Foliations in the Plane Uniquely Determined by Minimal Subschemes of its Singularities

Jorge Olivares ()
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Jorge Olivares: Centro de Investigación en Matemáticas, A.C.

A chapter in Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, 2018, pp 135-143 from Springer

Abstract: Abstract Let ℙ n $$ \mathbb {P}^n $$ be the projective space over an algebraically closed ground field K. In a previous paper, we have shown that the space of foliations by curves of degree greater than or equal to two which are uniquely determined by a subscheme of minimal degree of its scheme of singularities, contains a nonempty Zariski-open subset and hence, that the set of non-degenerate foliations with this property contains a Zariski-open subset. Moreover, we posed the question whether every non-degenerate foliation in ℙ 2 $$ \mathbb {P}^2 $$ has this property. In this paper, we prove that this is true, in ℙ 2 $$ \mathbb {P}^2 $$ , in degrees 4 and 5.

Keywords: Nonempty Zariski Open Subset; Minimum Degree; Special Subschemes; Polynomial Test; Dim kA (search for similar items in EconPapers)
Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-96827-8_6

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DOI: 10.1007/978-3-319-96827-8_6

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