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Invariant Curves by Vector Fields on Algebraic Varieties
Author(s) -
Campillo A.,
Carnicer M. M.,
García de la Fuente J.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700008978
Subject(s) - mathematics , complete intersection , pure mathematics , gravitational singularity , algebraic variety , quotient , invariant (physics) , intersection theory , algebraic curve , projective space , degree (music) , algebraic geometry , hilbert series and hilbert polynomial , algebraic surface , algebraic number , mathematical analysis , hilbert space , projective test , mathematical physics , differential algebraic equation , ordinary differential equation , physics , acoustics , differential equation
If C is a reduced curve which is invariant by a one‐dimensional foliation F of degree d F on the projective space then it is shown that d F −1+ a is a bound for the quotient of the two coefficients of the Hilbert–Samuel polynomial for C , where a is an integer obtained from a concrete problem of imposing singularities to projective hypersurfaces, and so a bound is obtained for the degree of C when it is a complete intersection. Concrete values of a can be derived for several interesting applications. The results are presented in the form of intersection‐theoretical inequalities for one‐dimensional foliations on arbitrary smooth algebraic varieties.