Enumerative geometry of legendrian foliations

Abstract: Foliations, or more generally, distributions, provide a geometric viewpoint in the theory of differential equations. Grosso modo, a foliation of dimension one, is a (polynomial) recipe to draw a line at each point. We’ll stick to 3-dim projective space. Similarly, a distribution of codimension one assigns a plane at each point. We assume the coefficients of the equation of the plane are of degree one. As syzygies trained minds will recognize, this entails the distribution is specified by an anti-symmetric 4×4 matrix. Those of maximal rank correspond to the distributions of contact. A foliation is called legendrian whenever tangent to some distribution of contact. Our goal is to describe the calculation of the dimension and degree of the subvariety of legendrian foliations (and friends). It turns out that the answer is given by Athus polynomials. This fits into a Schubert Calculus like programme of exploring the geometry of parameter spaces of foliations.

algebraic geometry

Audience: researchers in the topic

Brazilian algebraic geometry seminar

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