We study the geometry of standard graded Hilbert schemes of polynomial rings and exterior algebras. We are motivated by a famous theorem of Reeves and Stillman for the Grothendieck Hilbert scheme, which states that the lexicographic point is smooth... By contrast, we show that, in standard graded Hilbert schemes of polynomial rings and exterior algebras, the lexicographic point can be singular, and it can lie in multiple irreducible components. We settle questions of [I. Peeva, M. Stillman, Math. Ann. 339 (2007)] and of [D. Maclagan, G. Smith, Adv. Math. 223 (2010)]. (read more)