Quadratic forms and singularities of genus one or two

38 pages, 32 figuresInternational audienceWe study singularities obtained by the contraction of the maximal divisor in compact (non-kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or Gorenstein. A family of polynomials depending on the configuration of the curves compute the discriminants of the quadratic forms of these singularities. We introduce a multiplicative branch topological invarian