(Hypersurface)
A hypersurface of degree \(d\) in \(\PP^n\) is the closed subscheme \(V(F)\) of a nonzero homogeneous polynomial \(F\) of degree \(d\). Two polynomials define the same hypersurface exactly when they differ by a nonzero scalar, so the hypersurfaces of degree \(d\) are the points of \[ \PP^{N} = \PP H^0(\PP^n, \OO(d)), \qquad N = \binom{n+d}{d} - 1 . \]
(General and very general)
A property holds for a general hypersurface of degree \(d\) if it holds for every hypersurface whose point lies outside a proper closed subset of \(\PP^N\). It holds for a very general hypersurface of degree \(d\) if it holds for every hypersurface whose point lies outside a countable union of proper closed subsets of \(\PP^N\).
Over an uncountable algebraically closed field, a countable union of proper closed subsets does not cover \(\PP^N\), so a very general hypersurface exists. Over \(\overline{\mathbb{Q}}\) or \(\overline{\mathbb{F}}_p\) the countably many hypersurfaces defined over the field can exhaust the parameter space, and “very general” can be vacuous.
Smoothness is a general property: the singular hypersurfaces form the discriminant hypersurface in \(\PP^N\). A property such as \(\Pic(X) = \ZZ \cdot \OO_X(1)\) for a surface \(X \subseteq \PP^3\) of degree \(d \geq 4\) holds only for a very general surface: the Noether–Lefschetz locus where the Picard number jumps is a countable union of proper closed subsets.