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We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a ℂ*-action. For varieties with an isolated singularity, covered by a family of rational curves with a general member not passing
through the singular point, we show that this singularity is rational. In particular, this provides an explanation of classical
results due to H. A. Schwartz and G. H. Halphen on polynomial solutions of the generalized Fermat equation.
Received: 7 May 2002 /
Published online: 16 May 2003
Mathematics Subject Classification (2000): 14J17, 14L30, 13H10 相似文献
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Eduardo Cattani Raymond Curran Alicia Dickenstein 《Proceedings of the American Mathematical Society》2007,135(2):329-335
We present examples that show that in dimension higher than one or codimension higher than two, there exist toric ideals such that no binomial ideal contained in and of the same dimension is a complete intersection. This result has important implications in sparse elimination theory and in the study of the Horn system of partial differential equations.
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Eduardo Casas 《manuscripta mathematica》1983,43(2-3):167-190
Let be a given equisingularity class of irreducible algebroid plane curves with just one characteristic exponent. We first obtain the equisingularity type of the general polar of a curve of with general moduli and we also determine the Newton-Cramer polygon of the general polar of all curves in . 相似文献
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Gilcione Nonato Costa 《Bulletin of the Brazilian Mathematical Society》2016,47(3):883-910
Let F be a holomorphic foliation on Pn by curves such that the components of its singular locus are curves Ci and points pj. We compute the Baum-Bott indices BBφ(F, Ci) in terms of the main invariants of F and Ci. We also determine the sum of the BBφ(F, pi) in terms of the same invariants.When φ corresponds to the determinant, the latter result generalizes, from special to all holomorphic foliations, a formula for the number of isolated singularities of F, counted with multiplicities. 相似文献
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It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the
bifurcation set has singularities at the corresponding energy level.We formulate certain genericity assumptions for two degrees
of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy
levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under
our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing
the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate
integrable two degrees of freedom systems.
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Augusto Nobile 《Rendiconti del Circolo Matematico di Palermo》1996,45(3):437-452
Si comparano diverse nozioni d’equisingolarità per le curve tracciate sopra una superficie complessa liscia, e per gli ideali di dimensione finita dell’anello locale d’uno dei suoi punti. Si vede che, in generale, questi nozioni sono diverse, ma (per le curve) queste coincidono se esse fossero membri della stessa famiglia continua. 相似文献
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We find minimal generators for the ideals of secant varietiesof Segre varieties in the cases of k(1 x n x m) for all k, n,m, 2(n x m x p x r) for all n, m, p, r (GSS conjecture for fourfactors), and 3(n x m x p) for all n, m, p and prove they arenormal with rational singularities in the first case and arithmeticallyCohen–Macaulay in the second two cases. 相似文献
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Throughout this paper we study the existence of irreducible curves on smooth projective surfaces with singular points of prescribed topological types . There are necessary conditions for the existence of the type for some fixed divisor on and suitable coefficients , and , and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case . An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up of the form , deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in , and K3-surfaces.
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Marilda A. Simões 《Monatshefte für Mathematik》1985,99(3):235-244
Starting from a result of Pietsch on the Grothendieck ideal of strictly nuclear locally convex spaces, we classify all those Grothendieck ideals of nuclear locally convex spaces that are generated by a unique operator ideal on the class of Banach spaces. 相似文献
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Let C be a treelike curve, that is, a reduced curve whose irreducible components meet transversally at disconnecting points ofC. AssumeC has planar singularities. In this note we prove an Autoduality Theorem for C. More precisely, in [6] a natural Abel map A: C → $\bar J$ to a certain compactification $\bar J$ of the generalized Jacobian of C has been constructed. In this note we prove that the pullback map $A^* :Pic^0 (\bar J) \to Pic^0 (C)$ is an isomorphism. 相似文献