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Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(A_g\) be the coarse parameter space for principally polarized abelian varieties (p.p.a.v) of dimension \(g\) over an algebraically closed field of characteristic \( k \neq 2,3\) and \(\chi_g\) the universal p.p.a.v over \( A_g\). It has been shown by several authors as for example \textit{E. Freitag} [J. Reine Angew. Math. 296, 162--170~(1977; Zbl 0366.10023)] that for \( g \geq 7 \) , \( A_g\) is of general type thus the cases \( g \leq 6 \) are open. In particular it has been shown by several authors, as for example see in \textit{C. H. Clemens} [Adv. Math. 47,~107--230~(1983; Zbl 0509.14045)] and \textit{S. Mori} and \textit{S. Mukai} [Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982, Lect. Notes Math. 1016, 334--353 (1983; Zbl 0557.14015)] that \( A_4\) (resp. \( A_5\)) are unirational.
The author's main result is to prove that \( \chi_4\) is unirational and the universal theta divisor over \(A_4\) is unirational.~The paper is organized as follows. In section two, the author introduces basic notation and preliminaries on Prym varieties. The author considers the parameter space of pairs \( (\pi, d)\) where \(\pi: \tilde{C} \rightarrow C\) is a non-split étale double cover of a smoooth irreducible cover of genus 5 and \(d \) is an isolated, effective divisor on \(\tilde{C}\) such that \( {\pi}_{*}d \in | \omega_C|\) which exists by proposition 5.6 .~In section 6, the author introduces the construction of a moduli space which dominates \( \chi_4\) in definition 6.1 and in proposition 6.4 using Brill-Noether theory, from which it follows that there exists a line bundle \(L\) such that the Petri map has corank one for a general \(\tilde{C}\) and hence defines an embedding \( d \subset \tilde C \subset \mathbb{P}^2 \times \mathbb{P}^2 \) and uses in sections 3, 4 and 5 this embedding to construct a rational space for the family of 0-cycles \(d\) as above. The author in definition 4.1 introduces the rational family of triples \( (o, \tilde{S}, d)\) satisfying:
1)~\(o = ( y, B)\) where \( B \subset \mathbb{P}^2 \times \mathbb{P}^2 \) is a smooth conic and \( y= ( a, b_1, b_2)\) is a triple such that: \( (b_1, b_2) \in B \times B\), \(a\) is a set of 6 independent points in \(\mathbb{P}^2 \times \mathbb{P}^2\), \(q(a)\) is a set of 6 coplanar points where \(q : \mathbb{P}^8 \rightarrow \mathbb{P}^5 \) is the projection from the 2-dimensional projectivized eigenspace of \(\imath\) where \(\imath\) is the projective involution exchanging the factors of \(\mathbb{P}^2 \times \mathbb{P}^2\).
2)~\(\tilde{S}\) is a smooth \(K3\) surface which is a complete intersection of a quadratic and a hyperplane section of \(\mathbb{P}^2 \times \mathbb{P}^2\). Moreover, \(\imath \) restricts to a Nikulin involution on \(\tilde{S}\) and \( a \cup B \subset \tilde{S} \).
3)~\( a \cup b_1 \cup b_2 \) is contained in a unique smooth hyperplane section \( \tilde{F}\) of \( \tilde{S}\). Moreover \(\imath\) restricts to a fixed point-free involution on \(\tilde{F}\) and finally \( d \in | O_{\tilde{F}}( a + b_1 + \imath(b_2))| \). A triple with such properties is called a marking 0-cycle of type 1.~The author shows that the latter linear system is a pencil of divisors of degree 8 and that the family \(D_1\) of marking 0-cycles of type 1 is rational by proposition 4.4. In definition 5.1 the author considers the family of all 4-tuples \( ( o, \tilde{S}, d, \tilde{C})\) denoted as \( \tilde{C}_1\) and the map \(\phi: \tilde{C}_1 \rightarrow \chi_4\), such that \( \phi ( o, \tilde{S}, d, \tilde{C}) = (P(\pi), d) \).~In proposition 5.7, the author shows that \( \tilde{C}_{1}\) is a \(\mathbb{P}^1\)-bundle over \(D_1\) hence it is rational.~In lemma 6.9, he shows that \(\phi \) is dominant so that \(\chi_4\) is unirational. In the last section, in the proof of proposition 6.10 the unirationality of the universal theta divisor over \(A_4\) is proven similarly. The author introduces the family of marking 0-cycles of type 2 which are triples \( (o, \tilde{S}, d)\) defined by the same conditions 1),2),3) as above except that \( d \in | O_{ \tilde{F}}(a + b_1+ b_2)| \). algebraic moduli of abelian varieties; rational and unirational varieties; Jacobians; Prym varieties; families Verra A.: On the universal principally polarized abelian variety of dimension 4. In: Curves and abelian varieties, Proceedings of Internation Conference at Athens, Georgia, 2007, Contemporary Mathematics, vol. 345, pp. 253--274 (2008) Algebraic moduli of abelian varieties, classification, Rational and unirational varieties, Jacobians, Prym varieties, Families, moduli of curves (algebraic) On the universal principally polarized abelian variety of dimension 4 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In [\textit{G.-M. Greuel, C. Lossen} and \textit{E. Shustin}, Int. Math. Res. Not. 2001, No.11, 543--550 (2001; Zbl 0982.14018)] the authors gave a general sufficient numerical condition for the T-smooth-ness (smoothness and expected dimension) of equisingular families of plane curves. This condition involves a new invariant \(\gamma^\ast\) for plane curve singularities, and it is conjectured to be asymptotically proper. In [\textit{T. Keilen}, Trans. Am. Math. Soc. 357, No. 6, 2467--2481 (2005; Zbl 1070.14029)], similar sufficient numerical conditions are obtained for the T-smoothness of equisingular families on various classes surfaces. These conditions involve a series of invariants \(\gamma^\ast_\alpha\), \(0 \leq \alpha \leq 1\), with \(\gamma^\ast-1 = \gamma^\ast\). In the present paper we compute (respectively give bounds for) these invariants for semiquasihomogeneous singularities. Christoph Lossen and Thomas Keilen, The \(\gamma_\alpha\)-invariant for plane curve singularities, Preprint, 2003. Singularities of curves, local rings, Families, moduli of curves (analytic), Families, moduli of curves (algebraic) A new invariant for plane curve singularities | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(V_{g,n}\) be the Weil-Petterson volume of the moduli space \(\mathcal{M}_{g,n}\) of complex algebraic curves of genus \(g\) with \(n\) punctures. The paper under review studies the asymptotic behaviour of \(V_{g,n}\) as \(g \rightarrow \infty\). The main result is theorem 1.2, which states that there exists a constant \(C \in (0,\infty)\), such that for any given \(k \geq 1, n \geq 0\),
\[
V_{g,n} = C \frac {(2g-3+n)! (4 \pi^2)^{2g-3+n}} {\sqrt g} \left(1 + \sum_{i=1}^{k} \frac{c_n^{(i)}} {g^i} + O\left(\frac {1} {g^{k+1}}\right)\right)
\]
\noindent as \(g \rightarrow \infty\). Each \(c_n^{(i)}\) is a polynomial in \(n\) of degree \(2i\) with coefficients in \(\mathbb{Q}[\pi ^{-2}, \pi ^2]\), effectively computable.
This result is consistent with the conjecture made by the second author in [``On the large genus asymptotics of Weil-Peterson volumes'', Preprint (2008) \url{arXiv:0812.0544}]. In fact, the conjecture being true, the constant \(C\) would be \(\frac {1}{\sqrt \pi}\).
The proof of the result is made by studying the quotient \(\frac {V_{g+1,n}} {V_{g,n}}\) for a given \(n\), and \(g \rightarrow \infty\). Estimates for this ratio were obtained by the first author in [J. Differ. Geom. 94, No. 2, 267--300 (2013; Zbl 1270.30014)].
In the final section of the paper, the authors consider the behaviour of \(V_{g,n(g)}\) when \(n(g) \rightarrow \infty\) as \(g \rightarrow \infty\) but \(n(g)^2/g \rightarrow 0\). Weil-Petersson volume; moduli space; algebraic curve Maryam, Mirzakhani.; Peter, Zograf., Towards large genus asymptotics of intersection numbers on moduli spaces of curves, Geom. Funct. Anal., 25, 1258-1289, (2015) Families, moduli of curves (algebraic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Towards large genus asymptotics of intersection numbers on moduli spaces 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 \(\overline{\mathcal M}_{g,n}\) be the moduli space of stable genus \(g\) curves with \(n\) marked points. The tautological ring of \(\overline{\mathcal M}_{g,n}\) is the minimal subring inside the Chow ring (or cohomology ring) that is stable under pullbacks and pushforwards via forgetful and attaching maps between these moduli spaces. The natural classes \(\kappa\), \(\psi\), \(\lambda\), and boundary classes are all contained in the tautological ring. The study of relations between tautological classes dates back to Mumford, and has been investigated intensively in the last three decades.
In this paper the author studies the equivariant cycle of the moduli space of stable maps to \([\mathbb C/\mathbb Z_r]\), or equivalently, the vanishing of high-degree Chern classes of a certain vector bundle over the moduli space of stable maps to \(B\mathbb Z_r\), where \(B\mathbb Z_r\) denotes the stack consisting of a single point with \(\mathbb Z_r\) isotropy. As a result, the author obtains relations in the Chow ring of \(\overline{\mathcal M}_{g,n}(B\mathbb Z_r, 0)\), and they push forward to yield relations on \(\overline{\mathcal M}_{g,n}\). tautological ring; moduli space of curves; orbifold stable maps 10.1090/proc/13344 Families, moduli of curves (algebraic) Relations on \(\overline{\mathcal{M}}_{g,n}\) via orbifold stable maps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 stack \(\overline{\mathcal M}_{g,n}\) of stable curves and its coarse moduli space \(\overline{\mathcal M}_{g,n}\) are defined over \(\mathbb Z\), and therefore over any field. Over an algebraically closed field of characteristic zero, in [Algebra Number Theory 2, No. 7, 809--818 (2008; Zbl 1166.14019)] \textit{P. Hacking} showed that \(\overline{\mathcal M}_{g,n}\) is rigid (a conjecture of Kapranov), in [J. Eur. Math. Soc. (JEMS) 15, No. 3, 949--968 (2013; Zbl 1277.14022); J. Lond. Math. Soc., II. Ser. 89, No. 1, 131--150 (2014; Zbl 1327.14133)] \textit{A. Bruno} and \textit{M. Mella} for \(g=0\), and the second author for \(g\geq 1\) showed that its automorphism group is the symmetric group \(S_n\), permuting marked points unless \((g,n)\in\{(0,4), (1,1), (1,2)\}\). The methods used in the papers above do not extend to positive characteristic. We show that in characteristic \(p>0\), the rigidity of \(\overline{\mathcal M}_{g,n}\), with the same exceptions as over \(\mathbb C\), implies that its automorphism group is \(S_n\). We prove that, over any field, \(\overline{\mathcal M}_{0,n}\) is rigid and deduce that, over any field, \(\mathrm{Aut}(\overline{\mathcal M}_{0,n})\cong S_n\) for \(n\geq 5\). Going back to characteristic zero, we prove that for \(g+n>4\), the coarse moduli space \(\overline{\mathcal M}_{g,n}\) is rigid, extending a result of Hacking who had proven it has no locally trivial deformations. Finally, we show that \(\overline{\mathcal M}_{1,2}\) is not rigid, although it does not admit locally trivial deformations, by explicitly computing his Kuranishi family. Families, moduli of curves (algebraic), Stacks and moduli problems, Fine and coarse moduli spaces On the rigidity of moduli of curves in arbitrary characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We show that the class of the locus of hyperelliptic curves with \(\ell\) marked Weierstrass points, \(m\) marked conjugate pairs of points, and \(n\) free marked points is rigid and extremal in the cone of effective codimension-\((\ell+m)\) classes on \(\overline{\mathcal{M}}_{2,\ell+ 2m+n}\). This generalizes work of \textit{D. Chen} and \textit{N. Tarasca} [Algebra Number Theory 10, No. 9, 1935--1948 (2016; Zbl 1354.14048)] and establishes an infinite family of rigid and extremal classes in arbitrarily-high codimension. subvarieties of moduli spaces of curves; effective cones; higher codimensional cycles; hyperelliptic curves Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Algebraic cycles Hyperelliptic classes are rigid and extremal in genus two | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians An algebraic family \(\{Y_{\lambda}:\) \(\Lambda\in \Lambda\}\) of subvarieties \(Y_{\lambda}\) of a variety X parametrized by a variety \(\Lambda\) is called effectively parametrized, if for all \(\lambda_ 0\in \Lambda\) the set \(\{\lambda \in \Lambda:\quad Y_{\lambda} = Y_{\lambda_ 0}\}\) is finite, it is called closed if \(\Lambda\) is complete. It is the aim of the paper to give upper bounds for the dimension of \(\Lambda\) for such a family. The first result is that if all the varieties \(Y_{\lambda}\) are smooth, then \(\dim (\Lambda)\leq (\dim Y_{\lambda}+1)co\dim (Y_{\lambda},X).\) This estimate is best possible in general as the example of linear subspaces in \({\mathbb{P}}_ n\) shows. The method of proof is to analyze the infinitesimal situation of subvarieties all passing through a fixed point P of X and to study families of curvilinear schemes supported at P. In the case of curves in \({\mathbb{P}}_ n\) the above estimate can be improved to give the following main result of the paper: If the curves \(Y_{\lambda}\) of the family are nondegenerate and nonsingular in \({\mathbb{P}}_ n\) then \(\dim(\Lambda)\leq n- 1\). In fact the result is slightly more general. Examples show that the estimate is almost best possible. dimension of family of subvarieties; effectively parametrized; family of subvarieties; closed family of subvarieties; families of curvilinear schemes Chang, M.C. and Ran, Z. , Closed families of smooth space curves , Duke Math. J. 52 (1985) 707-713. Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Structure of families (Picard-Lefschetz, monodromy, etc.) Closed families of smooth space curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper we study moduli problems of real algebraic geometry in two ways. We first show that the space of real isomorphism classes of smooth real algebraic curves of a given topological type is a connected real analytic space. Using the Torelli mapping we then identify this moduli space with a subset of the moduli space of principally polarized real abelian varieties. This subset corresponds to real abelian varieties with a fixed number of connected components in the real part and a fixed real polarization (orthosymmetric or diasymmetric). We finally show that such abelian varieties form a semialgebraic set. This result allows us to equip also the moduli space of smooth real algebraic curves (of a given topological type) with a semialgebraic structure. moduli problems of real algebraic geometry; Torelli mapping; principally polarized real abelian varieties Seppälä M., Math. Z. 201 pp 151-- (1989) Real algebraic and real-analytic geometry, Families, moduli of curves (algebraic), Algebraic moduli of abelian varieties, classification Moduli spaces for real algebraic curves and real abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Surfaces of degree s and with a line of multiplicity s-2 are easily seen to be rational. In this note the author shows that they can be obtained as blow-ups of Hirzebruch surfaces of degree at most s-1 blowing up at most 3s-4 points. - Special attention is given to quartics with a double line. Curves on such quartics were exploited by Peskine and Gruson to classify all pairs of degree and genus that can occur for space curves.
The author studies the k-normality of smooth irreducible curves on such quartics, and gives a nice bound on k. blow-ups; Hirzebruch surfaces; quartics with a double line; normality Special surfaces, Projective techniques in algebraic geometry, Families, moduli of curves (algebraic) Courbes tracées sur les surfaces quartiques à droite double. (Curves on quartics with double 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 Denote by \(H_{d,g,r}\) the Hilbert scheme parametrizing projective smooth irreducible complex curves of degree \(d\) and genus \(g\) in \({\mathbb P^r}\). A natural question concerning \(H_{d,g,r}\), which goes back to [\textit{F. Severi}, Vorlesungen über algebraische Geometrie, Teubner (1921; JFM 48.0687.01)], is whether it is irreducible under the assumption \(d\geq g+r\).
In more recent years \textit{J. Harris} [see \textit{L. Ein}, Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, Part 1, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], and \textit{C. Keem} [Proc. Am. Math. Soc., 122, No.~2, 349--354 (1994; Zbl 0860.14003)] proved by examples that \(H_{d,g,r}\) may be reducible for \(d\geq g+r\) and \(r\geq 6\). Moreover \textit{L. Ein} [loc. cit.; Ann. Sci. Éc. Norm. Supér., IV. Sér., 19, No.~4, 469--478 (1986; Zbl 0606.14003)], and \textit{C. Keem} and \textit{S. Kim} [J. Algebra, 145, No.~1, 240--248 (1992; Zbl 0783.14002)] proved the irreducibility of \(H_{d,g,r}\) when \(d\geq g+r\), for \(r=3\) and \(r=4\). In their paper, C. Keem and S. Kim also proved the irreduciblity of \(H_{g+2,g,3}\) if \(g\geq 5\), and of \(H_{g+1,g,3}\) if \(g\geq 11\).
Continuing the quoted works of L. Ein, C. Keem and S. Kim, and using Brill-Noether Theory as developed by [\textit{E. Arbarello, M. Cornalba, P. Griffiths} and \textit{J. Harris}, ``Geometry of Algebraic Curves'', Grundlehren der Mathematischen Wissenschaften, 267 (1985; Zbl 0559.14017)], in the paper under review the author further refines the irreducibility range of \(H_{d,g,r}\) for \(3\leq r\leq 4\), proving that \(H_{g,g,3}\) is irreducible if \(g\geq 13\), and that \(H_{g+i,g,4}\) is irreducible for \(2\leq i\leq 3\) if \(g\geq 23-6i\). projective space curve; Brill-Noether Theory; line bundle; normal bundle; linear series \textsc{H. Iliev}, On the irreducibility of the Hilbert scheme of space curves, Proc. Amer. Math. Soc. \textbf{134} (2006), 2823-2832. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) On the irreducibility of the Hilbert scheme of space curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Lax operator algebras for the root system \(G_{2}\), and arbitrary finite genus Riemann surfaces and Tyurin data on them are constructed. O. K, S., No article title, Dokl. Math., 89, 151-153, (2014) Infinite-dimensional Lie (super)algebras, Families, moduli of curves (algebraic), Differentials on Riemann surfaces, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Lax operator algebras of type \(G_{2}\) | 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 See the preview in Zbl 0534.14014. moduli space of stable curves; homotopy type of moduli space; Satake compactification of the Siegel modular varieties; period mapping; Riemann surfaces; period matrices Families, moduli of curves (algebraic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Classification of homotopy type, Period matrices, variation of Hodge structure; degenerations, Compactification of analytic spaces Moduli space of stable curves from a homotopy viewpoint | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians With the notation of the previous review (Zbl 0932.13010) let \(G\) be a line-\(S_4.\) Then it is shown that the nondegenerate \(G\)-quartics are parametrized by \(F.\) For an appropriate \(G\) each nondegenerate \(G\)-quartic is a constant multiple of \(h_a = az^4 + (x^2 + yz)(y^2 + xz)\), \(a \in F.\) For each \(a\) the author attaches an integer \(l = l(a)\) that determines completely \(e_n(h_a).\) In particular, it turns out that \(c(h_a) = 3 + 4^{-2l}.\) The surprise is the definition of \(l = l(a).\) The author constructs a \(1\)-parameter family of dynamical systems parametrized by \(F.\) Then \(l(a)\) is defined to be an `escape time' for the system corresponding to \(a.\) Hilbert-Kunz function; Hilbert-Kunz multiplicity; plane quartic; dynamical systems Monsky, P.: Hilbert--kunz functions in a family: line-S4 quartics. J. algebra 208, 359-371 (1998) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Families, moduli of curves (algebraic), Polynomial rings and ideals; rings of integer-valued polynomials Hilbert-Kunz functions in a family: Line-\(S_4\) quartics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 a well-known configuration of 9 points and 12 lines in \(\mathbb P^2(k)\), the Hesse configuration, in which each point lies on 4 lines and each line contains 3 points. Such points can be chosen as the nine inflection points of a nonsingular plane cubic curve, and they can be as well taken as common inflection points of the Hesse pencil
\[
\lambda (x^3+y^3+z^3)+\mu xyz=0.
\]
The group of plane automorphisms preserving the Hesse pencil has order 216 and it is isomorphic to \((\mathbb Z/3\mathbb Z)SL)^2 \rtimes SL(2, \mathbb F_3)\). The algebra of invariant polynomials of one of its extensions to a subgroup of \(GL(3,\mathbb C)\) has a generator of degree 6 defining a plane sextic \(C_6\).
The double covers of \(\mathbb P^2\) branched over \(C_6\) and over the singular sextic \(C'_6\) with 8 cuspidal singularities are both \(K3\) surfaces and they are singular in the sense of Shioda, i.e. the subgroup of algebraic cycles in the second cohomology group is of maximal rank.
The authors compute the intersection form defined by the cup-product on these subgroups and describe the geometrical meaning of the set of intersection points \(C_6\) cuts each curve of the Hesse pencil at. configuration; pencil; double covers M. Artebani; I. Dolgachev, The Hesse pencil of plane cubic curves, L'Enseignement Mathématique. Revue Internationale. 2e Série, 55, 235-273, (2009) Families, moduli of curves (algebraic), Plane and space curves, Families, moduli, classification: algebraic theory The Hesse pencil of plane cubic 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 During the past fifteen years, there have been several attempts to solve a Schottky-type problem for Prym varieties, that is to characterize the locus of Prym varieties inside the corresponding moduli space of polarized abelian varieties either geometrically or by certain equations in theta constants. Some partial results concerning this problem have been obtained, in the meantime, by Shiota, Taimanov, Li, Mulase, Plaza-Martín, and others using different approaches. The paper under review also points in this direction and has two main objectives. First, the authors generalize some previous results of Shiota and Plaza-Martín to the more general case of Prym varieties associated with curves admitting an automorphism of prime order. Then they give an explicit description of the equations defining the moduli space of curves with an automorphism of prime order as a subscheme of Sato's infinite Grassmannian. Using the formal approach developed by two of the authors [\textit{J. M. Muñoz Porras} and \textit{F. J. Plaza Martín}, Equations of Hurwitz schemes in the infinite Grassmannian, Preprint http://arxiv.org/abs/math/0207091] in order to characterize Hurwitz schemes in the framework of inifinite Grassmannians, and extending it to their new concept of formal Prym varieties, the authors establish an analogue of the classical Krichever map as well as an explicit characterization of formal Prym varieties as subvarieties of the the Sato Grassmannian. Finally, in the last section of the present paper, explicit equations of the moduli spaces of curves with automorphisms of prime order are derived within the same framework. The latter formal approach is based on the results and methods of another foregoing work of two of the authors, and being concerned with the equations defining the moduli spaces of pointed curves in the infinite Grassmannian [\textit{J. M. Muñoz Porras} and \textit{F. J. Plaza Martín}, J. Differ. Geom. 51, No. 3, 431--469 (1999; Zbl 1065.14512)]. Jacobians; automorphisms of curves; infinite Grassmannians; moduli spaces; Krichever correspondence; formal schemes E. Gómez González, J. M. Muñoz Porras, and F. J. Plaza Martín, Prym varieties, curves with automorphisms and the Sato Grassmannian, Math. Ann. 327 (2003), no. 4, 609 -- 639. Jacobians, Prym varieties, Automorphisms of curves, Families, moduli of curves (algebraic), Infinite-dimensional manifolds, Theta functions and curves; Schottky problem, Generalizations (algebraic spaces, stacks) Prym varieties, curves with automorphisms and the Sato Grassmannian | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is a survey on recent research directly related to the other papers in the same volume.
After introducing the mapping class group and the Torelli group, the moduli space \({\mathcal M}_{g,n}\) of genus \(g,n\)-pointed, smooth curves is constructed as an analytic orbifold. The Deligne-Mumford-Knudsen compactification \(\overline{{\mathcal M}_{g,n}}\) of \({\mathcal M}_{g, n}\) is then regarded as the quotient of a smooth complete variety by a finite group. The \(k\)-th Chow group of \(\overline{{\mathcal M}_{g,n}}\) is then definable as the invariant part of the \(k\)-th Chow group of this smooth variety.
Next the authors recall the definition of the tautological classes and the results on the stability of the homology of the mapping class group as well as Mumford's conjecture. Finally the authors discuss the Witten conjecture, proved by Kontsevich, and its generalization to moduli spaces of stable maps.
The paper also contains an overview on general methods for finding complete subvarieties of \({\mathcal M}_g\). mapping class group; Torelli group; moduli space; Deligne-Mumford-Knudsen compactification; Chow group; tautological classes; Witten conjecture Carel Faber and Eduard Looijenga, Remarks on moduli of curves, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 23 -- 45. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, (Equivariant) Chow groups and rings; motives, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Remarks on moduli 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 Abramovich, Corti and Vistoli have studied modular compactifications of stacks of curves equipped with abelian level structures arising as substacks of the stack of twisted stable maps into the classifying stack of a finite group, provided the order of the group is invertible on the base scheme. Recently, Abramovich, Olsson and Vistoli [\textit{D. Abramovich} et al., J. Algebr. Geom. 20, No. 3, 399--477 (2011; Zbl 1225.14020)] extended the notion of twisted stable maps to allow arbitrary base schemes, where the target is a tame stack, not necessarily Deligne-Mumford. We use this to extend the results of Abramovich, Corti and Vistoli to the case of elliptic curves with level structures over arbitrary base schemes; we prove that we recover the compactified Katz-Mazur regular models, with a natural moduli interpretation in terms of level structures on Picard schemes of twisted curves. Additionally, we study the interactions of the different such moduli stacks contained in a stack of twisted stable maps in characteristics dividing the level. generalized elliptic curve; twisted curve; Drinfeld; structure; moduli stack Niles, A., Moduli of elliptic curves via twisted stable maps, Algebra Number Theory, 7, 2141-2202, (2013) Arithmetic aspects of modular and Shimura varieties, Algebraic moduli of abelian varieties, classification, Families, moduli of curves (algebraic), Stacks and moduli problems, Elliptic curves Moduli of elliptic curves via twisted stable maps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0614.00006.]
Let \(H(d,g)_ S\) be the open subscheme of the Hilbert scheme of curves of degree \(d\) and arithmetic genus g in \({\mathbb{P}}^ 3\) parametrizing smooth irreducible curves. The first author who pointed out the existence of irreducible non reduced components of \(H(d,g)_ S\), was \textit{D. Mumford} [Am. J. Math. 84, 642-648 (1962; Zbl 0114.131)] who found a non reduced component of \(H(14,24)_ S\), the general curve of which lies on a smooth cubic surface in \({\mathbb{P}}^ 3\). Mumford's example has been widely generalized by the author of the present paper in his thesis (``The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in 3-space'', Preprint no. 5-1981, Univ. Oslo). Among other things it turns out from his analysis that if \(W\subseteq H(d,g)_ S\) is a closed irreducible subset whose general point corresponds to a curve C lying on a smooth cubic surface, W is maximal under this condition and \(d>9\), then W irreducible, non reduced component of \(H(d,g)_ S\) yields \(g\geq 3d-18\) and \(H^ 1({\mathcal J}_ C(3))\neq 0\) (the latter inequality implying that \(g\leq (d^ 2-4)/8.\)
The author conjectures that these necessary conditions are also sufficient for W to be a non reduced component of \(H(d,g)_ S\), and he proves this conjecture in the ranges \(7+(d-2)^ 2/8<g\leq (d^ 2-4)/8\), \(d\geq 18\) and \(-1+(d^ 2-4)/8<g\leq (d^ 2-4)/8\), 17\(\geq d\geq 14\). The proof consists in an interesting analysis of the tangent and obstruction space to the so called Hilbert-flag scheme (parametrizing pairs (curve, surface), the first contained in the latter) in particular for curves lying on surfaces of degree \(s\leq 4.\) space curves; Hilbert scheme; degree; arithmetic genus; obstruction space; Hilbert-flag scheme J. O. Kleppe, Nonreduced components of the Hilbert scheme of smooth space curves. In Space curves (Rocca di Papa, 1985), volume 1266 of Lecture Notes in Math. (Springer, Berlin, 1987), pp. 181-207. Zbl0631.14022 MR908714 Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry Non-reduced components of the Hilbert scheme of smooth space curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be a projective k-scheme where k is an algebraically closed field and D a real divisor in the closure K of the cone \(N^ 1(X)\otimes_{{\mathbb{Z}}}{\mathbb{R}}\) generated by the classes of ample divisors. The authors show that D is exactly in the boundary of K if there is an irreducible closed subscheme \(Y\subset X\), say of dimension s, such that \(D^ s\cdot Y=0\). This is a ``real'' version of the Nakai- Moishezon criterion for ampleness of divisors. The method of proof depends on the classical approach used by Kleiman.
The second part of the paper deals with a more explicit description of the boundary of K and discusses some examples. real divisor; Nakai-Moishezon criterion for ampleness of divisors [2] Frédéric Campana &aThomas Peternell, &Algebraicity of the ample cone of projective varieties&#xJ. Reine Angew. Math.407 (1990), p.~160-MR~10 | &Zbl~0728. Divisors, linear systems, invertible sheaves, Real algebraic sets, Picard groups Algebraicity of the ample cone of projective varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be an algebraically closed field of characteristic \(p\geq 3\). Let \(f: X\to C\) be a non-isotrivial semistable family of curves of genus \(g\geq 2\) defined over \(k\). Let \(b\) be the genus of the base curve \(C\), and let \(s\) be the number of singular fibers of \(f\).
This paper provides the Arakelov inequality in positive characteristic, which is similar to the result of \textit{E. Viehweg} and \textit{K. Zuo} [J. Differ. Geom. 77, No. 2, 291--352 (2007; Zbl 1133.14010)] in characteristic zero. The result is obtained assuming that \(f\) is liftable to the ring \(W_2(k)\) of the second Witt vectors of \(k\).
Theorem 1: Let \(k\) be an algebraically closed field of characteristic \(\text{char}(k)=p\geq 3\). Let \(f:X\to C\) be a non-isotrivial semistable family of curves of genus \(g\geq 2\) defined over \(k\). Assume that \(f\) is liftable to \(W_2(k)\). Then
\[
\text{deg} f_*\omega_{X/C}\leq \frac{g-g_0}{2}(2b-2+s)
\]
where \(g_0\) is the rank of the kernel of the Higgs field associated to \(f\).
Proof of Theorem 1 is similar to that of Viehweg and Zuo in characteristic zero case, but with crucial use of the semistablity of Higgs bundles corresponding to families of curves to estimate the degrees of the kernel and the image of the Higgs field.
Theorem 2: Let \(f:X\to C\) be a non-isotrivial semistable family of curves of genus \(g\geq 2\) defined over \(k\). Assume that \(f\) is liftable to the ring \(W(k)\) of Witt vectors of \(k\). Then
\[
\text{deg} f_*\omega_{X/C}\leq \frac{g}{2}(2b-2+s_1)
\]
where \(s_1\) is the number of singular fibers with non-compact Jacobians.
Proof of Theorem 2 relies on the consistency of the relative invariants of \(f\) and its lifting.
As a byproduct, an analogue of Beauville's conjecture in positive characteristic is presented when \(f\) is liftable to \(W(k)\).
Corollary: Let \(f: X\to C={\mathbb{P}}^1\) as in Theorem 2. Then \(s\geq 5\). That is, there are at least \(5\) singular fibers for non-isotrivial semistable families of curves of genus \(g\geq 2\) over \({\mathbb{P}}^1\), which are liftable to \(W(k)\).
The definition of a lifting of a non-isotrivial semistable family of curves of genus \(g\geq 2\) over \(k\) is the standard one. A family of curves \(f:X\to C\) with singular fibers \(F_1,\dots F_s\) is said to be liftable to \(W_2(k)\) if there is a semistable family \(\tilde{f}: \mathcal{X}\to C\) over \(W_2(k)\) and a semistable curve \(\sum_{i=1}^s\tilde{F}_i\) on \(\mathcal{X}\) such that the reduction of \(\tilde{f}\) to \(k\) is \(f\) and \(\sum_{i=1}^s\tilde{F}_i\) is the lifting of \(\sum_{i=1}^s F_i\) to \(W_2(k)\). A similar definition for the liftablity of \(f\) to \(W(k)\). family of curves; Arakelov inequality; Higgs bundle; Beauville's conjecture Fibrations, degenerations in algebraic geometry, Families, moduli of curves (algebraic), Arithmetic ground fields for curves, Arithmetic varieties and schemes; Arakelov theory; heights On the Arakelov inequality in positive characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this paper is to obtain results on the moduli space \(\mathcal M_{15}\) of curves of genus \(15\). By Brill-Noether theory, a general such curve has a smooth model of degree 16 in \(\mathbb P^4\), so the author focuses his interest on the (unique) component \(\mathcal H \subset \mathrm{Hilb}_{16t+1-15}(\mathbb P^4)\) of the Hilbert scheme of curves of degree \(d=16\) and genus \(g = 15\) in \(\mathbb P^4\) which dominates the moduli space \(\mathcal M_{15}\). It is not hard to see that a general element \(C \in \mathcal H\) lies on a unique smooth cubic hypersurface \(X \subset \mathbb P^4\) defined by a homogeneous polynomial, say \(f\), of degree 3. The notion of a matrix factorization of \(f\) was introduced by \textit{D. Eisenbud} [Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)]; one produces a pair \((\phi,\psi)\) of matrices satisfying \(\psi \circ \phi =f \mathrm{id}\) and \(\phi \circ \psi = f \mathrm{id}\). From this one obtains a maximal Cohen-Macaulay module on the hypersurface ring \(R/f\). Setting \(\widetilde{\mathcal M}^4_{15,16} \) to be the component of \( \{(C,L) \mid C \in \mathcal M_{15}, L \in W_{16}^4 (C) \}\) which dominates \(\mathcal M_{15}\), the author proves that \(\widetilde{\mathcal M}^4_{15,16} \) is birational to a component of the moduli space of matrix factorizations of type \((\psi:\mathcal O^{18}(-3) \rightarrow \mathcal O^{15}(-1) \oplus \mathcal O^3(-2), \phi:\mathcal O^{15}(-1) \oplus \mathcal O^3(-2) \rightarrow \mathcal O^{18})\) of cubic forms on \(\mathbb P^4\). As a corollary he shows that a general cubic threefold in \(\mathbb P^4\) contains a 32-dimensional uniruled family of smooth curves of genus 15 and degree 16. He also shows that the moduli space \(\widetilde{\mathcal M}^4_{15,16} \) is uniruled, and produces a probabilistic algorithm to randomly produce curves of genus 15 over a finite field \(\mathbb F_q\) with \(q\) elements from a Zariski open subset of \(\mathcal M_{15}\) in running time \(O((\log q)^3)\). The original goal was to prove that \(\mathcal M_{15}\) is unirational, and he ends with a conjecture about the obstruction. He used the software package Macaulay2 in this work, and an important tool was Boij-Söderberg theory for a list of candidate Betti tables in the argument. matrix factorization; moduli of curves; unirationality Frank-Olaf Schreyer. Matrix factorizations and families of curves of genus 15. {Algebr. Geom.}, 2(4):489--507, 2015. DOI 10.14231/AG-2015-021; zbl 1342.14057; MR3403238; arxiv 1311.6962 Families, moduli of curves (algebraic), Syzygies, resolutions, complexes and commutative rings Matrix factorizations and families of curves of genus 15 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the paper under review the authors study a very natural question regarding the number of vertices of Newton-Okounkov polygons associated with big divisors on smooth algebraic surfaces.
Let \(S\) be a normal projective variety of dimension \(d\) and let \(D\) be a big divisor class on \(S\). We define the admissible flag for \(S\) which is
\[
Y_{\bullet} : S= Y_{0} \supset \dots \supset Y_{d} = \{pt\},
\]
where \(Y_{i}\) is an irreducible subvariety of codimension \(i\) and \(Y_{d}\) is smooth for each subvariety \(Y_{i}\). Let \(g_{i}\) be a local equation for \(Y_{i}\) in \(Y_{i-1}\) around the point \(Y_{d}\). Then the flag \(Y_{\bullet}\) determines a rank \(d\) valuation \(\nu_{Y_{\bullet}}\) on the field of rational functions of \(S\), namely \(\nu_{Y_{\bullet}}(f) = (v_{1}(f), \dots, v_{d}(f))\), where \(v_{i}\) are defined recursively setting \(f_{1}=f\) and
\[
v_{i}(f) = \mathrm{ord}(f_{i}), \quad i=1, \dots, d,
\]
\[
f_{i+1} = (f_{i}/g_{i}^{v_{i}(f)})_{|Y_{i}}, \quad i = 1, \dots, d-1.
\]
Based on that the Newton-Okounkov body of \(D\) with respect to \(Y_{\bullet}\) is the convex body
\[
\triangle_{Y_{\bullet}}(D) = \overline{\bigg\{\frac{\nu_{Y_{\bullet}}(s)}{k} : s \in H^{0}(S, \mathcal{O}_{S}(kD)) \bigg\}}.
\]
Whereas the volume of the Newton-Okounkov body is well-known, still its shape, and its dependence on \(Y_{\bullet}\), is an intriguing subject of study. The main aim of the paper is to understand this question when \(d=2\), where the prominent role is played by the Zariski decomposition of \(D\).
In order to formulate the main result of the paper under review, recall the following notations.
Let \(C_{1}, \dots, C_{k} \subset S\) be a configuration of negative curves. Assume that \(N\) is effective and the associated intersection matrix is negative definite. We denote by \(\mathrm{mc}(N)\) the largest number of irreducible components of a connected divisor contained in \(N\), and by \(\mathrm{mv}(N)\) we mean \(k+\mathrm{mc}(N)+4\) if \(k < \rho(S)-1\) and \(k+\mathrm{mc}(N)+3\) if \(k=\rho(S)-1\), where \(\rho(S)\) denotes the Picard number of \(S\).
Given a smooth projective surface \(S\), let
\[
\mathrm{mv}(S) =\max \{\mathrm{mv}(N):N = C_{1} +\dots+C_{k} \text{ is negative definite}\}.
\]
Main Result. On every smooth projective surface \(S\) and for every big divisor \(D\)
\[
\max_{Y_{\bullet}} \{ \# \text{ vertices of } \triangle_{Y_{\bullet}}(D)\} \leq \mathrm{mv}(S),
\]
where the maximum is taken over all admissible flags \(Y\). If \(D\) is ample, then for every \(3 \leq r \leq\mathrm{mv}(S)\) there exists a flag \(Y_{\bullet}\) such that \(\triangle_{Y_{\bullet}}(D)\) has exactly \(r\) vertices.
Corollary. Given a positive integer \(\rho\) there exists a projective smooth surface \(S\) with Picard number \(\rho = \rho(S)\), an admissible flag \(Y_{\bullet}\), and a divisor \(D\) such that \(\triangle_{Y_{\bullet}}(D)\) has exactly \(2\rho + 1\) vertices. Newton-Okounkov bodies; algebraic surfaces; toric geometry Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, Picard groups, Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry) On the number of vertices of Newton-Okounkov 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 Let \(D\subset\mathbb P^2\) be a general smooth complex plane curve of degree 5. The intersection of any line with \(D\) produces a configuration of 5 points on \(\mathbb P^1\) (up to \(\mathbb P GL(2)\)). How many times a fixed configuration occurs? The answer is 420 and the genericity assumption (any line intersects \(D\) at least in three distinct points) is necessary, for Fermat quintic this number equals 150.
In this very well written paper the authors address this classical question of enumerative geometry to illustrate the differences and advantages of various approaches.
The classical method is to consider the map \(\phi:\check{\mathbb P}^2-->\mathrm{Sym}^5\mathbb P^1//GL(2)\) defined by intersecting lines with the quintic. The map is finite, its indeterminacy locus corresponds to the lines tangent to \(D\) at the inflection points. The needed number is precisely the degree of the map. A natural way to compute this degree (after resolving the map) is to compare the self- intersection of divisors in the source and in the target: \((\phi^*\Delta)^2/\Delta^2\). The test divisor is the discriminant in \(Sym^5\mathbb P^1//GL(2)\) (i.e. the configurations with multiple roots).
The second method uses the moduli space \(\mathcal M_{0,5}(\mathbb P^2,1)\) of degree=1, genus=0 stable maps to \(\mathbb P^2\). Consider the closed subscheme \(\mathcal M_{0,5}^D(\mathbb P^2,1)\subset \mathcal M_{0,5}(\mathbb P^2,1)\) of all the configurations arising from the intersections with \(D\). Then the degree of the forgetfull map \(\mathcal M_{0,5}^D(\mathbb P^2,1)\to\mathcal M_{0,5}\) is precisely the needed number. The degree of this later map is computed by counting the preimages of a curve with three components. The essential step is to check that the map is étale over such a point of \(\mathcal M_{0,5}\).
The third method uses the Gromov-Witten theory for the stack \(\mathbb P^2_{D,r}\). The computation of the twisted Gromov-Witten invariant is straightforward, but one should prove that this invariant gives the actual degree of the map. Enumerative problems (combinatorial problems) in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Pencils, nets, webs in algebraic geometry, Families, moduli of curves (algebraic), Stacks and moduli problems, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Varieties of low degree Counting the hyperplane sections with fixed invariants of a plane quintic - three approaches to a classical enumerative problem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 moduli space of a punctured Riemann sphere. Let \(Y\) be the real part of the Grothendieck-Knudson compactification of \(X\). Then \(Y\) has the following two interesting properties [see \textit{M. Davis}, \textit{T. Januszkiewicz} and \textit{R. Scott}, Adv. Math. 177, 115--179 (2003; Zbl 1080.52512)]:
(1) \(Y\) is a stratified space, whose strata are labelled by planar trees.
(2) \(Y\) has fundamental group analogous to the Artin pure braid group, hence called a pure quasi-braid group.
Since Neretin's spheromorphism group \(N\), which contains Thompson's group \(V\), likewise depends on planar trees, the author manages to use a precise form of property (1) to construct an action of \(N\) (and hence also of \(V\)) on a suitable tower of such spaces \(Y\).
Passing to the universal cover of the towers, the author uses property (2) to obtain extensions of \(N\) and \(V\) by an infinite pure quasi-braid group. The author analyzes these extensions in terms of a certain cohomological Euler class. central group extensions; Euler class; moduli spaces of genus zero stable curves; Neretin's group of spheromorphisms; operads; quasi-braid groups; stabilization; Stasheff associahedron; Thompson's group C. Kapoudjian, From symmetries of the modular tower of genus zero real stable curves to an Euler class for the dyadic circle, math.GR/0006055. Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Groups acting on trees, Braid groups; Artin groups, Differential geometry of symmetric spaces, Topological methods in group theory From symmetries of the modular tower of genus zero real stable curves to an Euler class for the dyadic circle | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 deals with the moduli stack of \(SL(2)\)-bundles with connections on \(\mathbb{P}^1\). It is supposed that these connections have poles of first order at \(n\) points, and the conjugate classes of their residues are fixed. The authors compute the Picard group of such a stack. In the case of four points the cohomology groups of invertible sheaves on the stack are computed. invertible sheaves; bundles; cohomology group; moduli stack; Picard group Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Picard groups, Divisors, linear systems, invertible sheaves Invertible sheaves on the moduli space of \(SL(2)\)-bundles with connections on \(\mathbb{P}^1\) | 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 For an elliptic curve \(Y\) denote by \({\mathcal H}^0_{4,n}\) the Hurwitz space parametrizing fourfold coverings of \(Y\) with simple ramification in \(n\geq 2\) points, which do not factorize via an étale double covering of \(Y\). There is a smooth fibration \({\mathcal H}^0_{4,n}\to \text{Pic}^{n/2}(Y)\) whose fibre over \(A\) is denoted by \({\mathcal H}^0_{4,A}\). The main result of the paper is that \({\mathcal H}^0_{4,n}\) is connected and \({\mathcal H}^0_{4,A}\) is connected and unirational. This is an analogue of a result of \textit{E. Arbarello} and \textit{M. Cornalba} [Math. Ann. 256, 341--362 (1981; Zbl 0454.14023)] who showed that \({\mathcal H}_{4,n}(\mathbb{P}^1)\) is unirational for \(d\leq 5\). The proof uses heavily the description of fourfold coverings in terms of vector bundles by \textit{G. Casnati} and \textit{T. Ekedahl} [J. Algebr. Geom. 5, No. 3, 439--460 (1996; Zbl 0866.14009)] as well as Atiyah's theory of vector bundles on an elliptic curve.
An application of the result is a proof of the unirationality of the moduli spaces \({\mathcal A}(1,1,4)\) and \({\mathcal A}(1,4,4)\) of abelian 3-folds with polarization of type \((1,1,4)\) and \((1,4,4)\). unirationality Kanev, V.: Hurwitz spaces of quadruple coverings of elliptic curves and the moduli space of abelian threefolds \(A3(1,1,4)\). Math. nachr. 278, 154-172 (2005) Algebraic moduli of abelian varieties, classification, Families, moduli of curves (algebraic), Coverings of curves, fundamental group Hurwitz spaces of quadruple coverings of elliptic curves and the moduli space of abelian threefolds \(\mathcal A_{3}\)(1, 1, 4) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal M}_g\) denote the moduli space of smooth curves of genus \(g \geq 2\) over the field of complex numbers, and denote by \(\overline{{\mathcal M}}_g\) the Deligne-Mumford compactification consisting of stable curves of (arithmetic) genus \(g\). Let \({\mathcal H}_g \subset \overline{{\mathcal M}}_g\) be the moduli space of smooth hyperelliptic curves of genus \(g\). The closure \(\overline{{\mathcal H}}_g \subset \overline{{\mathcal M}}_g\) is the space of stable hyperelliptic curves. A smooth hyperelliptic curve of genus \(g \geq 2\) may be thought as a binary form of degree \(2 g + 2\), with nonzero discriminant, and there is a canonical isomorphism between \({\mathcal H}_g\) and \({\mathcal B}_{2 g + 2}\), the moduli space of such binary forms. Let \(\overline{{\mathcal B}}_{2 g + 2}\) be the usual one-point compactification of \({\mathcal B}_{2 g + 2}\).
The aim of the work under review is to study the relationship between \(\overline{{\mathcal H}}_g\) and \(\overline{{\mathcal B}}_{2 g + 2}\). Among many results, the main theorem states that the canonical isomorphism \({\mathcal H}_g \rightarrow {\mathcal B}_{2 g + 2}\) extends to a holomorphic map \(\overline{{\mathcal H}}_g \rightarrow \overline{{\mathcal B}}_{2 g + 2}\) (the authors describe this extension explicitly, it is a composition of a canonical homomorphism between \(\overline{{\mathcal H}}_g\) and the moduli space \(\overline{{\mathcal M}}_{0, 2g + 2} \) of stable \((2 g + 2)\)-marked curves of genus zero, and a holomorphic map \(\overline{{\mathcal M}}_{0, 2g + 2} \rightarrow \overline{{\mathcal B}}_{2 g + 2}\)). Avritzer, D., Lange, H.: The moduli spaces of hyperelliptic curves and binary forms, Math. Z. \textbf{242}(4), 615-632 (2002), arxiv:math/0109199v1 Families, moduli of curves (algebraic), Fine and coarse moduli spaces, Geometric invariant theory, Special algebraic curves and curves of low genus The moduli spaces of hyperelliptic curves and binary forms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0624.00007.]
The author reviews Dwork's preprint ``On the Tate constant'' [see \textit{B. Dwork}, Compos. Math. 61, 43-59 (1987; Zbl 0622.14016) and Groupe Étud. Anal. Ultramétrique, 11e Année 1983/84, Exposé No.11 (1985; Zbl 0614.14009)] describing the Picard-Fuchs equation and the Hasse-Witt matrix of a family of elliptic curves by considering the formal group of the Tate curve. The author states generalizations to the case of curves of higher genus; the details will appear in the author's thesis. Tate constant; hypergeometric differential equation; Legendre family of elliptic curves; Tate curve; Picard-Fuchs equation; Hasse-Witt matrix; formal group Local ground fields in algebraic geometry, Families, moduli of curves (algebraic), Algebraic functions and function fields in algebraic geometry, Formal groups, \(p\)-divisible groups, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) On the Tate-matrix | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0685.00007.]
From the authors' abstract: Several facts about \(SK_ 0\) and \(SK_ 1\) are presented both for commutative rings and schemes. If A is the homogeneous coordinate ring of a projective variety over a field k, then Pic(A), \(SK_ 0(A)\) and \(SK_ 1(A)\) are naturally modules over the ring W(k) of Witt vectors over k. If A is any commutative ring, NPic(A), \(NSK_ 0(A)\) and \(NSK_ 1(A)\) are naturally modules over W(A). The K- theory transfer map defined when B is an A-algebra which is a finite projective A-module, sends \(SK_ 0(B)\) to \(SK_ 0(A)\) and \(SK_ 1(B)\) to \(SK_ 1(A)\). \(SK_ 0\); \(SK_ 1\); coordinate ring of a projective variety; Pic; Witt vectors; K-theory transfer map B.H. Dayton and C.A. Weibel, On the naturality of Pic, SK0 and SK1, to appear. Grothendieck groups, \(K\)-theory and commutative rings, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Picard groups On the naturality of Pic, \(SK_ 0\) and \(SK_ 1\) | 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 We prove in this paper that the
subset of real \(p\)-gonal Riemann surfaces, \(p\geq 3\), is not a connected subset of \(M_g\) in general. This generalizes a result of Gross and Harris for real trigonal algebraic curves. A. F. Costa and M. Izquierdo, On the locus of real algebraic curves, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), no. suppl., 91 -- 107. Dedicated to the memory of Professor M. Pezzana (Italian). Real algebraic sets, Families, moduli of curves (algebraic), Word problems, other decision problems, connections with logic and automata (group-theoretic aspects), Compact Riemann surfaces and uniformization On the locus of real algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let C be a non-singular curve in the complex projective plane \({\mathbb{P}}^ 2\) and E a rank-2 stable bundle over \({\mathbb{P}}^ 2\) (E is stable if and only if \(c_ 1(L)<c_ 1(E)/2\) for all line subbundles L of E, \(c_ 1\) being the first Chern class).
The author establishes that if the degree of the curve is high enough compared with the number x such that \(\ell \geq x\Rightarrow H'(E(\ell))=0\), then the restriction bundle is stable. She also proves that if the degree is high compared to the numbers x, x' associated to bundles E and E' then their restriction bundles are equivalent if and only if they are equivalent. stable restriction bundle; rank-2 stable bundle Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic) Stability of some bundles over curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\overline{{\mathcal M}}_{g,n}({\mathbb{P}}^r,d)\) be the moduli space of stable maps from \(n\)-pointed, genus \(g\) curves to \({\mathbb{P}}^r\) of degree \(d\). In this paper the author computes the Poincaré polynomial of \(\overline{{\mathcal M}}_{0,2}({\mathbb{P}}^r,2)\), using what are called Serre characteristics in [\textit{E. Getzler} and \textit{R. Pandharipande}, J. Algebr. Geom. 15, No. 4, 709--732 (2006; Zbl 1114.14032)]. Serre characteristics are defined for varieties \(X\) over \(\mathbb{C}\) via the mixed Hodge theory of Deligne as follows. For a mixed Hodge structure \((V,F,W)\) over \(\mathbb{C}\), set \(V^{p,q}= F^p\text{gr}^W_{p+q} V\cap\overline F^q\text{gr}^W_{p+q} V\) and let \({\mathcal X}(V)\) be the Euler characteristic of \(V\) as a graded vector space. Then the Serre characteristic \(\text{Serre}(X)\) of \(X\)is defined to be \(\text{Serre}(X)= \sum^\infty_{p,q=0} u^p v^q{\mathcal X}(H^\bullet_c(X,\mathbb{C})^{p,q}))\). He employs the fact that \(\overline{{\mathcal M}}_{0,2}({\mathbb{P}}^r,2)\) is stratified according to the degeneration type of the stable maps. The compatibility of Serre characteristics with stratification allows him to compute of each stratum and add up the results to obtain \(\text{Serre}(\overline{{\mathcal M}}_{0,2}({\mathbb{P}}^r,2))\). In the final section, he also gives an additive basis for the Chow ring of \(\overline{{\mathcal M}}_{0,2}({\mathbb{P}}^r,2)\). Chow ring; moduli space; stable map; Betti numbers (Equivariant) Chow groups and rings; motives, Fine and coarse moduli spaces, Families, moduli of curves (algebraic) An additive basis for the Chow ring of \(\overline {\mathcal M}_{0,2}(\mathbb P^r,2)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \((x,y,z,w)\) be homogeneous coordinates in the complex projective space \(\mathbb{P}^3\). The subgroup \(H\) of \(\mathrm{Aut}(\mathbb{P}^3)\) generated by the following four transformations: \((x,y,z,w) \mapsto (z,w,x,y)\), \((x,y,z,w) \mapsto (y,x,w,z)\), \((x,y,z,w) \mapsto (x,y, -z,-w)\), and \((x,y,z,w) \mapsto (x,-y, z,-w)\) is named Heisenberg group. Clearly, \(H \cong (\mathbb{Z}/2\mathbb{Z})^4\). Quartic surfaces in \(\mathbb{P}^3\), invariant under \(H\), are called Heisenberg invariant quartics. Their family is parameterized by \(\mathbb{P}^4\) and its general member is smooth.
This family and some of its special loci have been widely studied in the literature, both in classical treatises as well as in the past century. In particular, \textit{W. Barth} and \textit{I. Nieto} [J. Algebr. Geom. 3, No. 2, 173--222 (1994; Zbl 0809.14027)] studied the locus of Heisenberg invariant quartics containing a line. Understanding which Heisenberg invariant quartics contain a conic has been the motivation of the paper under review, as the author says in the Introduction.
A first result is that a general invariant quartic contains \(320\) smooth conics. They are found by a direct computation relying the geometry of the family. Further results deal with the very general element of the family. Actually, the author shows that for a very general Heisenberg invariant quartic the Picard number is \(16\) and as a corollary of this fact he obtains that the sublattice of invariant divisor classes is generated by the class of the hyperplane section. In particular, any invariant curve on such a surface is a complete intersection.
Next the author determines the Picard lattice of a very general Heisenberg invariant quartic and shows that it is generated by the \(320\) smooth conics. quartic surface; conic; finite Heisenberg group; Picard group Special surfaces, \(K3\) surfaces and Enriques surfaces, Picard groups Curves on Heisenberg invariant quartic surfaces in projective 3-space | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathscr H_g\) be the moduli stack of hyperelliptic curves of genus \(g\) over an algebraically closed field. In [Trans. Am. Math. Soc. 370, No. 3, 1885--1906 (2018; Zbl 1421.14002)] the author introduced the concept of cohomological invariant of a smooth algebraic stack and in [Algebr. Geom. 4, No. 4, 424--443 (2017; Zbl 1397.14040)] he computed the cohomological invariants of \(\mathscr H_g\) for even \(g\). The main result of this paper computes the cohomological invariants with coefficients in \(\mathbb Z/p \mathbb Z\) of \(\mathscr H_3\). cohomological invariants; hyperelliptic curve; moduli stack; equivariant Chow rings Stacks and moduli problems, Families, moduli of curves (algebraic), Étale and other Grothendieck topologies and (co)homologies Cohomological invariants of genus three hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The study of the extrinsic geometry of the Jacobian locus \(\overline{J_g}\) in the moduli space \(A_g\) of principally polarized abelian varieties, namely of the closure of the locus of Jacobian varieties in \(A_g\), has been stimulated by different many problems throughout its history (two of the best known are the Torelli and Schottky problems). In particular, the theory of generalized Prym varieties as developed by [\textit{A. Beauville}, Invent. Math. 41, 149--196 (1977; Zbl 0333.14013)] highlighted a natural inclusion of the Jacobian locus inside boundary of the Prym locus \(\overline{P_{g+1}}\), which is the closure in \(A_g\) of the locus of Prym varieties associated to étale coverings of degree \(2\) onto smooth projective curves of genus \(g+1\). In terms of the study of the extrinsic geometry of \(\overline{J_g}\subset A_g\) such a theory suggested a refined study of \(\overline{J_g}\subset \overline{P_{g+1}}\) by using the parametrization of Jacobian varieties as generalized Prym varieties.
Let \(M_g\) be the moduli space of curves of genus \(g\) and let \(R_{g+1}\) be the moduli space of degree \(2\) étale coverings of curves of genus \(g+1\). For any \([C]\in M_g\), let \(JC\) be the Jacobian variety and let \(\mathrm{Cliff }C\) denote the Clifford index of \(C\) (see [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Berlin: Springer (2011; Zbl 1235.14002)]). For any \(\zeta \in T_{[JC]}J_g\), let \(\xi \in H^1(C, T_C)\) such that \(\zeta \) can be identified with the cup product map \(\cup \xi: H^0(C, \omega_C)\to H^0(C, \omega_C)^\vee\), under the isomorphism \(T_{[JC]}A_g\simeq \mathrm{Sym}^2H^0(C, \omega_C)^\vee\). We define the rank of \(\zeta\) as the rank of \(\cup \xi\). The following is one of the main results of the paper.
Theorem 1. Let \(JC\) be a general Jacobian variety of dimension \(g\geq 7\). Then for any \(\zeta\in T_{[JC]}J_g\) of rank \(k=rk\zeta <\mathrm{Cliff }C-3\), the local geodesic in \(A_g\) at \([JC]\) with direction \(\zeta\) (defined with respect to the Siegel metric) is not contained in the Prym locus \(\overline{P_{g+1}}\) (in particular, also in \(\overline{J_g}\subset \overline{P_{g+1}}\)).
Assume \(g\geq 7\).
Let \(M_g\) be the moduli space of curves of genus \(g\) and let \(A_g\) be the moduli space of principally polarized abelian varieties.
The Torelli morphism
\[
j: M_g\to A_g , \quad [C]\mapsto [JC]
\]
sends a genus \(g\) smooth projective curve \(C\) to its Jacobian variety \(JC\) (up to isomorphism). Its image \(J_g=j(M_g)\subset A_g\) defines a proper locus for \(g>3\) and its closure \(\overline{J_g}\subset A_g\) is called the Jacobian locus. By using the isomorphisms
\[
T_{[C]}M_g\simeq H^1(C,T_C), \quad T_{[JC]}A_g\simeq \mathrm{Sym}^2H^0(C, \omega_C)^{\vee},
\]
the differential of the Torelli map
\[
dj:H^1(C,T_C)\to \mathrm{Sym}^2H^0(C, \omega_C)^{\vee},\quad \xi \mapsto \zeta
\]
is given by \(\zeta=\cup\xi:H^0(C, \omega_C)\to H^0(C,\omega_C)^\vee,\) the cup product map with \(\xi.\)
Let \(R_{g+1}\) be the moduli space parametrizing \(2\)-sheeted étale coverings \(\pi: \tilde{C}\to C'\) between smooth projective curves of genus \(\tilde{g}=g(\tilde{C})=2g(C')-1\) and \(g'=g(C')=g+1\), respectively (modulo isomorphism). A point in \(R_{g+1}\) is an isomorphism class assigned equivalently by
\begin{itemize}
\item[(i)] a pair \((\tilde{C}, i)\), where \(i:\tilde{C}\to \tilde{C}\) is an involution such that \(\tilde{C}/(i)=C'\),
\item[(ii)] a pair \((C', \eta)\), where \(\eta \in \mathrm{Pic}^0(C')\setminus \{\mathcal O_{C'}\}\) is a \(2\)-torsion point and \(\tilde{C}= \mathrm{spec} (\mathcal O_C\oplus \eta)\).
\end{itemize}
One can reconstruct the data \((i)\) from \((ii)\) and conversely (see [\textit{A. Beauville}, Invent. Math. 41, 149--196 (1977; Zbl 0333.14013)], [\textit{D. Mumford}, in: Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers, 325--350 (1974; Zbl 0299.14018)]) and so with a little abuse of notations we will describe a point of \(R_{g+1}\) in both ways depending on the setting.
The Prym morphism
\[
Pr: R_{g+1}\to A_g, \quad [(C', \eta)]\mapsto [P(C',\eta)]
\]
maps a pair \((C', \eta)\) to its Prym variety \(P(C',\eta)\)(up to isomorphisms).
By definition, the Prym variety is \(P(C',\eta)=\ker Nm^0,\) namely the connected component containing zero in the kernel of the norm map \(Nm: J\tilde{C}\to JC'\), and it is a principally polarized abelian variety of dimension \(g\), with the principal polarization given by one half of the restriction of that on \(J\tilde{C}\).
The image \(P_{g+1}=Pr(R_{g+1})\subset A_g\) of the Prym morphism defines a proper locus for \(g\geq 6\) and its closure \(\overline{P_{g+1}}\) is called the Prym locus.
By construction, the forgetful functor \(\pi_{R_{g+1}}: R_{g+1}\to M_{g+1},\) sending \([(C',\eta)]\mapsto [C']\), is an étale finite covering and so we can identify \(T_{[(C', \eta)]}R_{g+1}\simeq T_{[C']}M_g.\) Using the isomorphisms
\[
T_{[C', \eta]}R_{g+1}\simeq H^1(C', T_{C'}), \quad T_{[P(C, \eta)]}A_g\simeq \mathrm{Sym}^2H^1(C',\eta)\simeq \mathrm{Sym}^2 H^0(C', \omega_{C'}\otimes \eta)^{\vee},
\]
the differential of the Prym map
\[
d Pr:H^1(C', T_{C'})\to \mathrm{Sym}^2H^1(C',\eta), \quad \xi' \longmapsto \zeta',
\]
is given by \(\zeta'=\cup\xi':H^0(C', \omega_{C'}\otimes \eta)\to H^0(C', \omega_{C'}\otimes \eta)^{\vee}\), the cup product with \(\xi'.\)
Let \(C\) be a smooth projective curve. The gonality and the Clifford index of \(C\) are defined as
\begin{align*}
&\operatorname{gon} C= \min\{n\in\mathbb N \,| \, C \mbox{ has a } g^1_n\} ;\\
&\mathrm{Cliff }C = \min\{\mathrm{Cliff }D=\deg D- 2 (h^0(D)-1)\, | \, D\subset C \mbox{ divisor}, h^0(D)\geq 2, h^1(D)\geq 2 \}.
\end{align*}
By [\textit{M. Coppens} and \textit{G. Martens}, Compos. Math. 78, No. 2, 193--212 (1991; Zbl 0741.14035)], these are related by
\[
\operatorname{gon} C -3 \leq\mathrm{Cliff }C \leq \operatorname{gon} C- 2 ,
\]
where the second inequality is an equality on a general \(C\) in \(M_g\), while the first one is conjectured to be extremely rare [\textit{D. Eisenbud} et al., Compos. Math. 72, No. 2, 173--204 (1989; Zbl 0703.14020)]. The followings are well known.
\begin{itemize}
\item[1.] \(\operatorname{gon} C \leq \left\lfloor \frac{g+3}{2}\right \rfloor\) (and so \(\mathrm{Cliff} C \leq \left\lfloor \frac{g-1}{2}\right \rfloor\)) and the equality holds for a general curve \([C]\in M_g\) (see [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Berlin: Springer (2011; Zbl 1235.14002)]);
\item[2.] let \(f:\mathcal C\to B\) be a fibration of projective curves over a complex curve \(B\) and such that the general fibre is smooth. The invariants \(\operatorname{gon} C_b \) and \(\mathrm{Cliff} C_b\) are maximal on a general fibre \(C_b\) of the set of smooth fibres ([\textit{K. Konno}, J. Algebr. Geom. 8, No. 2, 207--220 (1999; Zbl 0979.14004)]).
\end{itemize}
An \textit{admissible covering} of degree \(k\) with \(m\) ramification points is the data of
\begin{itemize}
\item[1.] a stable \(m\)-pointed reduced connected curve \((E, x_1, \dots ,x_m)\) of arithmetic genus \(0\) (i.e. a curve with ordinary double points where any rational component is smooth and stable and the dual graph is connected);
\item[2.] a reduced connected curve \(X\) with ordinary double points and a morphism \(\pi : X \to E\) of degree \(k\) (everywhere) such that over any marked point \(x_i\) of \(E\), \(X\) is smooth and \(f\) has a unique simple ramification point \(y_i\), on any smooth point of \(E\), \(f\) is étale and over double points \(p\) of \(E\), \(X\) has an ordinary double point \(q\) and \(f\) is locally described as
\[
X\,:\, xy=0; \quad E\,:\, uv=0;\quad \, f\,:\, u=x^{k'}\quad ,\, v= y^{k'},
\]
for some \(k'\leq k\).
\end{itemize}
We have the following
Lemma. Let \(C'\) be a stable nodal curve of genus \(g'\), let \(\nu': C^{\nu'}\to C'\) be its normalization and let \(C\subset C^{\nu'}\) be a smooth connected curve of genus \(g\). Consider \(f':\mathcal C'\to \Delta\), a family of smooth projective curves of genus \(g'\) over \(\Delta \setminus \{0\}\), the complex disk minus zero, such that \(C'=f^{-1}(0)\). Then any family of pencils \(g^1_k(t)\) over \( \Delta\setminus\{0\}\) compatible with \(f\) (i.e. \(g^1_k(t)\) is a pencil on \(C'_t=f^{-1}(t)\)) defines a pencil \(g^1_{k'}\) on \(C\) for some \(k'\leq k\). In particular, \(\operatorname{gon} C\leq \operatorname{gon} C'_t\) and \(\mathrm{Cliff }C\leq \mathrm{Cliff }C'_t+1\), for a general fibre \(C'_t=f^{-1}(t)\).
Let \(Y\) be a smooth complex variety and let
\((\mathbb H_{\mathbb Z}, \mathcal H^{1,0}, \mathcal Q) \) be a polarized variation of Hodge structures (pvhs, in short) of weight 1 on \(Y\). Namely, \(\mathbb H_Z\) is a local system of lattices, \(\mathcal H^{1,0}\) is a Hodge bundle of type \((1,0)\) (equivalently, the Hodge filtration in this case) and \(\mathcal Q\) is a polarization. Let \(\mathbb H_{\mathbb C}=\mathbb H_{\mathbb Z}\otimes_{\mathbb Z}\mathbb C\) and let \(\mathcal H=\mathbb H_{\mathbb C} \otimes \mathcal O_Y\) be the holomorphic flat bundle with the holomorphic flat connection \(\nabla\) defined by \(\ker \nabla \simeq \mathbb H_{\mathbb C}\), the Gauss-Manin connection. The holomorphic inclusion \(\mathcal H^{1,0}\subset\mathcal H\) of vector bundles induces the short exact sequence
\[
\begin{tikzcd}
0 \arrow{r} & \mathcal H^{1,0} \arrow{r} & \mathcal H \arrow{r}{\pi^{{0,1}}} &
\mathcal H/\mathcal H^{1,0} \arrow{r}& 0.
\end{tikzcd}
\]
Let \(\pi^{{0,1}'}: \mathcal H\otimes \Omega^1_Y\to\mathcal H/\mathcal H^{1,0}\otimes \Omega^1_Y\) be the map induced by \(\pi^{0,1}\) and \(\sigma =\pi^{{0,1}'}\circ \nabla: \mathcal H^{1,0}\to\mathcal H/\mathcal H^{1,0}\otimes \Omega_Y^1\) the second fundamental form of \(\mathcal H^{1,0}\subset\mathcal H\) with respect to \(\nabla\).
Following [\textit{A. Ghigi} et al., Commun. Contemp. Math. 23, No. 3, Article ID 2050020, 13 p. (2021; Zbl 1455.14013)], let \(\mathbb U=\ker \nabla_{|\mathcal H^{1,0}}\) and define
\[
\mathcal U:=\mathbb U\otimes
\mathcal O_Y,\quad \quad
\mathcal K:=\ker ( \sigma : \mathcal H^{1,0}\longrightarrow\mathcal H /\mathcal H^{1,0} \otimes
\Omega_Y^1).
\]
Then \(\mathcal U\) is a holomorphic vector bundle and \(\mathcal K\) is a coherent sheaf which is a vector bundle when \(\sigma\) is of constant rank.
Definition 1. We call \(\\mathcal U\) and \(\mathcal K\) as defined in (1), \textit{the unitary flat bundle} and \textit{the kernel sheaf} of the variation, respectively
Proposition 1. We have \( \mathcal U\subset \mathcal K\) and if \(\tau\equiv 0\), then \(\mathcal U=\mathcal K\).
For any \(B\subset A_g\) smooth complex curve, let \(f: \mathcal A\to B\) be the family of abelian varieties (defined up to finite base change) and let \(A_b\) be the fibre over \(b\in B\). Then the p.v.h.s. is defined by
\[
\mathbb H_{\mathbb Z}\simeq R^1f_\ast \mathbb Z, \quad \mathcal H^{1,0}= f_\ast \Omega^1_{\mathcal A/B}\subset \mathcal H= R^1f_\ast \mathbb C\otimes \mathcal O_{B};
\]
\[
\mbox{where }\quad (R^1f_\ast \mathbb Z)_b\simeq H^1(A_b,\mathbb Z), (f_\ast\Omega^1_{\mathcal A/B })_b\simeq H^0(A_b, \Omega^1_{A_b}), {(R^1f_\ast \mathbb C\otimes \mathcal O_{B})}_b\simeq H^1(A_b, \mathbb C).
\]
Let \(H_g\) denote the Siegel upper half space. As a symmetric space of non-compact type it is endowed by a symmetric metric \(h^s\), called the Siegel metric, defining a metric connection \(\nabla^{LC}\) on the tangent bundle \(TH_g\). As a parametrizing space of weight \(1\) p.v.h.s., it carries a universal p.v.h.s. \((\mathbb H_{\mathbb Z},\mathcal H^{1,0}, \mathcal Q)\) with its Hodge metric defined by \(Q\), inducing a metric \(h\) together with a metric connection \(\nabla^{hdg}\) on \(\mathrm{Hom} (\mathcal H^{1,0}, \mathcal H/\mathcal H^{1,0})\). There is a natural inclusion
\[
(TH_g, \nabla^{LC})\subset (\mathrm{Hom} (\mathcal H^{1,0}, \mathcal H/\mathcal H^{1,0}), \nabla^{hdg})
\]
compatible with the metric structure (see e.g. a classical reference [\textit{P. A. Griffiths}, in: Actes Congr. internat. Math. 1970, 1, 113--119 (1971; Zbl 0227.14008)] or some more recent references [\textit{A. Ghigi}, Boll. Unione Mat. Ital. 12, No. 1--2, 133--144 (2019; Zbl 1444.14028)], [\textit{A. Ghigi} et al., Commun. Contemp. Math. 23, No. 3, Article ID 2050020, 13 p. (2021; Zbl 1455.14013)]).
Consider the universal covering \(\psi: H_g\to A_g\) and the metric properties introduced before on \(H_g\). Let \([A]\in A_g\) and let \(\zeta\in T_{[A]}A_g\).
Take \(\tilde{A}\in \psi^{-1}([A])\) and consider \(\zeta\) as \(\zeta \in T_{\tilde{A}}H_g\simeq T_{[A]}A_g\). Then in \(H_g\) a (local) geodesic at \((\tilde{A}, \zeta)\) is simply a curve \(\gamma:(-\epsilon, \epsilon)\to H_g\) such that \(\gamma(0)=\tilde{A}\) and \(\gamma'(0)=\zeta\) satisfying \(\nabla^{LC}_{\gamma'}\gamma'=0\). Working locally, we can assume w.l.o.g that \(\gamma((-\epsilon, \epsilon))\) is contained in one sheet of \(\psi\).
Definition 2. Let \([A]\in A_g\) and let \(\zeta\in T_{[A]}A_g\)
A (local) geodesic associated to \(([A], \zeta)\) is a map \(\psi\circ \gamma: (-\epsilon, \epsilon)\to A_g\), where \(\gamma\) is a local geodesic in \(H_g\) defined as above.
We are interested in points of \(J_g\subset A_g\) and directions in \(T_{[A]}J_g\subset T_{[A]}A_g\). If \([A]=[JC]\), namely the Jacobian of some \([C]\in M_g\), and \(\zeta = \cup \xi\), for some \(\xi \in H^1(C, T_C)\), under the isomorphisms \(T_{[C]}M_g\simeq H^1(C, T_C)\) and \(T_{[J_g]}A_g\simeq \mathrm{Sym}^2H^1(C, \mathcal O_C)\simeq \mathrm{Sym}^2H^0(C, \omega_C)^\vee\), we will also refer to the geodesic at \((JC, \zeta)\) as the geodesic at \((C, \xi)\). This is admitted since the Torelli map is an immersion outside the Hyperelliptic locus and we are not considering hyperelliptic curves.
We have the following (see [\textit{A. Ghigi} et al., Commun. Contemp. Math. 23, No. 3, Article ID 2050020, 13 p. (2021; Zbl 1455.14013), Lemma 3.3 and Lemma 3.4] for the proof)
Lemma 2. Let \(\gamma:(-\epsilon, \epsilon)\to A_g\) be a local geodesic associated to \(([A], \zeta)\). Then there exists a complex curve \(B\subset H_g\) containing the geodesic and such that \(\mathcal K =\mathcal U\).
We can shrink \(B\) around \(\gamma((-\epsilon, \epsilon))\) in such a way that it is contained in one sheet on \(\psi\) and so we can then look at it as a curve in \(A_g\).
Let \(C\) be a smooth projective curve, let \(\mathcal F\) be a rank \(2\) vector bundle over \(C\) and let \(\alpha: \bigwedge^2H^0(C,\mathcal F)\to H^0(C,\det \mathcal F)\) be a linear map. A subspace \(W\subset H^0(C,\mathcal F) \) is called \textit{isoptropic} with respect to \(\alpha\) if \(\alpha_{|\bigwedge^2 W}\equiv 0\).
Let \(\mathcal L, \mathcal L'\) be two line bundles on \(C\), let \(0\to\mathcal L \to\mathcal F \to\mathcal L'\to 0\) be a s.e.s. associated to \(\xi\in \mathrm{Ext}^1_{\mathcal O_C}(\mathcal L', \mathcal L)\) and let \(\alpha: \bigwedge^2H^0(C,\mathcal F)\to H^0(C,\det \mathcal F)\simeq H^0(C,\mathcal L\otimes\mathcal L')\). A subspace \(V\subset H^0(C,\mathcal L')\) is called \textit{isotropic} with respect to \(\alpha\) if it lifts to a subspace \(W\subset H^0(C,\mathcal F)\) isotropic with respect to \(\alpha\).
Note that a subspace \(V \subset H^0(C,\mathcal L')\) lifts to \(W\subset H^0(C,\mathcal F)\) if and only if it lies in the kernel of the coboundary morphism \(\delta: H^0(C,\mathcal L')\to H^1(C,\mathcal L)\) on the long exact sequence in cohomology.
Theorem 2. Let \(f: \mathcal C\to B\) be a fibration of smooth projective curves over a smooth complex curve \(B\) and let \(\mathcal U\subset f_\ast \omega_{\mathcal C/B}\) be the associated unitary flat bundle (Definition 1). Assume that the fibres \(U_b\subset H^0(\omega_{C_b})\) are isotropic subspaces with respect to \(\alpha_b\) for any \(b\in B\). Then (up to a finite base change) there exists a smooth projective curve \(\Sigma\) of genus \(g'=rk \mathcal U\) and a non constant fibre-preserving map \(\varphi: \mathcal C \to \Sigma\) such that \(U_b\simeq \varphi^\ast H^0(\Sigma,\omega_{\Sigma})\). moduli space of curves and abelian varieties; geodesics; Prym locus; Jacobian locus; generalized Prym varieties; admissible coverings Jacobians, Prym varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Period matrices, variation of Hodge structure; degenerations, Subvarieties of abelian varieties On the Jacobian locus in the Prym locus and geodesics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(\bar M_{0,n\cdot \epsilon}\) be the moduli space of weighted pointed stable rational curves with symmetric weights \(n\cdot \epsilon = (\epsilon, \ldots, \epsilon)\) (see [\textit{B. Hassett}, Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)]). In the paper under review the authors reconstruct \(\bar M_{0,n\cdot \epsilon}\) as the GIT quotient of the moduli space of degree one weighted pointed stable maps to \(\mathbb P^1\). They also exhibit a sequence of blow-ups from \(\bar M_{0,n}\) to the GIT quotient and describe explicitly the center for each blow-up. As an application, the authors can identify \(\bar M_{0,n\cdot \epsilon}\) as intermediate models that appear in the log canonical model program for \(\bar M_{0,n}\), which was first proved by \textit{M. Simpson} [``On log canonical models of the moduli space of stable pointed curves'', \url{arXiv:0709.4037}] assuming Fulton's conjecture, and later proved unconditionally by \textit{M. Fedorchuk} and \textit{D. I. Smyth} [J. Algebr. Geom. 20, No. 4, 599--629 (2011; Zbl 1230.14034)] and by \textit{V. Alexeev} and \textit{D. Swinarski} [``Nef divisors on \(bar{M}_{0,n}\) from GIT'', \url{arXiv:0812.0778}]. moduli space of curves; GIT; log canonical model Y.-H. Kiem and H.-B. Moon, Moduli spaces of weighted pointed stable rational curves via GIT, Osaka J. Math. 48 (2011), no. 4, 1115-1140. Families, moduli of curves (algebraic), Geometric invariant theory Moduli spaces of weighted pointed stable rational curves via GIT | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 irreducibility of the family of all plane curves of given degree with a prescribed number of nodes and a further singular point of a special type (quasi-ordinary singularity). This result generalizes the irreducibility of Severi varieties [\textit{J. Harris}, Invent. Math. 84, 445-461 (1986; Zbl 0596.14017)] and answers in particular a conjecture of Enriques. The proof uses a special type of reducible surfaces (fans) whose components are blowing ups of projective planes. Fans were already used by the author in his own proof of the irreducibility of Severi varieties [Invent. Math. 86, 529-534 (1986; Zbl 0644.14009)]. Most of the paper is devoted to developing a degeneration theory for fans and families of curves on fans: such theory is the main tool for the proof and should have many other applications. irreducibility of the family of all plane curves of given degree; nodes; quasi-ordinary singularity; Severi varieties; fans Ziv Ran, Families of plane curves and their limits: Enriques' conjecture and beyond, Ann. of Math. (2) 130 (1989), 121-157 Zbl0704.14018 MR1005609 Families, moduli of curves (algebraic), Grassmannians, Schubert varieties, flag manifolds Families of plane curves and their limits: Enriques' conjecture and beyond | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \({\mathcal A}_ 5\) be the moduli space of 5-dimensional, principally polarized abelian varieties and let \({\mathcal R}_ 6\) be the Prym moduli space parametrizing double covers of curves of genus 6. By \textit{W. Wirtinger} [''Untersuchungen über Thetafunctionen'' Teubner, Leipzig 1895), there is the Prym map \(p : {\mathcal R}_ 6\to {\mathcal A}_ 5\) assigning to a double cover \(\pi : \tilde C\to C\) the neutral component of \(Ker(\pi_* : J(\tilde C)\to J(C)),\) where \(J(\;)\) denotes the Jacobian variety, and p is known to be dominant. A main result of the present article is to show that \({\mathcal R}_ 6\) and \({\mathcal A}_ 5\) are unirational. The idea is to use the theory of nets of quadrics. A net of quadrics in \({\mathfrak P}^ 6\) is a family of quadrics \(\{\) A(t);t\(\in \Pi \}\) in \({\mathfrak P}^ 6\) parametrized by a projective plane \(\Pi\). Such a net is invertible if a general member is nondegenerate and if every member has rank\(\geq 6\). Associated to such a net, we have a pair (C,L) of the discriminant locus \(C=\{t\in \Pi;\;A(t) \text{is degenerate}\}\), i.e., the locus of singular quadrics, and a non-vanishing theta characteristic L. Let \(\bar N_ 0\) be the space of projective equivalence classes of invertible nets whose discriminant locus is \(S\cup \ell\), where S is a sextic in \({\mathfrak P}^ 2\) with 4 double points and \(\ell\) is a line. Then there is a dominant map \(f : \bar N_ 0\to {\mathcal R}_ 6\) via the discriminant map \(\{A(t);t\in \Pi \}\mapsto (S\cup \ell,L)\mapsto (\nu (S),\pi),\) where \(\nu\) (S) is the normalization of S and \(\pi\) is an étale double cover of \(\nu\) (S) given by a line bundle \(L\otimes {\mathcal O}(-2)\). More precisely, f is dominant on each component of \(\bar N_ 0\). By a detailed analysis of \(\bar N_ 0\), the author shows that \({\mathcal R}_ 6\) and \({\mathcal A}_ 5\) are unirational. We note that the unirationality of \({\mathcal A}_ 4\) was proved by H. Clemens. unirationality of moduli space of 5-dimensional principally; polarized abelian varieties; Prym moduli space parametrizing double covers of curves of; genus 6; nets of quadrics; theta characteristic; Prym moduli space parametrizing double covers of curves of genus 6 R. Donagi, The unirationality of
\[
\mathcal{A}_{5}
\]
. Ann. Math. 119, 269--307 (1984) Rational and unirational varieties, Algebraic moduli of abelian varieties, classification, Pencils, nets, webs in algebraic geometry, Theta functions and abelian varieties, Families, moduli of curves (algebraic) The unirationality of \({\mathcal A}_ 5\). | 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 We study nonnegative (psd) real sextic forms \( q(x_0,x_1,x_2)\) that are not sums of squares (sos). Such a form has at most ten real zeros. We give a complete and explicit characterization of all sets \( S\subset \mathbb{P}^2(\mathbb{R})\) with \( | S|=9\) for which there is a psd non-sos sextic vanishing in \( S\). Roughly, on every plane cubic \( X\) with only real nodes there is a certain natural divisor class \( \tau _X\) of degree~\( 9\), and \( S\) is the real zero set of some psd non-sos sextic if and only if there is a unique cubic \( X\) through \( S\) and \( S\) represents the class \( \tau _X\) on \( X\). If this is the case, there is a unique extreme ray \( \mathbb{R}_{+} q_S\) of psd non-sos sextics through \( S\), and we show how to find \( q_S\) explicitly. The sextic \( q_S\) has a tenth real zero which for generic \( S\) does not lie in \( S\), but which may degenerate into a higher singularity contained in \( S\). We also show that for any eight points in \( \mathbb{P}^2(\mathbb{R})\) in general position
there exists a psd sextic that is not a sum of squares and vanishes in the given points. Real algebraic sets, Picard groups, Special algebraic curves and curves of low genus Extreme positive ternary sextics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0626.00011.]
This paper is a survey on the curves and their moduli: It begins by studying the moduli space \(M_ g\) of isomorphism classes of smooth complete curves of genus \(g;\) its topology is analyzed so as the complete subvarieties, the divisors and line bundles. Then the author describes the fibers of the natural map \(\phi\) from the space \(H_{d,g,r}\) of curves of degree \(d,\) and genus \(g\) in \({\mathbb{P}}^ r\) to \(M_ g\), at least over a general point of \(M_ g\). Further he studies the properties of \(H_{d,g,r}\) for the families of plane curves, and finally he illustrates by means of examples how the classes of divisors in the moduli space of stable curves may be computed. divisors; moduli space of stable curves Harris, J.: Curves and their moduli, Proc. sympos. Pure math. 46, 99-143 (1987) Families, moduli of curves (algebraic) Curves and their moduli | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors identify a combinatorial origin of the topological recursion formula of \textit{B. Eynard} and \textit{N. Orantin} [Commun. Number Theory Phys. 1, No. 2, 347--452 (2007; Zbl 1161.14026)] as the operation of edge removal from a ribbon graph. Among the other things, the authors obtain a new proof of the Kontsevich constants for the ratio of the Euclidean and symplectic volumes of the moduli space of curves. Kontsevich constant; topological recursion; ribbon graphs; symplectic and Euclidean volume; moduli of curves Chapman, K.; Mulase, M.; Safnuk, B.: Topological recursion and the kontsevich constants for the volume of the moduli of curves, Commun. number theory phys. 5, 643-698 (2011) Families, moduli of curves (algebraic), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) The Kontsevich constants for the volume of the moduli of curves and topological recursion | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(\mathcal{M}_g\) be the moduli space of closed Riemann surfaces of genus \(g\geq 2\). The space \(\mathcal{M}_g\) admits an involution \(\sigma\) which maps each Riemann surface to its complex conjugate. The fixed point set \(\text{Fix}(\sigma)\) of \(\sigma\) consists of the closed Riemann surfaces admitting an anticonformal automorphism and includes \(\mathcal{M}_g(\mathbb{R})\), the set of real Riemann surfaces. The complement \(\text{Fix}(\sigma)\setminus\mathcal{M}_g(\mathbb{R})\) consists of the pseudo-real Riemann surfaces. It is known that \(\mathcal{M}_g(\mathbb{R})\) is connected while the connectedness of \(\text{Fix}(\sigma)\setminus\mathcal{M}_g(\mathbb{R})\) depends on the genus \(g\). In the paper under review, by constructing a real Riemann surface \(\hat{S}\) with an anticonformal automorphism \(\hat{\tau}\) which is topologically conjugate to a given pseudo-real Riemann surface \(S\) with an anticonformal automorphism \(\tau\), it is proved that \(\text{Fix}(\sigma)\) is connected. Riemann surfaces; moduli space; real Riemann surfaces; pseudo-real Riemann surfaces Teichmüller theory for Riemann surfaces, Klein surfaces, Families, moduli of curves (algebraic) On the connectedness of the set of Riemann surfaces with real moduli | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We construct explicit \(K3\) surfaces over \(\mathbb{Q}\) having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces constructed. DOI: 10.1112/S1461157014000199 \(K3\) surfaces and Enriques surfaces, Transcendental methods, Hodge theory (algebro-geometric aspects), Complex multiplication and moduli of abelian varieties, Picard groups Examples of \(K3\) surfaces with real multiplication | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0667.00008.]
The authors study the deformations of space curves which lie on rational ruled surfaces. Let \(W(m,d,g,n)\) be the subscheme of the Hilbert scheme of projective n-space corresponding to those smooth degree \(d,\) genus \(g\) curves which are m-secant curves lying on a two dimensional rational scroll. Using the properties of ruled surfaces, the authors are able to give an upper bound on the Zariski tangent space at a general point of \(W(m,d,g,n)\). From that they are able to conclude that the closure of \(W(m,d,g,n)\) is a component of the Hilbert scheme, if d, g, n and m satisfy certain inequalities. This gives a generalization of an earlier example of \textit{J. Harris} which shows that \(H(d,g,n)\), the Hilbert scheme of smooth curves of degree \(d\) and genus \(g\) in \({\mathbb{P}}^ n (n\geq 6)\), is reducible even if \(d\geq g+n.\)
The authors also formulate many different conjectures concerning about the irreducibility of \(H(d,g,n)\). For instance they conjecture that if the Brill-Noether number \(\rho(d,g,n)\) is non-negative then the open set of \(H(d,g,n)\) corresponding to linearly normal curves is an irreducible variety. gonality; deformations of space curves; Hilbert scheme; ruled surfaces; Zariski tangent space; Brill-Noether number E. Mezzetti and G. Sacchiero, Gonality and Hilbert schemes of smooth curves, in Algebraic Curves and Projective Geometry, (Trento, 1988), 183--194, Lecture Notes in Math. 1389, Springer, Berlin, 1989. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) Gonality and Hilbert schemes of smooth curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review introduces the group action of \(\operatorname{SL}(2, \mathbb{C})\) on a partition function of a cohomological field theory in terms of a certain Givental action. To be precise, this paper consists of two main theorems (Theorems 3 and 6). In Theorem 3, the author shows the isomorphism between certain two Frobenius manifolds \(M^A\) and \({\hat{R^\sigma}}\cdot M\), depending on \(A\in \operatorname{SL}(2, \mathbb{C})\), for a particular Givental's group element \(R^\sigma\), and then extends the \(\operatorname{SL}(2, \mathbb{C})\)-action to the higher genera. As a consequence, the author obtains a particular formula for the \(\operatorname{SL}(2, \mathbb{C})\)-action on the genus-\(g\) small phase-space potential of a cohomological field theory. Theorem 6 gives some connection between the total ancestor potential formula of Milanov-Ruan and the \(\operatorname{SL}(2, \mathbb{C})\)-action of the paper. That is, it shows that the \(\operatorname{SL}(2, \mathbb{C})\)-action developed in Theorem 3 is equivalent to the primitive form change for simple elliptic singularities. cohomological field theories; Givental's action; total ancestor potential; primitive forms; \(\operatorname{SL}(2\mathbb{,C})\)-actions; singularity theory A. Basalaev, \(SL(2,\mathbb{C})\) group action on Cohomological field theories, Lett. Math. Phys. 108 (2018), no. 1, 161--183. Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (algebraic) \(\operatorname{SL}(2, \mathbb {C})\) group action on cohomological field 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 This paper is devoted to the proof of the unirationality of the Prym moduli space of étale double covers of genus 5 curves, denoted \(\mathcal R_5\). The argument goes as follows. Given a general quartic surface in \(\mathbb P^3\) with six ordinary double points as singluarities, the discriminant of the projection from one of them is a plane sextic with five nodes (hence a curve of geometric genus 5). Via a geometric construction due to Clemens one can associate in a natural way to the quartic surface the normalization of the discriminant curve together with an étale double cover of it. The moduli space \(\mathcal Q\) of quartic surfaces with six double points -one marked- is easily shown to be unirational. It is then sufficient to prove that the above constructed map \(\mathcal Q\rightarrow \mathcal R_5\) is dominant. This is done by proving that the corresponding map to the moduli space \(\mathcal M_5\) of genus five curves is generically surjective. The authors provide two different arguments; the first, more geometrical, exhibits an inverse image for a general curve in \(\mathcal M_5\). The second method is computational, and reduces the proof to the computation of the rank of a certain matrix.
Using the fact that the Prym map from \(\mathcal R_5\) to the moduli space of principally polarized abelian four-folds \(\mathcal A_4\) is dominant, the result also implies the unirationality of \(\mathcal A_4\), thus providing an alternative proof to the original one given by Clemens via intermediate Jacobians. Prym variety; moduli; canonical curve E. Izadi, A. Lo Giudice, G. Sankaran, The moduli space of étale double covers of genus 5 curves is unirational. Pac. J. Math. 239, 39--52 (2009) Jacobians, Prym varieties, Rational and unirational varieties, Computational aspects of algebraic curves, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles The moduli space of étale double covers of genus 5 curves is unirational | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The subject of the paper under review is the cohomology of the moduli space \({\mathcal H}_{g,n}\) of smooth hyperelliptic curves of genus \(g\) with \(n\) distinct marked points. The action of the symmetric group \(S_n\) by permuting the marked points endows both the Betti and the \(\ell\)-adic cohomology with a natural structure of \(S_n\)-representation that respects mixed Hodge structures, respectively, the structure as Galois representation. In this paper, the \(S_n\)-equivariant Euler characteristic of the (étale, resp., Betti) cohomology of \({\mathcal H}_{g,n}\) in the Grothendieck group of rational Hodge structures (respectively, Galois representations) is computed for all \(g\) and for \(n\leq 7\).
This result is achieved by performing an \(S_n\)-equivariant count of the number of points of the moduli space \({\mathcal H}_{g,n}\) defined over finite fields. This approach leads to the discovery that the number of points of \({\mathcal H}_{g,n}\) satisfies recurrence relations, so that the \(S_n\)-equivariant count of points of \({\mathcal H}_{g,n}\) for \(n\) fixed and \(g\) small determines the \(S_n\)-equivariant count of points for all \(g\). In particular, for \(n\leq 7\) the formulas for the \(S_n\)-count of points of \({\mathcal H}_{g,n}\) are obtained starting from known results for genus \(0\) and \(1\). In all these cases, the count of points gives a polynomial in the number elements of the field. For \(n\) small, these polynomials are independent of the characteristic of the finite field; the dependence starts for \(n=6\). The results on the count of points determine the \(S_n\)-equivariant Euler characteristic of \(\ell\)-adic cohomology by the Lefschetz trace formula. The corresponding result for the Betti cohomology of the complex moduli space follows from comparison theorems.
As an application, the author computes the cohomology of the moduli space of stable curves of genus \(2\) and \(n\) marked points with \(n\leq 7\). This extends the results of \textit{E. Getzler} [in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30-July 4, 1997, and in Kyoto, Japan, July 7-11, 1997. Singapore: World Scientific. 73--106 (1998; Zbl 1021.81056)] for \(n\leq 3\). Furthermore, if \(g\) is sufficiently large, there is a uniform description of the part of sufficiently high weight of the Euler characteristic of the \(\ell\)-adic cohomology of \({\mathcal H}_g\) with coefficients in certain local systems, which suggests the existence of stabilization phenomena in the cohomology. cohomology of moduli spaces of curves; curves over finite fields; hyperelliptic curves J. Bergström, Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves, Doc. Math. 14 (2009), 259-296. Families, moduli of curves (algebraic), Curves over finite and local fields, Transcendental methods, Hodge theory (algebro-geometric aspects) Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(A\subseteq B\) be domains. Let \(M\text{Pic}(A)\) denote \(\text{cokernel}(\text{Pic}(A)\to \text{Pic}(A[X,X^{-1}]).\) If \(M\text{Pic}(A)=0,\) \(A\) is called quasinormal. If \(M\text{Pic}(A)\to M\text{Pic}(B)\) is injective, Ais called quasinormal in \(B\). \(A\) is said to be \(u\)-closed in \(B\) if whenever \(\alpha\in B\) and \(\alpha^ 2-\alpha,\quad \alpha^ 3-\alpha^ 2\in A,\) then \(\alpha\in A\). Also, \(A\) is \(t\)-closed in \(B\) if whenever \(\alpha\in B\) and \(\alpha^ 2-a\alpha,\quad \alpha^ 3-a\alpha^ 2\in A\) for some \(a\in A\), then \(\alpha\in A\). This paper studies there lationship between these (and similar) definitions, showing, for instance, that \(t\)-closed \(\Rightarrow\) quasinormal \(\Rightarrow\) \(u\)-closed. Stability of these concepts under passage to polynomial (and other types of) extensions, is studied. root closed domain; t-closed domain; u-closed domain; quasinormal domain Onoda, N., Sugatani, T., Yoshida, K.: Local quasinormality and closedness type criteria. Houst. J. Math. 11, 247--256 (1985) General commutative ring theory, Integral domains, Divisibility and factorizations in commutative rings, Picard groups Local quasinormality and closedness type criteria | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians These notes are supposed to be an introduction to the moduli of \(G\)-bundles on curves. The introductory sections 2 and 3 collect basic definitions connected to \(G\)-bundles and algebraic stacks and give the necessary background which is needed later. Next the lectures contain: a topological classification and a proof of the uniformization theorem for \(G\)-bundles; determinant and pfaffian line bundles on the moduli stack, affine Lie algebras and groups, the infinite grassmannian, the ind-group of loops coming from the open curve, line bundles on the moduli stack of \(G\)-bundles. \(G\)-bundle; moduli space Sorger, Christoph, Lectures on moduli of principal \(G\)-bundles over algebraic curves.School on Algebraic Geometry, Trieste, 1999, ICTP Lect. Notes 1, 1-57, (2000), Abdus Salam Int. Cent. Theoret. Phys., Trieste Vector bundles on curves and their moduli, Group actions on varieties or schemes (quotients), Families, moduli of curves (algebraic), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles Lectures on moduli of principal G-bundles over algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is concerned with the study of the singular homology with field coefficients of the moduli stack \(\overline{\mathcal M}_{g,n}\) of Deligne-Mumford-Kudsen stable curves of genus \(g\) with \(n\) labelled marked points. Particular emphasis is given to the study of the mod \(p\) homology rather then the rational homology, as the first has received little attention when compared with the second. As the mod \(p\) homology of \(\overline{\mathcal M}_{g,n}\) and of its coarse moduli space are not necessarily isomorphic (as it is the case for the rational homology), the authors concentrate on the study of the homology of the stack rather then the space. The mod \(p\) cohomology of the substack \(\mathcal M_{g,n}\) of smooth pointed curves was computed by \textit{S. Galatius} in [Topology 43, No. 5, 1105--1132 (2004; Zbl 1074.57013)] in the Harer-Ivanov stable range, identifying a large number of torsion classes.
The main tool used in the paper is the Pontrjagin-Thom collapse map, in particular a generalized version of it holding for differentiable local quotient stacks; this generalization is discussed by the authors in Appendix A. This technique is then applied to the clutching (or gluing) morphisms given by identifying two marked points to form a node and whose images cover the irreducible components of the boundary \(\overline{\mathcal M}_{g,n}\setminus \mathcal M_{g,n}\) of \(\overline{\mathcal M}_{g,n}\). The outcome of this version of Pontrjagin-Thom's construction is the production of a map from ``the homotopy type of the stack'' \(\overline{\mathcal M}_{g,n}\), which is a space with the same topological invariants as the stack, to a certain infinite loop space, whose homology is well known (and discussed in Appendix B of the paper). The main result of the paper states the surjectivity of those maps in certain stable ranges. Using this theorem the authors then use the injectivity of the pullback map to detect classes in \(\overline{\mathcal M}_{g,n}\). In particular, large families of torsion classes, which are not reduction of rationally nontrivial classes nor coming from \(\mathcal M_{g,n}\), are shown to exist in \(\overline{\mathcal M}_{g,n}\). Ebert, J.; Giansiracusa, J.: Pontrjagin-thom maps and the homology of the moduli stack of stable curves, Math. ann. 349, No. 3, 543-575 (2011) Families, moduli of curves (analytic), Stacks and moduli problems, Families, moduli of curves (algebraic), Homology of classifying spaces and characteristic classes in algebraic topology Pontrjagin-Thom maps and the homology of the moduli stack 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 In this paper the Picard numbers of several families of elliptic surfaces are computed. This is equivalent to the difficult problem of determining the rank of the Mordell-Weil group of certain elliptic curves over function fields. The method is to study the action induced by automorphisms of these surfaces on a relevant part of the cohomology. The cohomology classes are represented by certain inhomogeneous differential equations - on which the effect of the action is easily understood. Picard numbers; rank of the Mordell-Weil group; elliptic curves over function fields; automorphisms Peter F. Stiller, The Picard numbers of elliptic surfaces with many symmetries, Pacific J. Math. 128 (1987), no. 1, 157 -- 189. Picard groups, Special surfaces, Group actions on varieties or schemes (quotients) The Picard numbers of elliptic surfaces with many symmetries | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians As the title suggests, this is the sequel to part I [\textit{Z. Ran}, Isr. J. Math. 111, 109-124 (1999; Zbl 0958.14038)]. It continues with the business of counting irreducible plane curves by providing recursive formulae for the count. Here two sets of curves are counted.
First, recursive formulae are given for the number of irreducible plane curves of degree \(d\) passing through \(3d-g+1\) generic points and having a given tangent direction at one point, in the case where the curves are either rational (\(g=0\)) or elliptic (\(g=1\)). Second, such formulae are given for the number of plane curves of degree \(d\) with one cusp and passing through \(3d+g-1\) points, again for both rational and elliptic curves. counting irreducible plane curves; elliptic curve; cusp; plane curves Z. Ran, Bend, break and count II, Math. Proc. Camb. Phil. Soc. 127 (1999), 7-12. Zbl0972.14040 MR1692527 Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Bend, break and count. II: Elliptics, cuspidals, linear genera | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Equisingularity for families of space curve singularities of dimensionality type 1 in the sense of Zariski is studied using the two approaches, simultaneous resolution of the singularities and the equivalence of curve singularities for families.
The simultaneous resolutions of families are characterized by means of some invariants of the curves of the families. equisingularity; Zariski; space curves; simultaneous resolution Families, moduli of curves (algebraic), Singularities of curves, local rings, Plane and space curves Singularities of dimensionality type 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 A curve Z in \({\mathbb{P}}^ n\) enjoys the maximal rank property if the natural restriction map \(H^ 0({\mathbb{P}}^ n,{\mathcal O}_{{\mathbb{P}}^ n}(k))\to H^ 0(Z,{\mathcal O}_ Z(k))\) has maximal rank for every positive integer k. A smooth projective curve \(Y\subseteq {\mathbb{P}}^ n\) is said to be canonical if the hyperplane sections of Y are canonical. The author relates these properties to intrinsic properties of the curve, proving that for \(g>n\), \(n>3\) the general trigonal (bielliptic) canonical curve of genus g in \({\mathbb{P}}^ n\) has maximal rank. He also proves that for all \(n,d,g\in {\mathbb{N}},\) \(g>n>3,\) \(d>2g,\) a general embedding of degree d in \({\mathbb{P}}^ n\) of a general hyperelliptic curve of genus g has maximal rank. Both results are proven by degeneration arguments. postulation; gonality; maximal rank property; canonical curve; embedding; degeneration Families, moduli of curves (algebraic), Curves in algebraic geometry, Projective techniques in algebraic geometry Postulation and gonality for projective curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The cohomology of modular varieties defined by congruence subgroups of \(\mathrm{Sp}_ 4(\mathbb Z)\) whose levels lie between 2 and 4 is studied. Using a counting argument and the techniques of zeta functions, the authors completely determine the cohomology of a particular variety of this type.
See the review of the announcement in Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 66, 35 p. (1986; Zbl 0601.14021). cohomology of modular varieties; congruence subgroups of Sp(4,Z); levels; zeta functions Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Compactification of analytic spaces, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Cohomology of arithmetic groups, Arithmetic aspects of modular and Shimura varieties Moduli spaces of Riemann surfaces of genus two with level structures. 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 The author's classic notes ``Introduction to Moduli Problems and Orbit spaces'' were based on a course of lectures he gave at the Tata Institute of Fundamental Research, Mumbai between January and March 1975. The published version of these lecture notes appeared in 1978 as Volume 51 of the series ``Tata Institute of Fundamental Research Lectures on Mathematics and Physics'' [Berlin-Heidelberg-New York Springer Verlag (1978; Zbl 0411.14003)], and were reviewed back then by P. Cherenack. The main goal of the author's course was to provide an introduction to the framework of geometric invariant theory (GIT) and its applications to the construction of various moduli spaces in algebraic geometry. This pioneering approach toward the classification theory of algebro-geometric objects had been developed by D. Mumford in the 1960s, that is, just about fifteen years prior to the author's lectures on the subject, and the first research monograph on GIT was \textit{D. Mumford's} book from 1965 [Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 34. Berlin-Heidelberg-New York: Springer-Verlag. VI, 145 p. (1965; Zbl 0147.39304)]. As Mumford's book is written in a highly concise, advanced and abstract style, it was (and still is) barely accessible to non-specialists in modern algebraic geometry. Due to this very fact, and in view of the crucial significance of moduli theory in contemporary mathematics, the author's notes were an attempt to explain Mumford's ideas in a simplified context, thereby working with algebraic varieties instead od schemes, on the one hand, and concentrating on carefully selected topics on the other. Actuaily, apart from the brilliant survey article ``Introduction to the Theory of Moduli'' by \textit{D. Mumford} and \textit{K. Suominen} [Algebraic Geom., Oslo 1970, Proc. 5th Nordic Summer-School Math., 171--222 (1972; Zbl 0242.14004)], the author's booklet from 1978 has been the only down-to-earth introduction to geometric invariant theory and moduli problems in algebraic geometry for several decades, and as such it has become one of the timeless classics on the subject. Indeed, generations of researchers in this area acquired their basic knowledge through these lecture notes, which unfortunately have been out of print for many years, and further generations can still profit a great deal from the study of this masterpiece of expository writing in the field.
The book under review is the long-desired reprint of the author's classic ``Introduction to Moduli Problems and Orbit Spaces''. Having left the well-tried original text totally unaltered, the Tata Institute of Fundamental Research has re-issued this classic in a new, modern print which replaces the nostalgic typescript from thirty-five years ago. As for the precise contents, we may refer to the review of the first edition (Zbl 0411.14003, loc. cit.), since no changes have been made. However, it seems appropriate, after so many years, to recall the topics treated in the five chapters of the book:
1. The concept of moduli (families of algebro-geometric objects, fine and coarse moduli spaces, universal families).
2. Moduli of endomorphisms of vector spaces (families of endomorphisms, semisimple and cyclic endomorphisms, moduli and quotients).
3. Quotients (actions of algebraic groups on varieties, reductive groups and Nagata's theorem, affine quotients, projective quotients, linearizations of group actions, (semi-)stable points).
4. Examples of (semi-)stable points (Mumford's criterion for stability, binary forms, plane cubics, \(n\)-ordered points on a line, sequences of linear subspaces).
5. Vector bundles over a curve (coherent sheaves over a curve, locally universal families for semi-stable bundle, existence of a fine moduli space, bundles over a singular curve).
As one can see from this table of contents, the material covered in the author's classic is still as topical as it was thirty-five years ago. Although a number of texts on the subject have appeared in the last few years, the book under review will maintain its unique role in the relevant literature for further decades to come. Therefore it is more than gratifying that the overdue reprint of it finally has become available for further generations of students and researchers. geometric invariant theory; reductive groups; moduli problems; moduli spaces; geometric quotients; moduli of vector bundles Newstead P. E., Introduction to Moduli Problems and Orbit Spaces (2012) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Collected or selected works; reprintings or translations of classics, Geometric invariant theory, Group actions on varieties or schemes (quotients), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Vector bundles on curves and their moduli, Other algebraic groups (geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic) Introduction to moduli problems and orbit 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 This is the fourth lecture of these proceedings. Here the author sketches the construction of the coarse moduli space \({\mathcal M}_g\) of stable curves of genus \(g\) (\(g\geq 2\)) over \(\mathbb Z\) and studies some of its general properties. The semi-stable and stable curves are introduced in such a way that the family of stable curves is embedded by the tricanonical linear system. Then the coarse moduli space is obtained following Mumford's geometric invariant theory, by passing to the quotient via the natural action of the projective linear group. stable curve; geometric invariant theory; semistable curve; moduli space; tricanonical embedding; coarse moduli space Fine and coarse moduli spaces, Families, moduli of curves (algebraic), Local ground fields in algebraic geometry, Geometric invariant theory 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 We give a combinatorial and self-contained proof of the Harer-Zagier formula for the numbers \(\varepsilon_g(m)\) of ways of obtaining a Riemann surface of given genus \(g\) by identifying in pairs the sides of a \(2m\)-gon. This formula was the key combinatorial fact needed for the calculation of the Euler characteristic of the moduli space of curves of genus \(g\). The method developed here completes the original combinatorial approach imagined by Harer and Zagier and avoids using the integration over a Gaussian ensemble of random matrices. Our derivation is based upon the enumeration of arborescences and Euler circuits. Harer-Zagier formula; Riemann surface; Euler characteristic; moduli space; curves; genus; enumeration; Euler circuits Lass, B., Démonstration combinatoire de la formule de harer-Zagier, C. R. Acad. Sci. Paris, 333, 155-160, (2001) Exact enumeration problems, generating functions, Families, moduli of curves (algebraic) A combinatorial proof of the Harer-Zagier formula | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 most important open questions in the theory of algebraic cycles are the Hodge Conjecture, and its companion problem, the Tate Conjecture. Both these questions attempt to give a description of those cohomology classes on a nonsingular proper variety which are represented by algebraic cycles, in terms of intrinsic structure which is present on the cohomology of such a variety (namely, a Hodge decomposition, or a Galois representation). For the Hodge conjecture, the case of divisors (algebraic cycles of codimension 1) was settled long ago by Lefschetz and Hodge, and is popularly known as the Lefschetz \((1,1)\) theorem, though there is little general progress beyond that case. However, even this case of divisors is an open question for the Tate Conjecture, in general, even for divisors on algebraic surfaces. After giving an introduction to these problems, I will discuss the recent progress on the Tate Conjecture for \(K3\) surfaces, around works of M.~Lieblich, D. Maulik, F. Charles and K. Pera. algebraic cycles; Tate conjecture; \(K3\) surfaces Picard groups, Algebraic cycles, Finite ground fields in algebraic geometry, \(K3\) surfaces and Enriques surfaces The Tate conjecture for \(K3\) surfaces -- a survey of some recent progress | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a curve of irreducible components \(C_1,\dots,C_n\) in a smooth projective complex threefold \(X\). One assumes that \(C_i\) is rational nonsingular and \((K_X\cdot C_i)=0\), \(i=1,\dots,n\) (where \(K_X\) is the canonical class of \(X\)). Denote by \(\hat X\) the formal completion of \(X\) along \(C\). Under some additional technical assumption (i.e. each \(C_i\) has Kollár length \(1\)) the author discusses questions of the following type: when the curve \(C\) contracts to a (singular) point, when \(C\) deforms, and when \(C\) neither contracts, nor deforms in \(\hat X\). These questions are closely related with the deformation theory of the compound \(A_n\) singularity.
The results of this paper generalize results due to \textit{H. B. Laufer} [Ann. Math. Stud. 100, 261--275 (1981; Zbl 0523.32007)], \textit{M. Reid} [in: Algebraic varieties and analytic varieties, Adv. Stud. Pure Math. 1, 131--180 (1983; Zbl 0558.14028)], \textit{J. Jiménez} [Duke Math. J. 65, 313--332 (1992; Zbl 0781.32025)], \textit{S. Katz} and \textit{D. Morrison} [J. Algebr. Geom. 1, 449--530 (1992; Zbl 0788.14036)], and others. The paper is part of the author's Ph.D dissertation supervised by S. Katz. contractibility; formal completion Zerger, T.: Contraction criteria for reducible rational curves with components of length one in smooth complex threefolds. Pacific J. Math. 212, 377-394 (2003) \(3\)-folds, Families, moduli of curves (algebraic) Contraction criteria for reducible rational curves with components of length one in smooth complex threefolds. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0682.00009.]
Let C be an irreducible smooth curve of genus 3, defined over \({\mathbb{C}}\) of general moduli. To each non zero halfperiod \(\sigma\) of the jacobian of C is associated an elliptic-hyperelliptic (e.h.) curve \(\tilde X=W_ 2(C)\cdot (W_ 2(C)+\sigma)\) and the fixed point free involution \(i_{\sigma}: x\to x+\sigma\). The authors prove that \(\tilde X\) is irreducible, smooth of genus 7 and define a rational map \(\rho: R_ 3\to R_ 4^{eh}\), where \(R_ 3\) and \(R_ 4^{eh}\) are the moduli spaces of étale double covers of genus 3 curves and étale e.h. double covers of genus 4 curves.
In this paper the authors prove that \(\rho\) is birational and construct explicitly the inverse map. These results are intimately related to the question of whether \(R_ 3\) is rational or not. The paper contains also interesting remarks on the projective geometry of e.h. double covers of a curve of genus \( 4\). genus 3 curves; genus 4 curves; elliptic-hyperelliptic curve; Kummer variety; moduli spaces; étale double covers Families, moduli of curves (algebraic), \(K3\) surfaces and Enriques surfaces On a property of the Kummer variety and a relation between two moduli spaces 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 determine the singular locus of the moduli \(W=W_ 1\) of rational Weierstrass fibrations over \(\mathbb{P}^ 1_ \mathbb{C}\). The singular locus has 7 components of dimensions 5, 4, 3, 1, 0, 0 and 3. We also compute explicitly the general singularities which turn out to be cyclic quotient singularities. singular locus of moduli of rational Weierstrass fibration; quotient singularities; elliptic surface Singularities in algebraic geometry, Families, fibrations in algebraic geometry, Families, moduli of curves (algebraic), Singularities of curves, local rings The moduli of rational Weierstrass fibrations over \(\mathbb{P}^ 1\): Singularities | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Tautological rings are central objects in the study of the geometry of the moduli space \(\overline{\mathcal M}_{g,n}\) of Deligne--Mumford stable curves of genus \(g\) with \(n\) marked points. They are subrings of either the Chow ring or the cohomology ring of \(\overline{\mathcal M}_{g,n}\), constructed in such a way to contain certain ``geometrically natural'' classes, called tautological classes. For an introduction to tautological rings, see e.g. [\textit{R.~Vakil}, in: Enumerative Invariants in Algebraic Geometry and String Theory, Lecture Notes in Mathematics 1947, Berlin: Springer (2008; Zbl 1156.14043)].
Motivated by Gromov--Witten theory, \textit{Y.-P.~Lee} introduced [J. Eur. Math. Soc. (JEMS) 10, No. 2, 399--413 (2008; Zbl 1170.14021) and ``Invariance of tautological equations. II. Gromov--Witten theory'', preprint, \url{arXiv:math/0605708}, J. Am. Math. Soc. 22, 331--352 (2009; doi:10.1090/S0894-0347-08-00616-4)] an algorithm that should conjecturally compute all relations in the tautological ring.
In the paper under review, the algorithm is applied to give a uniform derivation of all known tautological relations in genus \(2\). Note that a geometric interpretation of this algorithm was given in [\textit{C.~Faber, S.~Shadrin} and \textit{D.~Zvonkine}, ``Tautological relations and the \(r\)-spin Witten conjecture'', preprint, \url{arxiv:math/0612510}]. moduli space of curves; Gromov--Witten theory; tautological rings; tautological relations D. Arcara and Y.-P. Lee, Tautological equations in genus 2 via invariance constraints, Bull. Inst. Math. Acad. Sin. (N.S.) 2 (2007), no. 1, 1 -- 27. Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Tautological equations in genus 2 via invariance constraints | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt). logarithmic geometry; moduli of curves; intersection theory; Jacobians of curves Families, moduli of curves (algebraic), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Rational and birational maps, Jacobians, Prym varieties Logarithmic intersections of double ramification cycles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth irreducible curve of genus \(g\) and let \(C_ d\) be its \(d\)-fold symmetric product. For a curve \(C\) with general moduli its Néron-Severi group is generated by the classes of two divisors. If \(\vartheta\) and \(x\) stand for those classes, the slope of a divisor on \(C_ d\), whose class is \(a\vartheta-bx\), is defined by \(b/a\). The author looks for lower and upper bounds for the slope of effective and ample divisors on \(C_ d\). curve; genus; symmetric product; Néron-Severi group; slope of a divisor Kouvidakis, A.: Divisors on symmetric products of curves. Trans. Am. Math. Soc. 337, 117--128 (1993) Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves Divisors on symmetric products of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Picard group Pic(R) of a (not necessarily commutative) ring R is the set of biisomorphism classes R-bimodules P that define a (Morita) equivalence (that is, the functor \(P\otimes_ R\)- is an equivalence of categories from the category of left R-modules \({}_ RM\) to itself). Pic(R) is a group under \(\otimes\). A (Morita) duality between \({}_ SM\) and \(M_ R\) is, roughly speaking, a pair of contravariant functors \((M\to M^*):_ SM\to M_ R\) and \((N\to N^*): M_ R\to_ SM\) of the form \(Hom_ S(\)-,U), respectively \(Hom_ R(\)-,U), for a given S-R bimodule \({}_ SU_ R\), together with natural transformations \(\sigma\) : 1\({}_ SM\to^{**}\) respectively \(\sigma\) : 1\({}_{M_ R}\to^{**}\) satisfying certain other conditions, including that \(\sigma\) is an isomorphism for modules of finite type.
The author discusses the relationship between Pic(R), certain subgroups thereof and Pic(R/J) (J the Jacobson radical of R), and the existence and properties of various types of duality and self-duality for the ring R. A typical result is the following (Theorem 10, part 1): Let D-Pic(S,R) be the set of biisomorphism classes of those S-R-bimodules that define a duality. Then for \(U\in D\)-Pic(S,R) \(\chi_ U: Pic(R)\to D\)-Pic(S,R), \((P)\to (U\otimes_ RP)\) and \(\psi_ U: Pic(S)\to D\)-Pic(S,R), \((Q)\to (Q\otimes_ SU)\) are bijections, and \(\psi_ U^{-1}\chi_ U: Pic(R)\to Pic(S)\) is an isomorphism of groups. Morita equivalence; Picard group; biisomorphism classes R-bimodules; equivalence of categories; category of left R-modules; natural transformations; self-duality Kraemer, J.: (Self)-duality and the Picard group. Comm. alg. 16, 2283-2311 (1988) Module categories in associative algebras, Picard groups (Self-)duality and the Picard group | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We prove an explicit formula for the total Chern character of the Verlinde bundle of conformal blocks over \(\overline{\mathcal{M}}_{g,n}\) in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Teleman's classification of semisimple CohFTs, there exists an element of Givental's group transforming the fusion algebra into the CohFT. We determine the element using the first Chern class of the Verlinde bundle on the interior \({\mathcal{M}}_{g,n}\) and the projective flatness of the Hitchin connection. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles on curves and their moduli, Families, moduli of curves (algebraic) The Chern character of the Verlinde bundle over \(\overline{\mathcal{M}}_{g,n}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0743.00011.]
Let \(k\) be an algebraically closed field, \(\text{char}(k)=0\), and \(f\in k[x,y]\), let \((X,k)\) be the canonical embedded resolution of the plane curve \(f=0\). Let \(E_ i\), \(i\in T\), be the irreducible components of \(h^{-1}(f^{-1}(\{0\}))\) (inverse image of \(f=0)\). To each \(i\in T\), associate \((N_ i,\nu_ i)\) where \(N_ i\) and \(\nu_ i\) are the multiplicity of \(E_ i\) in the divisor of \(f\circ h\) and \(h^*(dx\bigwedge dy)\) on \(X\). Fix one component \(E\) with its invariants \((N,\nu)\), denote by \(E_ 1,\dots,E_ k\) the other components of \(h^{-1}(f^{-1}(\{0\}))\) intersecting \(E\), then, you have: \((*)\;\sum^ k_{i=1}(\alpha_ i-1)+2=0\), where \(\alpha_ i=\nu_ i- {\nu\over N}N_ i\), \(1\leq i\leq k\).
In this conferences article, the author gives (without proofs) some generalizations of \((*)\) for hypersurfaces in \(\mathbb{A}_ n(k)\), \(3\leq n\). This problem is much deeper than in the case of plane curves:
(1) The components of \(h^{-1}(f^{-1}(\{0\}))\) may happen not to be linear projective spaces;
(2) If you blow-up a smooth variety inside several components of \(h^{- 1}(f^{-1}(\{0\}))\), you may happen not to separate them and you modify their Pic.
The author finds two formulas \(B_ 1\), \(B_ 2\) generalizing \((*)\) and a special one \(A\) for the components of \(h^{-1}(f^{-1}(\{0\}))\) created by blowing-ups of type 2. intersections; embedded resolution; multiplicity; Pic Embeddings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Picard groups Relations between numerical data of an embedded resolution | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Étant donné \(g,k\geq0\) et une partition \(\mu=(m_{1},\dots,m_{n})\) de \(k(2g-2)\), on définit l'espace \(\mathcal{H}_{g}^{k}(\mu)\subset \mathcal{M}_{g,n}\) paramètrant les \((C;p_{1},\dots,p_{n})\) tels que \(\mathcal{O}_{C}(\sum m_{i}p_{i})\) est le fibré canonique de \(C\). \textit{G. Farkas} et \textit{R. Pandharipande} [J. Inst. Math. Jussieu 17, No. 3, 615--672 (2018; Zbl 1455.14056)] ont introduit pour \(k=1\) un espace des modules propre paramétrisant les diviseurs twistés canoniques contenant \(\mathcal{H}_{g}^{1}(\mu)\). Cet article introduit et étudie la généralisation naturelle de ces espaces \(\tilde{\mathcal{H}}_{g}^{k}(\mu)\) à tout \(k\geq 0\).
La dimension des composantes irréductibles de \(\tilde{\mathcal{H}}_{g}^{k}(\mu)\) est obtenue en calculant la dimension des composantes connexes de \(\mathcal{H}_{g}^{k}(\mu)\), c.f. théorème 1.1 et proposition 1.2. Notons que cette question à été abordé par de nombreux auteurs à différents niveaux de généralité et on pourra consulter la remarque 1.3 pour un panorama sur cette question.
Cet article donne une formule conjecturale entre la classe fondamentale de \(\tilde{\mathcal{H}}_{g}^{k}(\mu)\) et un élément explicite de l'anneau tautologique de la compactification de Deligne-Mumford \(\bar{\mathcal{M}}_{g,n}\), voir conjectures A et A'. Cette conjecture est prouvée dans quelques cas en petit genre.
Enfin, l'auteur montre que les pluridifférentielles stables dont la restriction à chaque composante est non nulle est lissable sans changer les ordres des zéros et des pôles de ces restrictions. Notons que c'est un cas particulier du résultat principal de l'article de \textit{M. Bainbridge} et al. [``Strata of \(k\)-differentials'', Preprint, \url{arXiv:1610.09238}]. strata of \(k\)-differentials; deformation theory; tautological classes; double ramification cycles Schmitt, J., Dimension theory of the moduli space of twisted \textit{k}-differentials Families, moduli of curves (algebraic), Differentials on Riemann surfaces Dimension theory of the moduli space of twisted \(k\)-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 Let \(g,n\) be non-negative integers with \(2g-2+n>0\) and let \(S_{g,n}\) be a closed orientable surface of genus \(g\) with \(n\) deleted points. The condition \(2g-2+n>0\) asserts that each (analytically finite) Riemann surface structure on \(S_{g,n}\) comes from a Fuchsian group.
The space that parametrizes isomorphic marked (analytically finite) Riemann surface structures on \(S_{g,n}\) is the Teichmüller space \(T_{g,n}\), this being a simply-connected complex manifold of dimension \(3g-3+n\). The modular group \(\Gamma_{g,[n]}\) of \(S_{g,n}\) is the quotient of the group \({\roman Hom}^{+}(S_{g,n})\) of orientation-preserving self-homeomorphisms of \(S_{g,n}\) by its normal subgroup \({\roman Hom}^{0}(S_{g,n})\) of homeomorphisms homotopic to the identity. The group \(\Gamma_{g,[n]}\) acts as a discrete group of holomorphic automorphisms on \(T_{g,n}\), and \({\mathcal M}_{g,[n]}=T_{g,n}/\Gamma_{g,[n]}\) is the space that parametrizes isomorphic unmarked (analytically finite) Riemann surface structures on \(S_{g,n}\), called the moduli space \({\mathcal M}_{g,[n]}\). It follows that \({\mathcal M}_{g,[n]}\) is a complex orbifold of dimension \(3g-3+n\) and that \(\Gamma_{g,[n]}\) is its orbifold fundamental group.
Each finite degree regular cover of \({\mathcal M}_{g,[n]}\) can be described by a finite index normal subgroup \(G\) of \(\Gamma_{g,[n]}\). Finite order elements of \(\Gamma_{g,[n]}\) act with fixed points on \(T_{g,n}\); so in order to obtain smooth finite degree regular covers, one need to ensure that \(G\) has no torsion.
A general construction of finite degree regular covers of \({\mathcal M}_{g,[n]}\) is by considering finite index normal subgroups \(K\) of \(\pi_{1}(S_{g,n})\) which are invariant under its automorphisms (for instance, characteristic subgroups). The normal subgroup \(K\) provides a finite degree regular cover \(P_{K}:S_{K} \to S_{g,n}\) with deck group \(G_{K}=\pi_{1}(S_{g,n})/K\). The invariance property of \(K\) by the automorphisms of \(\pi_{1}(S_{g,n})\) asserts that every self-homeomorphism \(\phi\) of \(S_{g,n}\) lifts to a self-homeomorphism \(\psi\) of \(S_{K}\) so that \(P_{K} \circ \psi = \phi \circ P_{K}\), in particular, that \(\psi\) belongs to the normalizer of \(H\) (seen in \(\Gamma(S_{K})\), the modular group of \(S_{K}\)). The subgroup \(G\) of \(\Gamma_{g,[n]}\), consisting of those \(\phi\) with the property that \(\psi\) induces the identity automorphism of \(G_{K}\), is a finite index normal subgroup of \(\Gamma_{g,[n]}\). Unfortunately, the group \(G\) may have torsion.
\textit{E. Looijenga} [J. Algebr. Geom. 3, No. 2, 283--293 (1994; Zbl 0814.14030)] provided a construction, for the same \(K\) as above, of another finite index normal subgroup of \(\Gamma_{g,[n]}\) which is torsion free. For this, he considers the natural homomorphism \(\rho_{m}:\Gamma(S_{K}) \to \mathrm{Sp}(H_{1}(\overline{S_{K}},{\mathbb Z}/m{\mathbb Z})\), where \(m \geq 2\) is an integer and \(\overline{S_{K}}\) is the compactification of \(S_{K}\) (by adding all the punctures). If \(m \geq 3\), the restriction of \(\rho_{m}\) to \(G_{K}\) is injective (if \(m=2\), one needs to take care that \(G_{K}\) does not contains an hyperelliptic involution). Consider the homomorphism
\[
\rho_{K,(m)}:\Gamma_{g,[n]} \to N_{{\roman Sp}(H_{1}(\overline{S_{K}},{\mathbb Z}/m{\mathbb Z}))}(\rho_{m}(G_{K}))/\rho_{m}(G_{K}),
\]
which is defined by sending each \([\phi] \in T_{g,[n]}\) first to \([\psi] \in \Gamma(S_{K})\) and then applying \(\rho_{m}\) and later the corresponding projection to the quotient by \(\rho_{m}(G_{K})\). Then the kernel \(G\) of \(\rho_{K,(m)}\) will provide a finite index normal subgroup of \(T_{g,[n]}\) without torsion.
In the reviewed paper, the author generalizes Looijenga's construction to provide finite degree regular covers of \(\overline{\mathcal M}_{g,[n]}\), the Deligne-Mumford compactification of \({\mathcal M}_{g,[n]}\). This last space is constructed by adding stable Riemann surfaces (obtained from \(S_{g,n}\) by pinching a finite set of suitable pairwise disjoint simple loops). The main idea to see why the extension is possible is in essence the following. Let \(K\) be a normal subgroup of \(\pi_{1}(S_{g,n})\) of finite index which is invariant under the automorphisms of \(\pi_{1}(S_{g,n})\). In order to see a point in the boundary, we consider a set \(F\) of pairwise disjoint simple loops in \(S_{g,n}\); let \(S_{g,n}^{F}\) be the (topological) stable surface obtained by pinching the loops in \(F\). Use \(P_{K}\) to lift the loops in \(F\) to obtain a set \(F_{K}\) of pairwise disjoint loops in \(S_{K}\); let \(S_{K}^{F_{K}}\) be the stable surface obtained by pinching the loops in \(F_{K}\). The map \(P_{K}\) induces a natural map \(P_{K}^{F}:S_{K}^{F} \to S_{g,n}^{F}\). Then each structure of stable Riemann surface given to \(S_{g,n}^{F}\) lifts under \(P_{K}^{F}\) to provide a structure of a stable Riemann surface to \(S_{K}^{F}\). The group \(\pi_{1}(S_{g,n})/K\) can be seen as a group of automorphisms of \(S_{K}^{F}\) whose quotient is \(S_{g,n}^{F}\), and \(P_{K}^{F}\) its natural cover map. Moduli space of curves; Galois coverings of moduli spaces; smooth compactifications of Galois coverings; Teichmüller theory; congruence subgroup problem for the Teichmüller group Boggi, Marco, Galois coverings of moduli spaces of curves and loci of curves with symmetry, Geom. Dedicata, 168, 113-142, (2014) Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Coverings of curves, fundamental group, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Teichmüller theory for Riemann surfaces Galois coverings of moduli spaces of curves and loci of curves with symmetry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 central concept discussed in this research monograph is that of ``local moduli suite'' of an algebro-geometric object X. Roughly speaking this is a collection \(\{M_{\tau}\}\) of algebraic spaces each \(M_{\tau}\) prorepresenting the \(\tau\)-constant deformations of X occurring in the family \(\pi_{\tau}\) obtained by restricting an algebraisation \(\pi:\quad \tilde X\to H\) of the formal versal family of X to the ``\(\tau\)-constant stratum''.
An abstract theorem is proved first which guarantees the existence of the local moduli suite in the presence of certain axioms on the formal versal family. Results along this line were independently obtained by Palamodov and Saito. A conjecture of \textit{J. Wahl} [cf. Topology 20, 219-246 (1981; Zbl 0484.14012)] on the dimension of a smoothing component is then proved [this was independently proved by \textit{G.-M. Greuel} and \textit{E. Looijenga} in Duke Math. J. 52, 263-272 (1985; Zbl 0587.32038)].
Next the case of hypersurface singularities is investigated in detail (some modifications of the general setting of local moduli suites are here necessary). Here one is interested in the dimensions of the loci \(M_{\mu \tau}\) of all points in \(M_{\tau}\) corresponding to singularities with a given Milnor number \(\mu\).
Quite precise results are obtained in the case of weighted homogeneous plane curve singularities; in particular a coarse moduli space is proved to exist for all plane curve singularities with given semigroup \(\Gamma =<a_ 1,a_ 2>\), \((a_ 1,a_ 2)=1\) and minimal Tjurina number \(\tau\) and a computation is given for its dimension.
Part of these results are joint work of the authors with \textit{B. Martin}; they are a remarkable contribution to Zariski's program on the moduli problem for curve singularities. local moduli suite; formal versal family; hypersurface singularities; homogeneous plane curve singularities; coarse moduli space; Tjurina number O. A. Laudal and G. Pfister, ''Local moduli and singularities,'' In: Lecture Notes in Math., Vol. 1310, Springer, Berlin (1988). Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities in algebraic geometry, Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties, Families, moduli of curves (algebraic) Local moduli and singularities. Appendix (by B. Martin and G. Pfister): An algorithm to compute the kernel of the Kodaira-Spencer map for an irreducible plane curve singularity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(M(c_1, c_2)\) be the moduli space of stable holomorphic vector bundles of rank two and Chern classes \(c_1\) and \(c_2\) over the smooth quadric \(Q_3\). It is known that for \(c_1 = - 1\), \(c_2 = 1\) this space consists exactly of the so-called spinor bundle (or more precisely its dual), while for \(c_1 = 0\) this moduli is empty for odd values of \(c_2\). In the paper under review, the authors study the moduli space of normalized (i.e. \(c_1 = 0\) or 1) such vector bundles for small values of \(c_2\), more precisely, for \((c_1, c_2) = (0,2)\), \((- 1,2), (- 1,3)\), \((0, 4)\). They focus their attention on \(M(0,2)\), which they study from different points of view. More precisely, they show that \(M(0,2)\) is isomorphic to \(\mathbb{P}^0\) minus a quartic hypersurface that they explicitly determine when identifying \(\mathbb{P}^9\) with the space of all \(5 \times 5\) skew-symmetric matrices or with the space of all quadrics in \(\mathbb{P}^3\). They also describe in three different ways \(M(0,2)\) as a nontrivial fibration over \(\mathbb{P}^4 \backslash Q_3\) with fibers \(P^5 \backslash Q_4\). For the other moduli spaces they prove that \(M (- 1,2)\) is a locally trivial fibration over \(Q_4 \backslash Q_3\) with fibers \(\mathbb{P}^2 \backslash Q_1\) and that \(M (- 1,3)\) and \(M (0,4)\) are unirational and reduced of respective dimensions 12 and 21.
In the appendix, \textit{N. Manolache} classifies all curves (possibly reducible or non-reduced) \(Y \subset Q_3\) of degree 6 whose canonical line bundle is \(\omega_Y = {\mathcal O}_Y (-1)\). He obtains exactly nine families. This classification is used in the main paper for the study of \(M(0,4)\). moduli space over smooth quadric; moduli space of normalized vector bundles; spinor bundle Ottaviani , G. Szurek , M. On Moduli of Stable 2-bundles with Small Chern Classes on \(\mathbb{Q}\) 3 Families, moduli of curves (algebraic), Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On moduli of stable 2-bundles with small Chern classes on \(Q_ 3\). Appendix by Nicolae Manolache: The curves \(Y\) of degree 6 with \(\omega_ Y= {\mathcal O}_ Y(-1)\) on a smooth quadric \(Q_ 3\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a projective scheme (over an algebraically closed field of characteristic \(0\)), and \(\mathrm{Hilb}^{sc}(X)\) the Hilbert scheme of smooth connected curves in \(X\). For \(X={\mathbb P}^3\) or \(X\) a prime Fano \(3\)-fold it is known that \(\mathrm{Hilb}^{sc}(X)\) contains a generically non-reduced irreducible component (see the Introduction of the paper under review and references therein). Moreover, in this second case, the general element in this component is contained in a smooth element of the anticanonical linear system, which is a smooth \(K3\)-surface. The paper under review deals with the case of Enriques-Fano \(3\)-folds \(X \subset {\mathbb P}^N\), that is, normal projective \(3\)-folds whose general hyperplane section is an Enriques surface, and which are not a cone. In Theorem 1.1, sufficient conditions on \(X\) are provided to get the existence of a generically non-reduced component \(W\) of \(\mathrm{Hilb}^{sc}(X)\) whose general member is contained in an Enriques surface in \(X\). To be precise, the condition is the existence of a half pencil \(E\) (\(2E\) is an elliptic pencil) of anticanonical degree \(\geq 2\) on a smooth hyperplane section \(S \subset X\) and such that \(h^1 (E,N_{E/X}(E))=0\) (\(N_{E/X}\) stands for the normal bundle of \(E\) in \(X\)). The dimension of \(W\) and the linear class of the general curve in its surface are also provided. This result is a consequence of Theorem 1.2 where, under some hypotheses (see 1.2 for details), the (un)obstructedness of curves in smooth hyperplane sections of Enriques-Fano \(3\)-folds is studied and a computation of the dimension of \(\mathrm{Hilb}^{sc}(X)\) is provided. Hilbert scheme; obstruction; Enriques surface; Enriques-Fano threefold Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Formal methods and deformations in algebraic geometry Obstructions to deforming curves on an Enriques-Fano 3-fold | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \({\mathcal M}_{0,n}\) be the space of \(n\) (ordered) distinct points \((x_1,\dots, x_n)\) on \(\mathbb{P}^1\) modulo \(\text{PSL}_2(\mathbb{Z})\). There is a natural compactification of this space by adding \(n\)-pointed singular stable curves to the boundary. The compactified space \(\overline{\mathcal M}_{0,n}\) is called Deligne-Mumford moduli space. Each point \(x_i\) induces a line bundle on \({\mathcal M}_{0,n}\) with fibre \(T_{x_i}^* \mathbb{P}^1\). This line bundle can be extended to \(\overline{\mathcal M}_{0,n}\), which we denote by \(L_i\).
Denote by \(\chi_{d_1,\dots, d_n}\) the holomorphic Euler characteristics of the line bundles
\[
\bigotimes_{i=1}^n L_i^{\otimes d_i}: \chi_{d_1,\dots, d_n}= \sum_k (-1)^k \dim_{\mathbb{C}} H^k (\overline{\mathcal M}_{0,n} \bigotimes_{i=1}^n L_i^{\otimes d_i}).
\]
Let \({\mathfrak q}_i\)'s be formal variables. Introduce the generating function
\[
G({\mathfrak q}_1,\dots, {\mathfrak q}_n)= \sum_{(d_1,\dots, d_n)} \chi_{d_1,\dots, d_n} {\mathfrak q}_1^{d_1}\dots {\mathfrak q}_n^{d_n}= \chi \Biggl( \bigoplus_{i=1}^n \frac{1}{1-{\mathfrak q}_i L_i} \Biggr).
\]
Our main result is the following theorem:
\[
G({\mathfrak q}_1,\dots, {\mathfrak q}_n)= \Biggl( 1+ \sum_{i=1}^n \frac{{\mathfrak q}_i}{1-{\mathfrak q}_i} \Biggr)^{n-3} \prod_{i=1}^n \frac{1} {1-{\mathfrak q}_i}.
\]
In fact, several examples show that this formula might actually give the dimensions of the spaces of holomorphic sections of the line bundles, i.e., all higher cohomology groups vanish. We will indicate some of them in \S 3. Deligne-Mumford moduli space; holomorphic Euler characteristics; line bundles Lee, Y-P, A formula for Euler characteristics of tautological line bundles on the Deligne-Mumford moduli spaces, IMRN, 8, 393-400, (1997) Families, moduli of curves (algebraic), Topological properties in algebraic geometry, Divisors, linear systems, invertible sheaves, Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli A formula for Euler characteristics of tautological line bundles on the Deligne-Mumford moduli 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 In 1990s, Faber proposed a series of remarkable conjectures on the tautological rings of moduli spaces of curves. Faber's conjecture was partly inspired by the Witten-Kontsevich theorem which asserts that the integrals of \(\psi\) classes on moduli spaces of curves are governed by the KdV hierarchy. As pointed out by Faber, the explicit form of \(n\)-point functions, a formal power series whose coefficients encode integrals of \(\psi\) classes, is useful in studying the tautological ring. Such explicit formulas were obtained by Witten, Dijkgraaf and Zagier respectively when \(n=1,2,3\). This paper under review shows that the Witten-Kontsevich theorem can be equivalently formulated as a recursive formula of \(n\)-point functions, which can be effectively used to obtain some closed formulas of integrals of \(\psi\) classes. These observations are used to give a direct proof of Faber's intersection number conjecture in [J. Differ. Geom. 83, No. 2, 313--335 (2009; Zbl 1206.14079)]. Liu, K., Xu, H.: The \(n\)-point functions for intersection numbers on moduli spaces of curves. Adv. Theor. Math. Phys. \textbf{15}(5), 1201-1236 (2011). arXiv:math/0701319 Families, moduli of curves (algebraic), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) The \(n\)-point functions for intersection numbers on moduli spaces 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 In the previous paper [\textit{M. Mase}, Bull. Braz. Math. Soc. (N.S.) 52, No. 3, 499--536 (2021; Zbl 1476.14070)], the author showed that any coupling pairs in Yonemra's list [\textit{T. Yonemura}, Tôhoku Math. J. (2) 42, No. 3, 351--380 (1990; Zbl 0733.14017)] extend to the polytope dual except a few cases. In the present paper, the author shows that coupling pairs which admit polytope-duality with trivial toric contribution extend to lattice duality of the families of \(K3\) surfaces associated to the polytopes. families of \(K3\) surfaces; coupling of weight systems; duality of Picard lattices; toric hypersurfaces determined by lattice polytopes \(K3\) surfaces and Enriques surfaces, Singularities of surfaces or higher-dimensional varieties, Picard groups, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Lattice duality for coupling pairs admitting polytope duality with trivial toric contribution | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 aim of the paper is the description of the stratification given by Weierstrass points on the moduli space \(M_ 3\) of the isomorphism classes of Riemann surfaces of genus 3. The weighted number of Weierstrass points on such a Riemann surface is 24 and the weight can be 1, 2 or 3. A hyperelliptic curve of genus 3 has exactly 8 Weierstrass points of weight 3 and a curve containing at least a point of weight 3 is necessarily hyperelliptic. A non hyperelliptic curve of genus 3 can be embedded in \({\mathbb{P}}^ 2\) as curve of degree 4 and its Weierstrass points are exactly the points of inflexion. Thus the core of the question will be the study of the inflexion points of smooth plane curves of degree 4. A point of inflexion p on such a curve C with tangent line \(\ell\) is called an ordinary flex resp. a hyperflex) if \(\ell\) intersects C at p with multiplicity three (resp. four); the first kind of points are the Weierstrass points of weight two and the second kind of weight three. Denoting by f(C) the number of ordinary flexes and by a(C) the number of hyperflexes, on has \(f(C)+2a(C)=24.\) A very detailed analysis enables the author to obtain both the complete classification of smooth plane quartics with respect to their number of hyperflexes and their configurations (in particular one shows that a(C) can be any integer between 0 and 12 except 10 and 11) and the stratification of \(M_ 3\). Many comments in the paper refer to the history of the question, as well as to the current research. The paper details the author's preprint from 1981.
As the author says, similar results has obtained independently \textit{E. Lugert} [''Weierstraßpunkte kompakter Riemannscher Flächen vom Geschlecht 3'' (Thesis, Erlangen-Nürnberg 1981)]. Weierstrass points on the moduli space; Riemann surfaces of genus 3; inflexion points; hyperflex A.M. Vermeulen. \textit{Weierstrass points of weight two on curves of genus three}. Universiteit van Amsterdam, Amsterdam, 1983. Dissertation, University of Amsterdam, Amsterdam, 1983;With a Dutch summary. Riemann surfaces; Weierstrass points; gap sequences, Special algebraic curves and curves of low genus, Compact Riemann surfaces and uniformization, Families, moduli of curves (algebraic) Weierstrass points of weight two on curves of genus three. (Thesis) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Lefschetz type theorems for Picard groups of quasi-projective varieties. In particular we prove a generalization of the Kodaira vanishing theorem that we understand as a Lefschetz theorem for the structural sheaf. Vanishing theorems in algebraic geometry, Divisors, linear systems, invertible sheaves, Picard groups, Local cohomology and algebraic geometry Vanishing theorems and Picard 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 Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere \(S^3\) and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all Stäckel systems in these varieties. This allows us to recover the known list of separation coordinates on \(S^3\) in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron \(K_4\). separation of variables; Killing tensors; Stäckel systems; integrability; algebraic curvature tensors; associahedron Schöbel, Konrad, The variety of integrable Killing tensors on the 3-sphere, SIGMA, 10, (2014) Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Families, moduli of curves (algebraic), Determinantal varieties, Applications of global differential geometry to the sciences, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests The variety of integrable Killing tensors on the 3-sphere | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Witt group of a general real projective curve is explicitly computed in terms of topological and geometrical invariants of the curve. The authors method is strongly inspired by \textit{R. Sujatha}'s [Math. Ann. 288, 89--101 (1990; Zbl 0692.14012)] computation of the Witt group of a smooth projective real surface and uses a comparison theorem between the graded Witt group and the étale cohomology groups. The second part of the paper concentrates on the torsion subgroup of the Picard group of a smooth geometrically connected (not necessarily complete) curve \(X\) over a real closed field \(R\). Let \(C\) be the algebraic closure of \(R\) and \(X_C: =X\times_{\text{Spec}\,R}\) Spec \(C\). We compute \(\text{Pic}_{\text{tors}}(X)\) and \(\text{Pic}_{\text{tors}} (X_C)\) using the Kummer exact sequence for étale cohomology. These computations depend on a new invariant \(\eta(X)\in\mathbb{N}\) (resp. \(\eta(X_C))\) which we introduce in this note. The author studies relations between \(\eta(X)\), \(\eta(X_C)\) and the level and Pythagoras number of curves using new results of \textit{J. Huisman} and \textit{L. Mahé} [J. Algebra 239, 647--674 (2001; Zbl 1049.14015)]. The last part is devoted to the study of smooth affine conics and hyperelliptic curves. For such curves we calculate the Witt group and the torsion Picard group determining the invariant \(\eta\). étale cohomology; Pythagoras number Monnier, J. -P.: Witt group and torsion Picard group of real curves. J. pure appl. Algebra 169, 267-293 (2002) Picard groups, Topology of real algebraic varieties, Algebraic theory of quadratic forms; Witt groups and rings, Real algebraic sets Witt group and torsion Picard group of real 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 and study the stack \(\mathcal{U}^{ns,a}_{g,g}\) of (possibly singular) projective curves of arithmetic genus \(g\) with \(g\) smooth marked points forming an ample nonspecial divisor. We define an explicit closed embedding of a natural \(\mathbb{G}_{m}^{g}\)-torsor \(\widetilde{\mathcal{U}}^{ns,a}_{g,g}\) over \(\mathcal{U}^{ns,a}_{g,g}\) into an affine space, and we give explicit equations of the universal curve (away from characteristics \(2\) and \(3\)). This construction can be viewed as a generalization of the Weierstrass cubic and the \(j\)-invariant of an elliptic curve to the case \(g>1\). Our main result is that in characteristics different from \(2\) and \(3\) the moduli space \(\widetilde{\mathcal{U}}^{ns,a}_{g,g}\) is isomorphic to the moduli space of minimal \(A_{\infty}\)-structures on a certain finite-dimensional graded associative algebra \(E_{g}\) (introduced by Fisette and Polishchuk). moduli space of curves; A-infinity-algebra; Hochschild cohomology; deformation theory Polishchuk, A.: Moduli of curves as moduli of \(A_\infty \)-structures. Duke Math. J. (to appear). arXiv:1312.4636 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Differential graded algebras and applications (associative algebraic aspects) Moduli of curves as moduli of \(A_\infty\)-structures | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, the authors fill in a gap in the proof of the main result of their previous paper [\textit{L. Caporaso} et al., J. Am. Math. Soc. 10, No. 1, 1--35 (1997; Zbl 0872.14017)]. Namely, the main result of [loc. cit.] asserts that assuming a strong form of Lang's conjecture, the following holds for any fixed genus \(g \geq 2\): There is a bound \(N(g)\), such that for any number field \(K\), there are only finitely many isomorphism classes of curves over \(K\) of genus \(g\) that have more than \(N(g)\) \(K-\)rational points. The proof used in [loc. cit.] establishes this only for curves defined over the algebraic closure \(\overline K\) of \(K\), and in the current paper the authors provide an argument fixing this gap. rational points; uniformity Rational points, Arithmetic ground fields for curves, Families, moduli of curves (algebraic) Uniformity of rational points: an up-date and corrections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth curve of genus \(g\). We say that a line bundle \(L\) on \(C\) is normally generated if \(L\) is very ample and \(C\) has a projectively normal embedding via its associated morphism. Recently \textit{T. Kato}, \textit{C. Keem} and \textit{A. Ohbuchi} [Abh. Math. Semin. Univ. Hamb. 69, 319--333 (1999; Zbl 0969.10417)] gave the necessary and sufficient conditions for nonspecial very ample line bundles of degrees \(2g-2\), \(2g-3\) and also for special line bundles of degree \(d\geq 2g-6\) being normally generated. In this paper we determine the conditions that nonspecial very ample line bundles of degrees \(2g-4,2g-5\), or special line bundles of degree \(2g-7\) on a smooth curve are normally generated. DOI: 10.1016/j.jpaa.2004.01.008 Families, moduli of curves (algebraic) Normal generation of line bundles on algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the review of the first edition (1975; Zbl 0317.14004). local moduli problem; moduli space of the equisingularity class; deformation theory Zariski O.: Le problème des modules pour les courbes planes. Hermann, Paris (1986) Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Singularities in algebraic geometry, Power series rings Le problème des modules pour les branches planes. Cours donné au Centre de Mathématiques de l'École Polytechnique. Nouvelle éd. revue par l'auteur. Rédigé par François Kmety et Michel Merle. Avec un appendice de Bernard Teissier | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 setting in this article fixes a simple complex Lie algebra \({\mathfrak g}\), a Cartan subalgebra \({\mathfrak h}\), a root system \(Q\) and a Killing form, normalized such that the longest root is of length 2. The set of dominant integral weights of level \(\ell\) is denoted \(P_{\ell}({\mathfrak g})\). For an \(n\)-tuple \(\vec{\lambda}\) of weights, the corresponding bundle of conformal blocks on the moduli stack \(\overline{\mathrm M}_{g,n}\) of genus \(g\) curves with \(n\) marked points is denoted \({\mathbb V}_{\vec{\lambda}}({\mathfrak g},\ell)\).
Rank-level duality is a duality in the special case \(g=0\) between certain conformal blocks of \({\mathfrak s}{\mathfrak l}(r)\) at level \(s\) and certain conformal blocks of \({\mathfrak s}{\mathfrak l}(s)\) at level \(r\). There exist similar results for the symplectic and odd orthogonal Lie algebras.
The goal of the article under review is to relate explicitely the conformal block divisors, i.e. the Chern classes \(c_1({\mathbb V}_{\vec{\lambda}}({\mathfrak g},\ell)) \), on \(\overline{\mathrm M}_{g,n}\) using the rank-level duality isomorphisms. Under some technical conditions on \(\vec{\Lambda}\in P_1({\mathfrak g})^n\), \(\vec{\lambda}\in P_{\ell_1}({\mathfrak g}_1)^n\) and \(\vec{\mu}\in P_{\ell_2}({\mathfrak g}_2)^n\) which are known to be satisfied in the case of conformal embeddings \({\mathfrak s}{\mathfrak l}(r)\oplus{\mathfrak s}{\mathfrak l}(s)\to {\mathfrak s}{\mathfrak l}(rs)\) and \({\mathfrak s}{\mathfrak p}(2r)\oplus{\mathfrak s}{\mathfrak p}(2s)\to {\mathfrak s}{\mathfrak o}(4rs)\), the main rsult of the article (Theorem 1.2) asserts the following relation among conformal block divisors in the Picard group \(\mathrm{Pic}(\overline{\mathrm M}_{0,n})\):
\[
c_1({\mathbb V}_{\vec{\lambda}}({\mathfrak g}_1,\ell_1))+ c_1({\mathbb V}_{\vec{\mu}}({\mathfrak g}_2,\ell_2)) = \mathrm{rk}{\mathbb V}_{\vec{\lambda}}({\mathfrak g}_1,\ell_1).\left\{ c_1({\mathbb V}_{\vec{\Lambda}}({\mathfrak g},1))+\sum_{j=1}^{n} n_{\lambda_j,\mu_j}^{\Lambda_j}\psi_j\right\}
\]
\[
- \sum_{i=2}^{\;lfloor\frac{n}{2}\rfloor}\epsilon_i\left\{\sum_{A\subset\{1,\ldots,n\},|A|=i} b_{A,A^c}[D_{A,A^c}]\right\},
\]
where \([D_{A,A^c}]\) denotes the class of the boundary divisor corresponding to the partition \(A\cup A^c=\{1,\ldots,n\}\), \(\epsilon_i=\frac{1}{2}\) if \(i=n/2\) and \(\epsilon_i=1\) otherwise, \(\psi_j\) is the \(j\)th psi class, \(n_{\lambda_j,\mu_j}^{\Lambda_j}\) is an integer related to the conformal embedding (a sum/difference of trace anomalies), and \(b_{A,A^c}\) is another integer (sum of products of ranks of conformal blocks with some \(n_{\lambda,\mu}^{\Lambda}\) as coefficients).
The author proposes two proofs of the above formula, one using a geometric approach (vertex algebra techniques) and one using Fakhruddin's Chern class formula. rank-level duality; vertex algebras; conformal blocks; Picard group of the moduli stack of stable curves; psi classes; conformal embedding S. Mukhopadhyay, Rank-level duality and conformal block divisors, preprint (2013), . Vector bundles on curves and their moduli, Families, moduli of curves (algebraic), Relationships between algebraic curves and physics, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Vertex operators; vertex operator algebras and related structures, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Rank-level duality and conformal block divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We define a partition of the moduli space \(\overline {M^n_g}\) of stable \(n\)-pointed complex curves of genus \(g\) and show that the cohomology of \(\overline {M^n_g}\) in a given degree admits a filtration whose respective quotients are isomorphic to the shifted cohomology groups of the parts if \(g\) is sufficiently large. This implies that the map \(H^k (\overline M^n_g) \to H^k (M^n_g)\) is onto and that the Hodge structure of \(H^k (M^n_g)\) is pure of weight \(k\) if \(g \geq 2k+1\). stable cohomology; partition of the moduli space; filtration Martin Pikaart, An orbifold partition of \overline\?\(^{n}\)_{\?}, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 467 -- 482. Families, moduli of curves (algebraic) An orbifold partition of \(\overline {M}^ n_ g\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(M_{0,n}\) be the moduli space of \(n\)-pointed smooth curves of genus zero and \(\bar{M}_{0,n}\) its Deligne-Mumford compactification. \textit{B. Hassett} [Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)] introduced the notion of moduli space of \textit{weighted} stable pointed curves: this is a family of compactifications of \(M_{g,n}\), parametrized by choices of weights \(a_1,\ldots,a_n \in [0,1]\), such that a subset of the markings may coincide if and only if the total weight of these markings is at most \(1\). When all weights equal \(1\) one recovers the Deligne-Mumford compactification.
The paper under review considers the ``weighted'' compactification of \(M_{0,m+n}\) in the case that the first \(m\) markings have weight \(1\) and the last \(n\) markings have weight \(\epsilon \ll 1\), so that any subset of the last \(n\) markings may coincide. This space is denoted he \(\bar M_{0,m| n}\). To get a Deligne-Mumford stack one assumes that \(m \geq 2\) and \(m+n \geq 3\). The main result is the calculation of the \(S_m \times S_n\)-equivariant Poincaré polynomial of \(\bar M_{0,m| n}\).
When \(n=0\) this recovers a result of \textit{E. Getzler} [Prog. Math. 129, 199--230 (1995; Zbl 0851.18005)] and the method of proof is a generalization of Getzler's. Let me briefly summarize the idea of Getzler. First he calculates the \(S_n\)-equivariant cohomology of \(M_{0,n}\). Then he uses that \(\bar M_{0,n}\) has a stratification in which all strata are products of smaller moduli spaces \(M_{0,n_i}\), so that one basically needs to sum the contributions of all strata. He does this by encoding the data in terms of symmetric functions, using the bijection between symmetric functions and virtual representations of the symmetric group. Since the strata of \(\bar M_{0,n}\) correspond to trees, he needs to compute a sum over trees, which can then be encoded in terms of an operation on symmetric functions which is an analogue of the classical Legendre transform.
In the situation of the paper under review, the author instead begins by computing the \(S_m \times S_n\)-equivariant Poincaré polynomial of \(M_{0,m | n}\), which denotes the Zariski open subset of \(\bar M_{0,m | n}\) parametrizing smooth curves. The equivariant Poincaré polynomials of \(M_{0,m | n}\) are now indexed instead by a bisymmetric function. Then \(\bar M_{0,m| n}\) has a stratification whose strata are all products of smaller spaces \(M_{0,m_i | n_i}\), and the strata are now indexed by \textit{bicolored} trees -- more specifically, trees whose legs and internal edges and legs can have two colors (corresponding to markings of weight \(1\) resp. \(\epsilon\)), where all internal edges correspond to weight \(1\). The operation of summing over such bicolored trees is interpreted by an operation that the author calls the ``partial Legendre transform'' of bisymmetric functions. moduli of curves; poincare polynomial; operads; tensor species Families, moduli of curves (algebraic), Representations of finite symmetric groups, Fine and coarse moduli spaces Equivariant cohomology of certain moduli of weighted pointed rational curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0655.00011.]
Let \(S\) be a non-singular affine surface over \({\mathbb{C}}\). Suppose \(Pic(S)=(0)\) and \(\Gamma (S)^*={\mathbb{C}}^*\). The problem is if these conditions imply \(\pi_ 1(S)=(1)\). The answer is negative but investigation of non-singular affine surfaces \(S\) with \(Pic(S)=(0)\) and \(\Gamma (S)^*={\mathbb{C}}^*\) lead to the following results:
(1) If the fundamental group of \(S\) at infinity is finite then \({\bar \kappa}\)(S)\(=-\infty\). This later condition is known to imply that \(S\cong {\mathbb{C}}^ 2\) (as an affine surface).
(2) If \({\bar \kappa}\)(S)\(=0\) then \(S\) is simply connected. Moreover an explicit description of \(S\) is given.
(3) If \({\bar \kappa}\)(S)\(=1\), \(S\) has a \({\mathbb{C}}^*\)-fibration \(\pi\) : \(S\to C\) with \(C\) isomorphic to \(A^ 1_{{\mathbb{C}}}\) or \(P^ 1_{{\mathbb{C}}}\). If \(C\cong P^ 1_{{\mathbb{C}}}\), \(\pi_ 1(S)\) is trivial iff \(\pi\) has at most two singular fibers with multiplicity \(>1.\)
The paper contains also an example of such \(S\) with \({\bar \kappa}\)(S)\(=2\) and \(\pi_ 1(S)=(1)\), \(S\) being non isomorphic to an example previously constructed by C. P. Ramanujam.
Editorial remark (2022): \textit{G. Freudenburg} et al. [``Smooth factorial affine surfaces of logarithmic Kodaira dimension zero with trivial units'', \url{arXiv:1910.03494}] identified a gap in this classification of smooth affine surfaces of logarithmic Kodaira dimension zero, whose coordinate ring is factorial and has trivial units, which was closed in [\textit{T. Pełka} and \textit{P. Raźny}, Pac. J. Math. 311, No. 2, 385--422 (2021; Zbl 1479.14076)] who identified an additional infinite series of such surfaces. logarithmic Kodaira dimension; trivial Picard group; affine surface; fundamental group R. V. Gurjar and M. Miyanishi, Affine surfaces with \overline\?\le 1, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 99 -- 124. Homotopy theory and fundamental groups in algebraic geometry, Picard groups, Moduli, classification: analytic theory; relations with modular forms Affine surfaces with \({\bar \kappa}\leq 1\) | 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 A classical result from Hurwitz asserts that the endomorphism ring of the Jacobian of a very general curve \(C\) of genus at least 2 is the smallest possible, i.e. it is \(\mathbb{Z}\). \textit{S. Lefschetz} [Am. J. Math. 50, 159--166 (1928; JFM 54.0410.02)] proved the same in the case of hyperelliptic curves. \textit{C. Ciliberto} et al. [J. Algebr. Geom. 1, No. 2, 215--229 (1992; Zbl 0806.14020)] studied curves whose Jacobian has endomorphism ring larger than \(\mathbb{Z}\). Finally, \textit{Y. G. Zarhin} [Math. Proc. Camb. Philos. Soc. 136, No. 2, 257--267 (2004; Zbl 1058.14064)] considered curves with automorphisms (of a specific type) and showed that the endomorphism ring of their Jacobians is as smallest as possible.
In the paper under review, the authors show that such a minimality property still holds for curves endowed with an action of a given (but arbitrary) finite group \(G\). Indeed, they show that \(\text{End}_{\mathbb{Q}}(JC)\cong \mathbb{Q}[G]\) for a very general \(G\)-curve, with quotient curve of genus at least 3.
As an application, they also obtain interesting consequences on the natural representation of the centralizer of \(G\) in \(\text{Mod}(C)\), \(\rho_G: \text{Mod}(C)^G\rightarrow \text{Sp}(H^1(C,\mathbb{Q}))^G\), where \(\text{Sp}(H^1(C,\mathbb{Q}))^G\) stands for the centralizer of \(G\) in \(\text{Sp}(H^1(C,\mathbb{Q}))\), regarded as a virtual
linear representation the mapping class group \(\text{Mod}(C/G)\). Indeed, let \(X(\mathbb{Q}[G])\) be the set of rational irreducible characters of \( G \), take the isomorphism \[
\text{Sp}(H^1(C,\mathbb{Q}))^G\cong \prod_{\chi \in X(\mathbb{Q}[G]) }\text{Sp}(H^1(C,\mathbb{Q})_{\chi})^G,
\]
which mirrors the isotypical decomposition of the Jacobian of a \(G\)-curve \(C\), and denote by \(Mon^0(C)\) the identity component of the Zariski closure of the
image of \( \rho_G \) in \(\text{Sp}(H^1(C,\mathbb{Q}))^G\) and by \(\text{Mon}^0(C)_\chi\) the projection of \(\text{Mon}^0(C)\) to the factor \( \text{Sp}(H^1(C,\mathbb{Q})_{\chi})^G \). Then, the authors deduce different interesting properties on \(\text{Mon}^0(C)\) and on its factors \(\text{Mon}^0(C)_\chi\). moduli of curves; automorphisms of curves and Jacobians Families, moduli of curves (algebraic), Automorphisms of curves, Group actions on manifolds and cell complexes in low dimensions, Jacobians, Prym varieties Curves with prescribed symmetry and associated representations of mapping class 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 \(\bar \mathcal M_{g.n}\) be the moduli stack parametrizing Deligne-Mumford stable \(n\)-pointed genus \(g\) curves and let \(\bar M_{g,n}\) be its coarse moduli space: the Deligne-Mumford compactification of the moduli space of n-pointed genus g smooth curves. We prove that the automorphism groups of \(\bar \mathcal M_{g.n}\) and \(\bar M_{g,n}\) are isomorphic to the symmetric group on \(n\) elements \(S_{n}\) for any \(g, n\) such that \(2g-2 + n \geqslant 3\), and compute the remaining cases. Massarenti, A., The automorphism group of \(\overline{M}_{g, n}\), J. Lond. Math. Soc. (2), 89, 1, 131-150, (2014), MR 3174737 Families, moduli of curves (algebraic), Fibrations, degenerations in algebraic geometry, Fine and coarse moduli spaces, Stacks and moduli problems The automorphism group of \(\bar M_{g,n}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We find an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL\(_2\)-Higgs moduli space on a Riemann surface. On one side we have the components of the Lagrangian brane of U\((1,1)\)-Higgs bundles, whose mirror was proposed by Nigel Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL\(_2\)-Higgs moduli space. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin's proposal. Mirror symmetry (algebro-geometric aspects), Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Mirror symmetry with branes by equivariant Verlinde formulas | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 tackles the problem of rationality for twisted forms of the moduli space \(\overline M_{0,n}\) of stable curves of genus \(0\) with \(n\) marked points. The moduli space \(\overline M_{0,5}\) is a del Pezzo surface of degree \(5\) and all its twisted forms over nonclosed fields are known to be rational by classical results of Enriques, Manin and Swinnerton-Dyer. The paper under review shows that for odd \(n\geq 5\) twisted forms of the moduli space \(\overline M_{0,n}\) over infinite fields \(F\) of characteristic \(\neq 2\) are all \(F\)-rational, while for even \(n\geq 5\) there exists an \(F\)-form of \(\overline M_{0,n}\) that is not \(F\)-rational (actually, not even retract rational) provided that \(F\) admits a nonsplit quaternion algebra.
For the proof the authors use the identification of the automorphism group of \(\overline M_{0,n}\) with \(S_n\) by \textit{B. Fantechi} and \textit{A. Massarenti} [Int. Math. Res. Not. 2017, No. 8, 2431--2463 (2017; Zbl 1405.14064)], and establish the equivalence between the (retract) rationality of twisted forms of \(\overline M_{0,n}\) and the (retract) rationality of suitable representations of certain twists of \((\mathrm{GL}_2\times\mathbb G_m^n)/\mathbb G_m\). rationality; moduli spaces of marked curves; twisted forms; Galois cohomology; Brauer group Rational and unirational varieties, Rationality questions in algebraic geometry, Families, moduli of curves (algebraic), Linear algebraic groups over arbitrary fields, Brauer groups (algebraic aspects) The rationality problem for forms of \(\overline{M}_{0,n}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians On sait que l'espace des modules des surfaces de Riemann \(\mathcal{M}_{g,n}\) est de type général pour \(n\geq0\) et \(g\geq 24\) par les travaux d'Eisenbud, \textit{J. Harris} and \textit{D. Mumford} [Invent. Math. 67, 23--86 (1982; Zbl 0506.14016); ibid. 90, 359--387 (1987; Zbl 0631.14023)] ou pour tout \(g\) lorsque \(n\) est suffisamment grand [\textit{A. Logan}, Am. J. Math. 125, No. 1, 105--138 (2003; Zbl 1066.14030)]. De très nombreux articles traitent des cas restants, mais la dimension de Kodaira \(\mathcal{M}_{g,n}\) n'est pas encore connue pour tout \((g,n)\).
Le résultat principal de cet article est que \(\mathcal{M}_{12,7}\), \(\mathcal{M}_{12,6}\), \(\mathcal{M}_{13,4}\) et \(\mathcal{M}_{14,3}\) sont uniréglés, la dimension de Kodaira de \(\mathcal{M}_{16}\), respectivement \(\mathcal{M}_{12,8}\), est bornée par la dimension de \(\mathcal{M}_{16}\), respectivement \(\mathcal{M}_{12,8}\), à laquelle on ôte \(2\). De plus, les auteurs montrent que l'espace des modules des courbes hyperelliptiques de genre \(g\) avec \(4g+5\) points marqués est recouvert par des surfaces rationnelles pour \(g\geq2\).
La preuve repose sur un résultat indépendamment intéressant. Ce résultat est un critère pour lisser des pinceaux sur une surface réductible. moduli space of pointed curves; Kodaira dimension; unirulness; moduli space of hyperelliptic curves Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Rational and unirational varieties Pencils on surfaces with normal crossings and the Kodaira dimension of \(\overline{\mathcal{M}}_{g,n}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study the hyperbolic metrics with conical singularities on compact Riemann surfaces and consider the variation of the hyperbolic conical metric in holomorphic families, and introduce a generalized Weil-Petersson metric. They discuss the variational properties of the unique conical metric of constant curvature \(-1\) associated to a compact Riemann surface together with a weighted divisor. hyperbolic cone metric; weighted divisor; Weil-Petersson metric; holomorphic quadratic differentials Schumacher G., Trapani S.: Variation of cone metrics on Riemann surfaces. J. Math. Anal. Appl. 311, 218--230 (2005) Families, moduli of curves (algebraic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Conformal metrics (hyperbolic, Poincaré, distance functions) Variation of cone metrics on 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 [For the entire collection see Zbl 0619.00007.]
In the last 25 years, vector bundles have been an important topic in algebraic geometry. In this article, the author gives a summary of some of the important results in this subject.
First he describes the relations between stable vector bundles and unitary representations of the fundamental group of the Riemann surface. Let \(U(r,d)\) be the moduli space of semistable vector bundles of rank \(r\) and degree \(d\) on a curve X. The author states the results of himself, Ramanan, and Seshadri on the singularities of \(U(r,d)\) and questions on the existence of a universal bundle. - Let \(S(L,r,d)\) be the subvariety of \(U(r,d)\) corresponding to those bundles satisfying the property \(\bigwedge rE=L\). The author gives a summary of results of Harder, Mumford, himself, Newstead, and Tjurin on the topology and geometry of \(S(L,r,d)\). It is known that \(S(L,r,d)\) is unirational. However, it is not known whether \(S(L,r,d)\) is rational in general. stable vector bundles; fundamental group of the Riemann surface; moduli space; universal bundle Narasimhan M.S. Lectures on theta-functions. Lecture delivered at the University of Kaiserslautem Kaiserslautem (1987) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Rational and unirational varieties Survey of vector 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 Here the author proves (in characteristic 0 and using algebraic tools) the following results of Lefschetz-Barth-Larsen type.
Let \(f : X \to \mathbb{P}^n \times \mathbb{P}^n\) be a finite morphism with \(X\) smooth and irreducible, \(\dim (X) \geq n + 2\). Let \(\Delta \subset \mathbb{P}^n \times \mathbb{P}^n\) be the diagonal and \(Y : = \dim (f^{-1} (\Delta))\). Assume \(\dim (Y) = \dim (X) - n\) and \(Y\) normal. Then the natural maps \(\text{Pic}^0 (X) \to \text{Pic}^0 (Y)\) and \(\text{Alb} (Y) \to \text{Alb} (X)\) are isomorphisms and if \(\dim (X) \geq n + 3\) there is a canonical exact sequence
\[
0 \to \mathbb{Z} \to \text{Pic}^0 (X) \to \text{Pic}^0 (Y) \to 0.
\]
If \(Y \subset X_{\text{reg}}\), then there is a canonical exact sequence of profinite groups
\[
\mathbb{Z}^\wedge \to \pi_1^{\text{alg}} (Y) \to \pi_1^{\text{alg}} (X_{\text{reg}}) \to 0.
\]
Corollary: Let \(g : A \to \mathbb{P}^n\) be a finite morphism with \(A\) irreducible and \(B\) a closed subvariety of \(\mathbb{P}^n\). Set \(a : = \dim (A)\) and \(b : = \dim (B)\). If \(a + b \geq n + 2\), \(2b \geq n + 2\) and \(C : = g^{-1} (B)\) is normal of dimension \(a + b - n\), then the natural maps \(\text{Pic}^0 (A) \to \text{Pic}^0 (C)\) and \(\text{Alb} (C) \to \text{Alb} (A)\) are isomorphisms. If also \(a + b \geq n + 3\), then the natural restriction map \(\text{Pic} (A) \to \text{Pic} (C)\) is an isomorphism. ramified covering; Lefschetz theorem; Picard group; Albanese variety; algebraic fundamental group; finite morphism Topological properties in algebraic geometry, Picard schemes, higher Jacobians, Coverings in algebraic geometry, Picard groups Lefschetz type results for proper 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 Given an integer k denote by \({\mathcal F}_{2k}\) the set of isomorphism classes of couples (S,\({\mathcal O}_ S(D))\) where S is a K-3 surface and D is a primitive effective and numerically effective divisor of degree 2k. It is known that \({\mathcal F}_{2k}\) has a structure of quasi-projective variety. The following theorem is proved: Given any number N there exists a k such that Pic(\({\mathcal F}_{2k})\) contains N linearly independent elements. dimension of Picard group; moduli space of K-3 surfaces; numerically effective divisor O'Grady, Kieran G, On the Picard group of the moduli space for \(K\)-\(3\) surfaces, Duke Math. J., 53, 117-124, (1986) Picard groups, Divisors, linear systems, invertible sheaves, Families, moduli, classification: algebraic theory, \(K3\) surfaces and Enriques surfaces On the Picard group of the moduli space for K-3 surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be a smooth projective curve. A stable pair (E,\(\phi\)) on X, as defined by \textit{N. J. Hitchin} [Proc. Lond. Math. Soc., III. Ser. 55, 59- 126 (1987; Zbl 0634.53045)], is a vector bundle E on X together with a morphism \(\phi: E\to \Omega_ X\otimes E\) of \({\mathcal O}_ X\)-modules such that for any \(\phi\)-invariant proper subbundle F of E, the inequality \(\mu (F)<\mu (E)\) holds where \(\mu = \)degree/rank. In the quoted paper, Hitchin proved that the set of all isomorphism classes of stable pairs of rank 2 over a compact Riemann surface can be given the structure of a complex manifold which has the coarse moduli property in the analytic category.
The author constructs a coarse moduli scheme within the algebraic category in the following more general set up. Instead of taking (E,\(\phi\)) with \(\phi: E\to \Omega_ X\otimes E\), he considers the more general situation where \(\phi: E\to E\otimes L\) takes values in any fixed line bundle L on X. Furthermore, he proves that for any line bundle L, there exists a coarse moduli scheme M(r,d,L) for (S-equivalence classes of) semistable pairs ((E,\(\phi\)): \(E\to L\otimes E)\) of rank r, degree d, on X. The scheme M(r,d,L) is quasi-projective, and has an open subscheme \(M'\) which is the moduli scheme of stable pairs. stable pairs of rank 2 over a compact Riemann surface; coarse moduli property; moduli scheme of stable pairs 30. Nitsure, Nitin Moduli space of semistable pairs on a curve \textit{Proc. London Math. Soc.}62 (1991) 275--300 Math Reviews MR1085642 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on curves and their moduli, Fine and coarse moduli spaces, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic) Moduli space of semistable pairs on a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\overline{f}: \overline{X}= \{((x,y,z); (\alpha,\beta))\in \mathbb{P}_\mathbb{C}^2\times \mathbb{P}_\mathbb{C}^1\); \(\beta x^3+\beta y^3+\beta z^3-3\alpha xyz=0\}\to \mathbb{P}_\mathbb{C}^1\) denote the Hessian family of elliptic curves. Let \(S= \mathbb{P}_\mathbb{C}^1- \{1,\rho,\rho^2, \infty\}\), where \(\rho\) is a primitive third root of unity. Then the family restricts to a smooth family \(f:X\to S\) with a general fiber \(X_s\). In this article the author shows that the mixed Hodge structure on the second cohomology group \(H^2 (X,X_s)\) is a non-splitting extension of \(\mathbb{Z}(-2)^4\) by \(H^1(X_s)\). The proof consists of the following three steps:
(i) To determine completely the mixed \(\mathbb{Q}\)-Hodge structures on \(H^\bullet (X,\mathbb{Q})\) by using the weight spectral sequence with respect to the compactification \(X\subset \overline{X}\);
(ii) to show that \(H^2(X,\mathbb{Z})\) is torsion free, and
(iii) to find an appropriate element in \(F^2H^2 (X,X_s)_c\cong H^0(\overline{X}, \Omega_{\overline{X}}^2 (D))\), where \(D= \overline{f}^{-1} (\mathbb{P}_\mathbb{C}^1-S)\) denotes the union of the singular fibers, in order to ensure the extension is nontrivial. mixed Hodge structure; second cohomology group; Hessian family of elliptic curves Variation of Hodge structures (algebro-geometric aspects), Families, moduli of curves (algebraic), Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic topology on manifolds and differential topology, Elliptic curves A geometric example of non-trivially mixed Hodge structures | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Improving the previous paper [\textit{T. Shioda} and \textit{Y. Umezu}, Comment. Math. Univ. St. Pauli 48, No.1, 35-47 (1999; see the preceding review Zbl 0958.14013)] the author constructs here an infinite family of curves of genus \(g\geq 2\) over \(\mathbb{Q}\) with Mordell-Weil rank of its Jacobian variety \(\geq 3g+7\) modifying a method of Néron. genus; Mordell-Weil rank; Jacobian variety Families, moduli of curves (algebraic), Jacobians, Prym varieties, Pencils, nets, webs in algebraic geometry, Curves of arbitrary genus or genus \(\ne 1\) over global fields On Néron's construction of curves with high rank. II | 0 |
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