown to be projective. projectivity of the moduli of stable curves; moduli of surfaces; compactification of the Picard schemes Kollár, J., Projectivity of complete moduli, J Differential Geom, 32, 235-268, (1990) Algebraic moduli problems, moduli of vector bundles, Families, moduli, classification: algebraic theory, Families, moduli of curves (algebraic), Projective techniques in algebraic geometry Projectivity of complete 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 Let \(X\) be an abelian surface and let \(L\) be a polarization of type \((1,3)\) on \(X\). The corresponding map \(\varphi_ L : X \to \mathbb{P}^ 2\) is a 6-fold covering and one has 4 isogenies \(f_ i(X,L) \to (Y_ i, P_ i)\), \(i = 1, \dots, 4\), onto principally polarized abelian surfaces; when \((Y_ i, P_ i)\) is the Jacobian of a curve \(H\) of genus 2, then \(C = f_ i^{-1} (H)\) is a smooth curve of genus 4 and \(C \to H\) is an étale cyclic 3-fold covering.
In this paper, the authors describe carefully the case \(X = E \times E\) \((E =\) elliptic curve) and \(L = {\mathcal O}_ X (E \times \{0\} + \{0\} \times E + A)\) where \(A\) is the antidiagonal. In particular, they find the equation of the ramification curve of \(\varphi_ L\) in terms of the \(j\)-invariant of \(E\) and describe when the resulting principally polarized surfaces \((Y_ i, P_ i)\) are Jacobians; in these cases, the covering curve \(C\) has \(\Aut (C) = S_ 3 \times S_ 3\) hence, varying the elliptic curve \(E\), the authors construct a 1-dimensional family of curves of genus 4, with automorphism group \(S_ 3 \times S_ 3\). product of elliptic curves; principally polarized abelian surfaces; Jacobians; family of curves Birkenhake, A family of abelian surfaces and curves of genus 4, Manuscripta Math. 85 pp 393-- (1994) Algebraic theory of abelian varieties, Jacobians, Prym varieties, Families, moduli of curves (algebraic) A family of abelian surfaces and curves of genus four | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X \subset \mathbb P^n_{\mathbb C}\) be a smooth complete intersection of \(c \leq n\) hypersurfaces of degrees \(d_i \geq 2\) having low degree in the sense that \(n \geq 2 \sum d_i - c + 1\). The Kontsevich moduli space \({\overline {\mathcal M}}_{0,2} (X,2)\) of stable maps parametrizes tuples \((C,f,p,q)\) where \(C\) is a proper connected nodal curve with \(p_a (C)=0\), \(p \neq q\) are smooth points of \(C\), \(f:C \to X\) is a morphism whose image has degree \(2\), and \((C,f,p,q)\) has finitely many automorphisms. In the paper under review the author gives a detailed description of the general fiber \(\mathcal F\) of the evaluation map \({\overline {\mathcal M}}_{0,2} (X,2) \to X \times X\) given by \((C,f,p,q) \mapsto (f(p),f(q))\).
Work of \textit{A. J. de Jong} and \textit{J. M. Starr} [``Low degree complete intersections are rationally simply connected'', Preprint (2006)]shows that \(\mathcal F\) is smooth of the expected dimension \(n+c+1-2 \sum d_i\) and that \(\mathcal F\) meets the boundary of \({\overline {\mathcal M}}_{0,2} (X,2)\) (the locus where \(C\) is reducible) in a simple normal crossing divisor \(\Delta\). Interpreting \(\mathcal F\) as the Hilbert scheme of conics in \(X\) passing through \(f(p)\) and \(f(q)\), each \(C \in \mathcal F\) gives rise to the unique \(2\)-plane \(\pi_C\) containing \(C\) and hence \(f(p)\) and \(f(q)\) as well: since the \(2\)-planes containing \(f(p)\) and \(f(q)\) are parametrized by \(\mathbb P^{n-2}\), one obtains a map \(\varphi: \mathcal F \to \mathbb P^{n-2}\) given by \(C \mapsto \pi_C\). The main theorem says that \(\varphi\) is a closed immersion whose image is a complete intersection and gives the degrees of the hypersurfaces whose intersection is \(\varphi (\mathcal F)\). For example, when \(X \subset \mathbb P^3\) is a quadric surface, the theorem says that the conics on \(X\) through two general fixed points \(P,Q \in X\) is parametrized by a \(\mathbb P^1\), which is clear because the conics intersections of \(X\) with a plane passing through \(P\) and \(Q\). A more interesting instance occurs when \(X \subset \mathbb P^7\) is the complete intersection of quadric and cubic hypersurface, when the theorem says that the conics passing through two points \(P,Q \in X\) is parametrized by a complete intersection in \(\mathbb P^5\) of surfaces of degrees \(1,1,2,2,3\), hence there are exactly twelve such conics. This confirms results of \textit{A. Beauville} obtained using quantum cohomology [Mat. Riz. Anal. Geom. 2, No. 3--4, 384--398 (1995; Zbl 0863.14029)]. complete intersections; conics; Kontsevich moduli space of stable maps Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) Spaces of conics on low degree complete intersections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The conjecture referred to in the title is the boundedness result for rational points of small height on a smooth curve \(C\) over a global field \(K=k(Y)\) proved by \textit{Z. Cinkir} [Invent. Math. 183, No. 3, 517--562 (2011; Zbl 1285.14029)]. The formula proved by Cinkir is rather precise and involves the modular invariants of the special fibres of \(f: C\to Y\): that is, the intersections of the image of \(Y\) in the moduli space of curves with the strata in the Deligne-Mumford boundary.
The result here is a more uniform (but, in general, weaker) version of Cinkir's result that depends only on the genus \(g\) of~\(C\). The bound, on the lim~inf over all rational points of the heights associated with degree~\(1\) divisors on~\(C\), is of order \(g^{-3}\).
The method is to examine in detail the possible fibres of \(f\) (after semi-stable reduction) and thus estimate the modular invariants: one should recall that the modular invariant \(\delta_i\) is essentially a count of nodes on the fibre occurring on components of genus~\(\geq i\). Néron-Tate height; Deligne-Mumford stratification; semi-stable curve Local ground fields in algebraic geometry, Heights, Algebraic functions and function fields in algebraic geometry, Families, moduli of curves (algebraic), Arithmetic varieties and schemes; Arakelov theory; heights Uniform bound for the effective Bogomolov conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is shown that the homology of the spin moduli spaces \({\mathcal M}_ g[\epsilon]\) of Riemann surfaces of genus g with spin structure of Arf invariant \(\epsilon\in {\mathbb{Z}}/2{\mathbb{Z}}\) (resp. of the corresponding spin mapping class groups) is stable, i.e. independent of g and \(\epsilon\) for sufficiently large g. As the author notes, the interest in these moduli spaces comes from fermionic string theory. For a second paper the computation of the first (integer coefficients) and second (rational coefficients) homology, and thus of the Picard group, of the spin moduli spaces is announced. All of this generalizes results and methods (constructing simplicial complexes from configuration of simple closed curves on a surface on which the mapping class groups act, then applying spectral sequence arguments) of two of the author's previous papers in which he obtained analogous results for the ordinary mapping class groups resp. moduli spaces. homology of the spin moduli spaces of Riemann surfaces with spin structure; Arf invariant; spin mapping class groups; fermionic string theory; Picard group; configuration of simple closed curves on a surface Harer J.L. (1990) Stability of the homology of the moduli spaces of Riemann surfaces with spin structure. Math. Ann. 287(2): 323--334 Topology of Euclidean 2-space, 2-manifolds, General low-dimensional topology, Teichmüller theory for Riemann surfaces, Homology of classifying spaces and characteristic classes in algebraic topology, Differential topological aspects of diffeomorphisms, Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) Stability of the homology of the moduli spaces of Riemann surfaces with spin structure | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is known that the ring of modular forms of the Picard modular group \(\Gamma_ K\subset U_ K\) of an imaginary quadratic field K which operates on the 1-ball \(B\subset {\mathbb{C}}^ 2\) is generated by three elements and that the values of certain quotients of these generators, i.e. modular functions, on so-called K-singular moduli \(\tau\in B\), i.e. fixed points of \(U_ K\), are algebraic.
As in the classical cases of the elliptic and Hilbert modular group these singular moduli correspond to ideals in orders of K (which in the classical cases at least furnish an algebraic equation with rational coefficients for the generators of the field of modular functions). Having this final result in mind the author first characterises all K- singular moduli by arithmetic data of K, secondly shows that a certain canonically given jacobian \(Jac(C_{\Phi}^{-1}{}_{(\tau)})\) is simple as long as \(\tau\in B\) is K-singular and finally expresses the number of K-singular moduli in terms of class-numbers of CM-extension, \(K\subset L\) of degree 3 following essentially the ideas of Hecke. elliptic modular group; Picard modular group; imaginary quadratic field; K-singular moduli; Hilbert modular group; jacobian; class-numbers of CM-extension Feustel, J, Eine klassenzahlformel für singuläre moduln der picardschen modulgruppen, Comp. Math, 76, 87-100, (1990) Picard groups, Algebraic moduli problems, moduli of vector bundles, Modular and Shimura varieties, Quadratic extensions, Picard schemes, higher Jacobians, Class numbers, class groups, discriminants Eine Klassenzahlformel für singuläre Moduln der Picardschen Modulgruppen. (A class number formula for singular moduli of Picard modular groups) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is well-known so far that very interesting connections with cryptography matters can be obtained by using hyperelliptic curves, specially in characteristic 2 or 3; see e.g. \textit{L. M. Adleman, J. DeMarrais} and \textit{M.-D. Huang} [Algorithmic number theory. 1st international symposium, ANTS--I, Ithaca, NY, USA. May 6--9, 1994. Proceedings. Berlin: Springer-Verlag, Lect. Notes Comput. Sci. 877, 28--40 (1994; Zbl 0829.11068)]. In characteristic 2, these curves were classified by \textit{G. Cardona, E. Nart} and \textit{Pujolás} [Curves of genus two over fields of even characteristic. \url{http:\slash\slash www.arxiv.org/math.NT/0210105}].
In this paper the authors consider non-isomorphic plane models of hyperelliptic curves of genus 2 admitting a Weierstrass point over an arbitrary finite field. As a matter of fact, they subsume results that were noticed and published by several authors by working with different methods. These plane models allow us to design and implement the so called hyperelliptic curve cryptosystems. finite fields; hyperelliptic curves; public-key cryptography García, J. Espinosa; Encinas, L. Hernández; Masqué, J. Muñoz: A review on the isomorphism classes of hyperelliptic curves of genus 2 over finite fields admitting a Weierstrass point, Acta appl. Math. 93, 299-318 (2006) Curves over finite and local fields, Families, moduli of curves (algebraic), Applications to coding theory and cryptography of arithmetic geometry, Special algebraic curves and curves of low genus, Plane and space curves, Cryptography A review on the isomorphism classes of hyperelliptic curves of genus 2 over finite fields admitting a Weierstrass point | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathcal M_{g, n}\) be the coarse moduli space of smooth complex algebraic curves of genus \(g\) with \(n\) marked points. The authors prove the rationality of \(\mathcal M_{6, n}\) for \(1\leq n\leq 8\) and the unirationality of \(\mathcal M_{g, n}\) for \(g=8\) and \(1\leq n\leq 11\), \(g=10\) and \(1\leq n\leq 3\), \(g=12\) and \(n=1\).
Reviewer's remark: There is a misprint in the formulation of Theorem B: \(1\leq n\leq 8\) should be replaced by \(1\leq n\leq 11\). moduli space of marked curves; rationality; unirationality Ballico E., Casnati G., Fontanari C.: On the birational geometry of moduli spaces of pointed curves (English summary). Forum Math. 21(5), 935--950 (2009) Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Rationality questions in algebraic geometry, Group actions on varieties or schemes (quotients) On the birational geometry of moduli spaces of pointed 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 \(V(d)\) be the \(\mathbb{C}\)-linear space of forms in \(x_1, x_2, x_3\) of degree d. Let \(\mathcal A_ d\) be the moduli space of plane curves of degree d. It is well known that the field of rational functions of \(\mathcal A_ d\) is isomorphic to the field (*) \((\mathbb{C}(V)(d))^{\text{GL}_ 3(\mathbb{C})})\mathbb{C}^*\). Using a method of F. A. Bogomolov, the author proves the following theorem:
If \(d\equiv 0\pmod3\) and \(d\geq 1821\), then the field (*) is rational.
Corollary. In the conditions of the previous theorem, the moduli space \(\mathcal A_ d\) is rational. rationality; moduli space of plane curves of degree d; field of rational functions Katsylo, P.I.: Rationality of moduli varieties of plane curves of degree 3k. Math. USSR Sbornik 64(2) (1989) Rational and unirational varieties, Families, moduli of curves (algebraic) Rationality of moduli varieties of plane curves of degree 3k | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study the space of rational curves of fixed degree on a smooth Fano hypersurface \(X\) of degree \(d\) in \(\mathbb P^n\). More precisely, they study the Kontsevich moduli space \(\overline{\mathcal M}_{0,0}(X,e)\) of stable maps of degree \(e\), which is a compactification of the space of rational curves of degree \(e\) on \(X\).
\textit{J. Harris, M. Roth} and \textit{J. Starr} [J. Reine Angew. Math. 571, 73--106 (2004; Zbl 1052.14027)] proved that for the \textit{general} \(X\) of degree \(d < (n+1)/2\) the space \(\overline{\mathcal M}_{0,0}(X,e)\) is irreducible of the expected dimension \(e(n+1-d)+n-4\).
The main results of the present paper are about the case \(d=3\), i.e. when \(X\) is a cubic hypersurface: if \(n >4\) then it is proved that for \textit{any} smooth \(X\) the space \(\overline{\mathcal M}_{0,0}(X,e)\) is irreducible of the expected dimension \(e(n-2)+n-4\).
In the case \(n=4\) it had been previously shown, in the Ph.D. thesis of the second author, that \(\overline{\mathcal M}_{0,0}(X,e)\) has two irreducible components of the expected dimension.
The methods used for cubic hypersurfaces also show that \(\overline{\mathcal M}_{0,0}(X,e)\) is irreducible of the expected dimension for a \textit{general} hypersurface \(X\) of degree \(d\) such that \(d \leq \min(n+4,2n-2)\) and \(n \geq 4\). I. Coskun, J. Starr, Rational curves on smooth cubic hypersurfaces. \textit{Int. Math. Res. Not}. (2009), no. 24, 4626-4641. MR2564370 Zbl 1200.14051 Families, moduli of curves (algebraic), Fano varieties Rational curves on smooth cubic 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 Let \(\Sigma\) be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers. In this paper, we prove that various objects that arise from the \textit{geometry} of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero may be \textit{reconstructed group-theoretically} from the pro-\(\Sigma\) \textit{fundamental group} of the configuration space. Let \(X\) be a hyperbolic curve of type \((g,r)\) over a field \(k\) of characteristic zero. Thus, \(X\) is obtained by removing from a proper smooth curve of genus \(g\) over \(k\) a closed subscheme [i.e., the ``divisor of cusps''] of \(X\) whose structure morphism to \(\operatorname{Spec} (k)\) is finite étale of degree \(r\); \(2g-2+r>0\). Write \(X_n\) for the \(n\)-th configuration space associated to \(X\), i.e., the complement of the various diagonal divisors in the fiber product over \(k\) of \(n\) copies of \(X\). Then, when \(k\) is \textit{algebraically closed}, we show that the \textit{triple} \((n,g,r)\) and the \textit{generalized fiber subgroups} -- i.e., the subgroups that arise from the various \textit{natural morphisms} \(X_n \to X_m [m < n]\), which we refer to as \textit{generalized projection morphisms} -- of the pro-\(\Sigma\) \textit{fundamental group} \(\Pi_n\) of \(X_n\) may be \textit{reconstructed group-theoretically} from \(\Pi_n\) whenever \(n \geq 2\). This result \textit{generalizes} results obtained previously by the first and third authors and A. Tamagawa to the case of \textit{arbitrary hyperbolic curves} [i.e., without restrictions on \((g,r)]\). As an application, in the case where \((g,r)= (0,3)\) and \(n \geq 2\), we conclude that there exists a \textit{direct product decomposition}
\[
\mathrm{Out}(\Pi_n) = \mathrm{GT}^{\Sigma} \times \mathfrak{S}_{n + 3}
\]
-- where we write ``\(\mathrm{Out}(-)\)'' for the group of outer automorphisms [i.e., \textit{without any auxiliary restrictions}!] of the profinite group in parentheses and \(\mathrm{GT}^{\Sigma}\) (respectively, \(\mathfrak{S}_{n + 3})\) for the pro-\(\Sigma\) \textit{Grothendieck-Teichmüller group} (respectively, symmetric group on \(n + 3\) letters). This direct product decomposition may be applied to obtain a \textit{simplified purely group-theoretic equivalent definition} -- i.e., as the \textit{centralizer} in \(\mathrm{Out}(\Pi_n)\) of the \textit{union of the centers of the open subgroups} of \(\mathrm{Out}(\Pi_n)\) -- of \(\mathrm{GT}^{\Sigma}\). One of the key notions underlying the theory of the present paper is the notion of a pro-\(\Sigma\) \textit{log-full subgroup} -- which may be regarded as a sort of \textit{higher-dimensional analogue} of the notion of a pro-\( \Sigma\) \textit{cuspidal inertia subgroup of a surface group} -- of \(\Pi_n\). In the final section of the present paper, we show that, when \(X\) and \(k\) satisfy certain conditions concerning ``\textit{weights}'', the pro-\(l\) log-full subgroups may be \textit{reconstructed group-theoretically} from the natural outer action of the absolute Galois group of \(k\) on the geometric pro-\(l\) fundamental group of \(X_n\). anabelian geometry; configuration space; generalized fiber subgroup; Grothendieck-Teichmüller group; hyperbolic curve; log-full subgroup Families, moduli of curves (algebraic), Coverings of curves, fundamental group Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let C be an irreducible, smooth, projective curve defined over the field of complex numbers. We call C elliptic-hyperelliptic (e.h. for short) if it admits a degree two morphism \(\pi: C\to E\) onto an elliptic curve. We denote by \(M_ g^{eh}\) the moduli space of e.h. curves of genus g. The aim of this note is to present a proof of the following theorem: \(M_ 4^{eh}\) is rational.
We proceed as follows: In section 1 the canonical model of a generic e.h. curve C (of genus \(4\)) is shown to be complete intersection of a unique cubic cone R and a unique quadric. By looking at the tangent space to the canonical space at the vertex of R, in section \(2,\) we associate to C a pair \((Z,\gamma)\), where Z and \(\gamma\) are smooth coplanar curves of degree 3 and 2 respectively, and we are able to show that \(M_ 4^{eh}\) is birational to \(((Z,\gamma))/PGL(3).\) - After fixing a quadratic form defining \(\gamma\) we can prove that \(\{(Z,\gamma)\}/PGL(3)\) is birational to \(H^ 0(P^ 1,{\mathcal O}_{{\mathbb{P}}^ 1}(6))/G_{\ell_ 0}\) where \(G_{\ell_ 0}\) is a \({\mathbb{C}}^*\)-extension of \({\mathbb{Z}}_ 2.\)
In section 3 we compute the representation of \(G_{\ell_ 0}\) on \(H^ 0(P^ 1,{\mathcal O}_{{\mathbb{P}}^ 1}(6))\) and we show that its \(G_{\ell_ 0}\)-invariant field is purely transcendental over \({\mathbb{C}}\) completing the proof of the theorem. genus four double covers of elliptic curves; rationality of moduli space of elliptic-hyperelliptic curves Bardelli, Fabio; Del Centina, Andrea: The moduli space of genus four double covers of elliptic curves is rational. Pac. J. Math. 144, No. 2, 219-227 (1990) Families, moduli of curves (algebraic), Complete intersections, Algebraic moduli problems, moduli of vector bundles The moduli space of genus four double covers of elliptic curves is rational | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 projective curve, \(C\subset\mathbb P^r\). For a choice of positive integers \(s_1=\deg(C), s_2,\dots,s_l\), \(1\leq l<r\), the curve \(C\) satisfies the flag condition \((r; s_1,\dots,s_l)\) if it is not contained in any subvariety of dimension \(i\) and degree smaller than \(s_i\), for \(i=2,\dots,l\). It is well understood, since Halphen's works on the genus of space curves, that the arithmetic genus of curves satisfying flag conditions is bounded by a function \(G\) which depends on \(r\) and the \(s_i\)'s.
The author investigates the structure of curves \(C\) of maximal genus, among those satisfying a fixed flag condition. Under the assumption \(s_1\gg s_2 \gg \dots\gg s_l\) (precise bounds are given through the paper), with a careful study of the Hilbert function of \(C\), he is able to prove that \(C\) is arithmetically Cohen-Macaulay. Furthermore, such curves fit in an interesting hierarchical description. They must indeed lie in a flag
\[
C\subset V_2\subset \dots \subset V_l
\]
where each \(V_i\), \(i=2,\dots, l\), is an integral variety of dimension \(i\) and degree \(s_i\), whose general curve section is again a curve of maximal genus among those satisfying the relative flag condition \((r-i+1; s_i,\dots, s_l)\). curves; genus Di Gennaro, V.: Hierarchical structure of the family of curves with maximal genus verifying flag conditions. Proc. Amer. Math. Soc. (to appear) Plane and space curves, Projective techniques in algebraic geometry, Families, moduli of curves (algebraic), Classical problems, Schubert calculus Hierarchical structure of the family of curves with maximal genus verifying flag conditions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(q_ 0,...,q_ 3\) be positive integers such that each triple among them has gcd equal to 1. Let \({\mathbb{P}}={\mathbb{P}}(q_ 0,...,q_ 3)\) be the associated weighted projective space. Then Pic(\({\mathbb{P}})\cong {\mathbb{Z}}\). Let \({\mathcal L}\) be an ample generator of Pic(\({\mathbb{P}})\). The authors give sufficient conditions (in terms of the weights) in order that a general element of the linear system \(| {\mathcal L}|\) on \({\mathbb{P}}\) is a surface with Picard number equal to 1, thus supplementing the result of \textit{D. A. Cox} [Math. Z. 201, No.2, 183-189 (1989; Zbl 0686.14041)] who showed that a general element of \(| {\mathcal L}^{\otimes 2}|\) either has \(p_ g=0\) or Picard number equal to 1. general element of linear system; surface with Picard number equal to 1 DOI: 10.1007/BF02571348 Picard groups, Divisors, linear systems, invertible sheaves Picard numbers of surfaces in 3-dimensional weighted projective 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 Concerns the article mentioned in the title [Algebr. Represent. Theory 11, No. 1, 25-42 (2008; Zbl 1155.16028)]. Lemma 1.1 and Lemma 1.2 are modified and some misprints noted. Picard groups; coinvariants; Doi-Hopf modules; grouplike elements; invertible comodules; exact sequences Hopf algebras and their applications, Picard groups, Actions of groups and semigroups; invariant theory (associative rings and algebras) Addenda and corrigenda: Picard group of rings of coinvariants. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(Y \subset \mathbb{P}^N\) be a closed subscheme containing zero-dimensional embedded points and let \(X\) be the closed scheme obtained by removing the embedded points. The main question of the paper is: when does \(Y\) belong to a flat irreducible family having \(X\) union isolated points as its general member?
The authors are able to give the following interesting answer: Suppose that the multiplicities of the embedded points are at most 3 and that \(X\) is locally a complete intersection of codimension 2, then \(Y\) is a flat specialization of \(X\) union isolated points. This result is optimal for the size of the multiplicity and the codimension, and also with respect to being a local complete intersection. Using \textit{S. Nollet} and \textit{E. Schlesinger} [Compos. Math. 139, No. 2, 169--196 (2003; Zbl 1053.14035)] the authors give examples of irreducible components of the Hilbert scheme \({\text{ H}}(d,g):={\text{ Hilb}}^{dz+1-g}(\mathbb{P}^3)\) of one-dimensional schemes of degree \(d=4\) and arithmetic genus \(g\) whose general point is a curve with an embedded point.
Using their theorems they show several results for the Hilbert scheme \({\text{ H}}(d,g)\) of high genus, e.g. that \({\text{ H}}(d,g)\) is irreducible for \(d \geq 6\) and \(g > -4 +(d-1)(d-2)/2\), and also smooth in the case \(g = -1 +(d-1)(d-2)/2\). Finally they prove that \({\text{ H}}(4,0)\) consists of four irreducible components and they describe the general curves. Hilbert scheme; embedded points; deformation; space curve 10.2140/ant.2012.6.731 Deformations of singularities, Families, moduli of curves (algebraic), Plane and space curves Detaching embedded 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 See the author's paper in Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 3, No. 1, 1--10 (2000; Zbl 0970.14015). [S] Sernesi, E.: Topics on families of projective schemes. Preprint Families, moduli of curves (algebraic), Surfaces of general type Families of curves on 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 We consider the moduli of elliptic curves with \(G\)-structures, where \(G\) is a finite 2-generated group. When \(G\) is abelian, a \(G\)-structure is the same as a classical congruence level structure. There is a natural action of \(\mathrm{SL}_2(\mathbb {Z})\) on these level structures. If \(\Gamma \) is a stabilizer of this action, then the quotient of the upper half plane by \(\Gamma \) parametrizes isomorphism classes of elliptic curves equipped with \(G\)-structures. When \(G\) is sufficiently nonabelian, the stabilizers \(\Gamma \) are noncongruence. Using this, we obtain arithmetic models of noncongruence modular curves as moduli spaces of elliptic curves equipped with nonabelian \(G\)-structures. As applications we describe a link to the Inverse Galois Problem, and show how our moduli interpretations explains the bad primes for the Unbounded Denominators Conjecture, and allows us to translate the conjecture into the language of geometry and Galois theory. \(G\)-structure; inverse Galois problem; elliptic curves Elliptic curves over global fields, Fourier coefficients of automorphic forms, Families, moduli of curves (algebraic), Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Moduli interpretations for noncongruence modular curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A Poncelet polygon is a polygon which is circumscribed about a conic \(Q\) and inscribed in a second conic \(Q'\). In case such a polygon exists, the pair \(Q,Q'\) is called a Poncelet pair. In the first part of the paper, a bijective correspondence is set up between Poncelet pairs and elliptic curves with a point of finite order. This is used to identify the corresponding moduli problems and to construct a moduli space for Poncelet polygons. Furthermore, a second construction for the moduli space of Poncelet pairs is given, by which explicit equations for the modular curves are obtained. Poncelet pair; moduli space for Poncelet polygons; equations for the modular curves; quadric B. Jakob. Moduli of Poncelet polygons.J. Reine Angew. Math 436, 33--44 (1993) Families, moduli of curves (algebraic) Moduli of Poncelet polygons | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Demailly-Siu criterion [\textit{Y. T. Siu}, J. Differ. Geom. 19, 431-452 (1984; Zbl 0577.32031); Lect. Notes Math. 1111, 169-192 (1985; Zbl 0577.32032), and \textit{J. P. Demailly}, Ann. Inst. Fourier 35, No. 4, 189-229 (1985; Zbl 0565.58017)] gives a sufficient condition in order that a compact complex analytic manifold \(X\) is Moishezon. The condition is not necessary as shown by J. Kollár and K. Oguiso by giving examples of Moishezon manifolds \(X\) with \(\text{Pic} (X)=\mathbb{Z} \cdot {\mathcal O}_X(1)\), \(m_X\leq 0\), where \(m_X\) is defined by \(K_X={\mathcal O}_X(m_X)\), without any big and nef bundle on them.
The paper deals with the problem of the existence of such manifolds with \(m_X>0\). From a result of \textit{J. Kollár} [Surv. Differ. Geom. 1, 113-199 (1991; Zbl 0755.14003)] such manifolds do not exist in dimension 3 since if the canonical bundle is big then it is nef. The author proves that in dimension 4 or larger than 4 there exist such Moishezon manifolds where \(K_X\) is big but it is not nef and, using Mori theory, he produces explicitly some interesting examples, which give also new manifolds not satisfying the Demailly-Siu criterion. Picard group; big canonical bundle; Moishezon manifolds; Mori theory L. Bonavero, Sur des variétés de Moishezon dont le groupe de Picard est de rang un, Bull. Soc. Math. France 124 (1996), no. 3, 503-521. Compact analytic spaces, Complex manifolds, \(n\)-folds (\(n>4\)), Picard groups On Moishezon manifolds with Picard group of rang one | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We describe the locus of stable bundles on a smooth genus \(g\) curve that fail to be globally generated. For each rank \(r\) and degree \(d\) with \(rg<d<r(2g-1)\), we exhibit a component of the expected dimension. We show, moreover, that no component has larger dimension and give an explicit description of those families of smaller dimension than expected. For large-enough degrees, we show that the locus is irreducible. moduli spaces of vector bundles; projective curves; stability Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Families, moduli of curves (algebraic), Rational and ruled surfaces, Algebraic moduli problems, moduli of vector bundles Non-globally generated bundles on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 75, 127-142 (Russian) (1978; Zbl 0477.13001). finite group of automorphisms; Pic; Brauer group; Picard group Galois theory and commutative ring extensions, Morphisms of commutative rings, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Brauer groups of schemes, Picard groups The seven-term sequence in the Galois theory of 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 This paper is based on results of \textit{N. Hitchin} [Duke Math. J. 54, 90- 114 (1987; Zbl 0627.14024) and Proc. Lond. Math. Soc., III. Ser. 55, 59- 126 (1987; Zbl 0634.53045)] extended by Simpson. Hitchin proved that for any simple compact Lie group \(G\) the cotangent bundle of the moduli space of stable principal \(G\)-bundles over a compact Riemann surface is a completely integrable system. He showed that when \(G\) is one of the classical groups then the generic level set of this integrable system can be compactified to a Jacobian or Prym variety. Analogous results are obtained in this paper for the geometry of the level sets when \(G\) is the exceptional Lie group \(G_ 2\). exceptional Lie group; moduli space over compact Riemann surface; compactification of level set of integrable system; Prym variety; Jacobian variety; completely integrable system Katzarkov, L.; Pantev, T., Stable \(G_2\) bundles and algebraically completely integrable systems, Compos. math., 92, 43-60, (1994) Families, moduli of curves (algebraic), Picard schemes, higher Jacobians, Lie algebras of linear algebraic groups, Dynamical systems and ergodic theory, Algebraic moduli problems, moduli of vector bundles, Differentials on Riemann surfaces, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Stable \(G_ 2\) bundles and algebraically completely integrable systems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\overline{\mathcal M}_{g,n}\) be the moduli space of stable genus \(g\) curves with \(n\) marked point and \(\psi_i\) be the cotangent line bundle class associated with the \(i\)-th marked points. The top intersection numbers of \(\psi\)-classes packaged in certain generating function satisfy a fascinating KdV hierarchy, conjectured by Witten and first proved by \textit{M. Kontsevich} [Commun. Math. Phys. 147, No. 1, 1--23 (1992; Zbl 0756.35081)]. One can change the form of such generating function to the so-called \(n\)-point function. In this paper the author exhibits a new explicit formula for the \(n\)-point function. The proof uses the formula of the intersection numbers of \(\psi\)-classes with the double ramification cycle on the moduli space of curves discovered in [\textit{A. Buryak} et al., Am. J. Math. 137, No. 3, 699--737 (2015; Zbl 1342.14054)]. moduli space of curves; intersection numbers Families, moduli of curves (algebraic), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Double ramification cycles and the \(n\)-point function for the moduli space of curves | 0 |
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