• James Alexander

      Articles written in Proceedings – Mathematical Sciences

    • Generic hypersurface singularities

      James Alexander André Hirschowitz

      More Details Abstract Fulltext PDF

      The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x1,...,xr be closed points in general position in projective spacePn, then the linear subspaceV ofH0 (⨑n,O(d)) (the space of homogeneous polynomials of degreed on ⨑n) formed by those polynomials which are singular at eachxi, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x1,...,xr. As such, the “expected” value for the dimension ofV is max(0,h0(O(d))−r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.

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