2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99888We consider symplectic singularities in the sense of A. Beauville as examples of Poisson schemes. Using Poisson methods, we prove that a symplectic singularity admits a finite stratification with smooth symplectic strata. We also prove that in the formal neighborhood of a closed point in some stratum, the singularity is a product of the stratum and a transversal slice. The transversal slice is also a symplectic singularity, and the product decomposition is compatible with natural Poisson structures. Moreover, we prove that the transversal slice admits a $C^*$-action dilating the symplectic form.Version 4, really final, to appear in Crelle. Corrected Corollary 2.8 (it was too strong)Algebraic Geometrytext