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    Algebraic quotients and Geometric Invariant Theory
    (Universidad Nacional de Trujillo, 2020-06-30)
    The quotient of an algebraic variety by action of an algebraic group does not always has a variety structure. The aim of this work is to describe a methodfor constructing good quotients, in the sense of Geometric invariant theory, in algebraicgeometry.
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    On the Zariski invariant of plane branches
    (Polish Academy of Sciences::will be referenced::ROR-ID, 2026-03-16)
    We show how to obtain the Zariski invariant of a plane branch employing the contact order or the intersection multiplicity with elements in a particular family of curves, and we present some consequences of this result.
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    On some indices of foliations and applications
    (Springer Science and Business Media Deutschland GmbH, 2026-03-01)
    In this paper, we establish a relationship between the Milnor number, the χ-number and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the Gómez-Mont–Seade–Verjovsky index, we prove that the difference between the multiplicity and the Tjurina number of a foliation with respect to a reduced curve is independent of the foliation. We also derive a local formula for the Tjurina number of a foliation with respect to a reduced curve. From a global point of view, these results lead to the following consequences: We provide a new proof of a global result regarding the multiplicity of a foliation due to Cerveau–Lins Neto and a new proof of a Soares’s inequality for the sum of the Milnor number of an invariant curve of a foliation. Additionally, we obtain bounds for the global Tjurina number of a foliation on the complex projective plane. Finally, we provide an answer to the conjecture posed by Alcántara and Mozo-Fernández about foliations on the complex projective plane having a unique singularity.
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