Abstract
Introduction Borel-fixed ideals Monomial orders Some algebraic lemmas Defining the higher initial ideals Representing the higher initial ideals Group action on $R^{s+1}(I)$ Describing the action on $R^{s+1}(I)$ Borel-fixedness Higher initial ideals of hyperplane sections Representing the higher initial ideals of general hyperplane sections Higher initial ideals as combinatorial structures Reading cohomological information Examples: Points and curves in $\mathbf P^3$ References.
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