Abstract
Let \( \mathbb {P}^n \) 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 \( \mathbb {P}^2 \) has this property. In this paper, we prove that this is true, in \( \mathbb {P}^2 \), in degrees 4 and 5.
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Olivares, J. (2018). Foliations in the Plane Uniquely Determined by Minimal Subschemes of its Singularities. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_6
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DOI: https://doi.org/10.1007/978-3-319-96827-8_6
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