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

In: Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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  • Jorge Olivares

    (Centro de Investigación en Matemáticas, A.C.)

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.

Suggested Citation

  • Jorge Olivares, 2018. "Foliations in the Plane Uniquely Determined by Minimal Subschemes of its Singularities," Springer Books, in: Gert-Martin Greuel & Luis Narváez Macarro & Sebastià Xambó-Descamps (ed.), Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, pages 135-143, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-96827-8_6
    DOI: 10.1007/978-3-319-96827-8_6
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