Inductively computable unions of fat linear subspaces
Abstract This paper introduces complexes of linear varieties, called inclics (for INductively Constructible LInear ComplexeS). By assigning an order of vanishing (i.e., a multiplicity) to each member of the complex, we obtain fat linear varieties (fat points if all of the linear varieties are points). The scheme theoretic union of these fat linear varieties gives an inclic scheme X. For such a scheme, we show there is an inductive procedure for computing the Hilbert function and a resolution of its defining ideal I X , regardless of the choice of multiplicities. As an application, we show how our results allow the computation of the Hilbert functions and of a resolution of fat points with all but one point having support in a hyperplane. We also explicitly compute the Waldschmidt constants α ˆ ( I X ) for galactic inclics X; these are special inclics constructed starting from a star configuration to which we add general points in a larger projective space.