$${\mathfrak {S}}_5$$-equivariant syzygies for the Del Pezzo surface of degree 5

The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$P5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $${\mathfrak {S}}_5$$S5. We give canonical explicit $${\mathfrak {S}}_5$$S5-invariant Pfaffian equations through a 6$$\times $$×6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$S5. Finally, we give $${\mathfrak {S}}_5$$S5-invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$(P1)5, and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.

SOLVING NOT COMPLETELY INTEGRABLE QUANTILE PFAFFIAN DIFFERENTIAL EQUATIONS

The present work deals with quantile Pfaffian differential equations which are constructed using two-dimensional conditional quantiles of multidimensional probability distributions. As it was shown in [3] in case when the initial probability distributions have reproducible conditional quantiles this kind of Pfaan equations is completely integrable and the integral manifold is the conditional quantile of maximum dimension. In this paper we discuss properties of integral manifolds of maximum possible dimension for quantile Pfaffian equations which are not completely integrable. Manifolds of this type are described in terms of conditional quantiles of intermediate dimensions.

The Pfaffian closure of an o-minimal structure

Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C1 coefficients are definable.