Abstract:On this note we prove that a holomorphic foliation of the projective plane with rich, but finite, automorphism group does not have invariant algebraic curves. Seja {mathcal F} uma folheação do plano projetivo complexo de grau d com grupo de automorfismo finito e cuja ação no espaço de cofatores não possui ponto fixo. Neste artigo mostramos que se {mathcal F} possui ao menos uma singularidade genérica então {mathcal F} não possui nenhuma curva algébrica invariante
“…We prove also (Proposition 9.3) that if d : k(x, y, z) → k(x, y, z) is a derivation such that d(x) = y p , d(y) = z q , d(z) = x r , where p, q, r ∈ N, then k(x, y, z) d = k if and only if pqr 2. Some similar questions are studied in the interesting paper [18]. (The authors wish to thank the referee for pointing out this paper.…”
We present some general properties of the field of constants of monomial derivations of k(x 1 , . . . , x n ), where k is a field of characteristic zero. The main result of this paper is a description of all monomial derivations of k(x, y, z) with trivial field of constants. In this description a crucial role plays the classification result of Moulin Ollagnier for Lotka-Volterra derivations with strict Darboux polynomials. Several applications of our description are also given in this paper.
“…We prove also (Proposition 9.3) that if d : k(x, y, z) → k(x, y, z) is a derivation such that d(x) = y p , d(y) = z q , d(z) = x r , where p, q, r ∈ N, then k(x, y, z) d = k if and only if pqr 2. Some similar questions are studied in the interesting paper [18]. (The authors wish to thank the referee for pointing out this paper.…”
We present some general properties of the field of constants of monomial derivations of k(x 1 , . . . , x n ), where k is a field of characteristic zero. The main result of this paper is a description of all monomial derivations of k(x, y, z) with trivial field of constants. In this description a crucial role plays the classification result of Moulin Ollagnier for Lotka-Volterra derivations with strict Darboux polynomials. Several applications of our description are also given in this paper.
“…Theorem 4.16 admits many generalizations, see for instance [41], [70], [18], [51]. Even the original statement has now many different proofs exploiting different aspects of the theory of algebraic foliations like automorphism groups of foliations [69,53]), global geometry of the space of foliations ( [50]), or arithmetic through Galois group actions ( [17]) to name a few.…”
We review properties of closed meromorphic 1-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, existence of separatrices, and resolution of singularities under the lenses of the theory of closed meromorphic 1-forms and flat meromorphic connections. We apply the theory to investigate the algebraicity separatrices in a semi-global setting (neighborhood of a compact curve contained in the singular set of the foliation), and the geometry of smooth hypersurfaces with numerically trivial normal bundle on compact Kähler manifolds.
“…The presence of symmetries has been used to understand some relevant problems in holomorphic foliation theory, especially concerning integrability. For instance, it is used in the construction of Jouanolou's foliations [11], which play an important role in his proof of the density of foliations on the complex projective plane P 2 without invariant algebraic curves; in [16] this relation between automorphism groups and integrability was explored and put into more concrete terms.…”
Let F denote a singular holomorphic foliation on P 2 having a finite automorphism group Aut(F ). Fixed the degree of F , we determine the maximal value that Aut(F ) can take and explicitly exhibit all the foliations attaining this maximal value. Furthermore, we classify the foliations with large but finite automorphism group.If this is the case, we also say that the vector field v is G-semi-invariant.
ExamplesIn this section we describe those foliations already mentioned in Theorem 1.2.3.1. The Jouanolou Foliation J d . The Jouanolou foliation of degree d, denoted by J d , is defined by the vector fieldor, in affine coordinates, by the 1-form (x d y − 1)dx + (y d − x d+1 )dy.
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