Tvariaties

Tvariaties is a Dutch-language term used in some mathematical texts to refer to t-varieties, namely families of algebraic varieties parameterized by a variable t. In this sense, a t-variety encodes how a given geometric object can vary continuously in a one-parameter family, often studied through deformation theory and moduli problems.

Formally, a t-variety over a field k is a pair (X, π) where π: X → Spec(k[t]) is a flat morphism of k-schemes. For each t0 in the base, the fiber X_{t0} is a variety over k. The fiber at t = 0 is frequently taken as a reference or special fiber, while fibers at other values of t describe deformations of that reference. In some contexts, the base is Spec(k[[t]]) for formal deformations, or Spec(k[t]/(t^n)) for truncated deformations.

Examples commonly involve curves or surfaces defined by equations with coefficients depending on t. For instance, a family of plane curves given by F_t(x,y,z) = 0, where F_0 defines a singular curve and F_t for t ≠ 0 yields smooth curves, illustrates how singularities can appear or disappear along a deformation.

Deformation theory provides tools to study t-varieties. The tangent space to the space of deformations often relates to H^1(X0, T_{X0}) for a smooth reference fiber X0, with obstructions in H^2(X0, T_{X0}) in higher dimensions. Rigid varieties have no nontrivial deformations.

Applications of t-varieties include the construction of moduli spaces, smoothing of singularities, and investigations into how geometric invariants behave in families. See also deformation theory, moduli spaces, and algebraic families.