A subscheme X of Pn admits an unexpected hypersurface of degree t and multiplicity m if requiring multiplicity m at a general point P fails to impose the expected number of conditions on the linear system of hypersurfaces of degree t containing X. Conditions which either guarantee the occurrence of unexpected hypersurfaces, or which ensure that they cannot occur have shown surprising connections to root systems, hyperplane arrangements among other diverse topics. In this talk we focus in particular on the connection between unexpected hypersurfaces and the generic initial ideal.
Based on a joint work with E. Guardo, B. Harbourne, J. Migliore.