A generalization of the space of complete quadrics

  • A. Al Ahmadieh University of Washington, USA
  • M. Kummer Technische Universitat Dresden, Germany
  • M. Ş. Sorea SISSA, Trieste, Italy and RCMA Lucian Blaga University Sibiu, Romania


To any homogeneous polynomial h we naturally associate a variety Ωwhich maps birationally onto the graph Γh of the gradient map ∇h and which agrees with the space of complete quadrics when h is the determinant of the generic symmetric matrix. We give a sufficient criterion for Ωh being smooth which applies for example when h is an elementary symmetric polynomial. In this case Ωh is a smooth toric variety associated to a certain generalized permutohedron. We also give examples when Ωh is not smooth.