(Generic, special and central fibres)
Let \(f \colon X \to S\) be a morphism. If \(S\) is irreducible with generic point \(\eta\), the generic fibre is \(X_\eta = X \times_S \Spec \kappa(\eta)\). If \(S = \Spec R\) for a discrete valuation ring \(R\) with fraction field \(K\) and residue field \(k\), the generic fibre is \(X_K\) and the special fibre is \(X_k = X \times_R \Spec k\), the fibre over the closed point. When \(S\) is a pointed curve or the spectrum of a local ring, the fibre over the distinguished point \(0\) is called the central fibre.
(Fat points)
An \(n\)-fold point over a field \(k\) is a scheme \(\Spec R\) with one point, where \(R\) is a local \(k\)-algebra with \(\dim_k R = n\).
(Scheme-theoretic intersection)
For closed subschemes \(X_1, X_2 \subseteq Y\), the scheme-theoretic intersection is the fibre product \(X_1 \times_Y X_2\), a closed subscheme of \(Y\). If \(Y = \Spec R\) and \(X_i = \Spec R/J_i\), it is \(\Spec (R/J_1 \otimes_R R/J_2) = \Spec R/(J_1 + J_2)\).
The family \(X = V(y^2 - x^3 - t) \subseteq \AA^2 \times \AA^1_t \to \AA^1_t\) over a field of characteristic \(0\) has smooth general fibres and the cuspidal cubic \(y^2 = x^3\) as central fibre. The line \(y = 0\) and the parabola \(y = x^2\) in \(\AA^2\) meet in \(\Spec k[x,y]/(y, y - x^2) = \Spec k[x]/(x^2)\), a \(2\)-fold point, which records the tangency that the set-theoretic intersection \(\{0\}\) forgets.