Hodge numbers of hypersurfaces in \(\mathbb{P}^4\) with ordinary triple points (Q2035036)

This paper was motivated by the study of quintics with ordinary triple points. For a hypersurface \( X\) with only ordinary triple points as singularities in a weighted projective space \(\mathbb P\) of dimension 4 the blow-up \(\tilde X\) of \( X\) at its singular locus is a crepant resolution. The main result is a formula for Hodge numbers: \[ h^{1,1}(\tilde X)= 1+ t + \delta, \qquad h^{1,2}(\tilde X)= h^{1,2}( X_s) -11 t +\delta \] where \(X_s\) is a smoothing of \(X\) (a smooth hypersurface of the same degree), \(t\) is the number of triple points and \(\delta\) is the \textit{defect}, the difference between the actual and expected number of conditions imposed by the singularities. The result also applies to triple solids. The proofs are based on a careful analysis of certain exact sequences of sheaves of differential forms.