Expecting the unexpected: Quantifying the persistence of unexpected hypersurfaces

The independence of imposed vanishing conditions is a foundational issue for wide range research in algebraic geometry. In this spirit, if X⊂Pn reduced subscheme, we say that X admits an unexpected hypersurface degree t and multiplicity m requiring at general point P fails to impose the expected number on linear system hypersurfaces containing X. Conditions which either guarantee occurrence hypersurfaces, or ensure they cannot occur, are not well understood. Research date has made surprising connections root systems, hyperplane arrangements, generic splitting types vector bundles, among other diverse topics. paper introduce new methods studying unexpectedness, such as use initial ideals partial elimination clarify when it can occur. We also exhibit geometric properties some cases preclude having certain kinds unexpectedness. addition, formulate way quantifying unexpectedness (our AV sequence), allows us detect extent persists increases but t−m remains constant. Finally, study what from Hilbert function