Beatriz MOLINA SAMPER – Invariant surfaces of non-degenerate foliations in dimension three

Date(s) : 24/10/2019   iCal
10 h 30 min - 11 h 30 min

Beatriz MOLINA SAMPER (Universidad de Valladolid)

More than forty years ago the recognized mathematician Rene Thom asked the following question : Is there always an invariant curve for a germ of foliation on (C2, 0) ? In 1982 Cesar Camacho and Paulo Sad gave a positive answer to this question. For higher dimension, there are results of Felipe Cano, Dominique Cerveau and Jean-Francois Mattei where they prove the existence of invariant hypersufaces for germs of foliations on (Cn, 0), in the non-dicritical frame. However, when the existence of dicritical components in the exceptional divisor is allowed, the classical collection of examples of Jouanolou provides foliations in (C3, 0) without invariant surface. We present here a result of existence of invariant surface for “non-degenerate foliations” on (C3, 0), possibly dicritical. In this context, with non-degenerated we mean foliations that admit a combinatorial procedure of reduction of singularities (these generalize Kouchnirenko’s classical non-degenerate hypersurfaces). To prove this assertion of local nature in dimension three, we pass through the following global statement in dimension two : “The isolated invariant branches of non-degenerate foliations over projective toric surfaces extend to global curves”.

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