Linear systems with points on a rational normal curve
Let L = Ln,d(m1, . . . , ms) be the linear system of hypersurfaces of degree d passing through s points p1, . . . , ps ∈ P n C , with multiplicities at least m1, . . . , ms, respectively. A classical question, known as the polynomial interpolation problem, asks to compute the dimension of such linear systems. In this talk we give a complete answer to the problem when the points p1, . . . , ps sit on a rational normal curve of degree n in P n , say C. From the perspective of commutative algebra, this configuration of points was first thoroughly studied by Catalisano together with Ellia and Gimigliano in the 90s. Indeed, they conjectured an algorithm that computes the Hilbert function of the ideal IZ ⊆ C[x0, . . . , xn] associated to the scheme of fat points Z = m1p1 + . . . + msps. That is to say, they conjectured an algorithm to compute the dimension of L.
We will point out how Castravet and Tevelev proved this conjecture when studying the finite generation of the Cox ring of the blown-up space Blp1,...,psP n . Finally, we will explain how this can be exploited to obtain a closed formula computing the dimension of any linear system with points sitting on C. We will see how this formula depends exclusively on the appearence of certain varieties in the base locus. With this viewpoint we provide a geometric understanding of the problem for this configuration of points, giving a new perspective to some known algebraic results. This talk is based on a preprint written with Antonio Laface and Elisa Postinghel
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