Let $X\subset\mathbb{P}^n$ be a (globally) complete intersection, let $(X_t)_{t\in\mathbb{C}^1}$ be a flat family, with $X_1=X$. Which types of schemes can we get as $X_0$?

Or, conversely, which (embedded, projective) schemes deform to complete intersections?

e.g. some non-ACM schemes (not arithmetically Cohen-Macaulay) deform to C.I.'s. Does every pure-dimensional subscheme of $\mathbb{P}^n$ deform to a C.I.? Two immediate obstructions are the degree and the arithmetic genus. Any other necessary conditions?

(Just for housekeeping, a somewhat related question on deformations of C.I.:)

upd: I meant the family over the germ $(\mathbb{C}^1,0)$, so there are no complications with the geometry of the base.