Numerical Computation of the dimensions of the cohomology of twists of Ideal Sheaves

This article presents several numerical algorithms for computations in sheaf cohomology. Let X be an algebraic set defined by a system of homogeneous multivariate polynomials with coefficients in C. Let C be a union of reduced, irreducible pure-dimensional curve components of X. The first algorithm computes the dimension of the first cohomology of any twist of the ideal sheaf of C. Let D be a reduced set of points on C. The second and third algorithms solve the Riemann-Roch problem of computing the dimension of the space of divisors on C which are linearly equivalent to D. Let S be a reduced, connected, locally Cohen-Macaulay pure-dimensional surface made up of components of X. The fourth algorithm computes the first and second cohomology of any twist of the ideal sheaf of S. Furthermore, as the algorithms are based on homotopy continuation, they take advantage of the natural parallelism underlying continuation methods.