doublecover

Double cover denotes a map p: Y -> X that is a covering of degree two. For most points x in X the fiber p^{-1}(x) consists of two points; when branches occur, over a branch locus B ⊂ X the fiber has one point with multiplicity two, and the map is not locally a disjoint union of two sheets.

In the topological setting, double covers are two-sheeted covering spaces, classified by a Z/2Z principal bundle, equivalently by a cohomology class in H^1(X, Z/2Z). They arise by taking a universal cover and factoring by a free Z/2 action. An example is the map from the circle to itself given by z -> z^2, which is unbranched.

In algebraic geometry, a double cover is a finite morphism f: Y -> X of degree 2. Locally it looks like Spec(O_X[u]/(u^2 - f)) with a local function f. Globally one common construction uses a line bundle L on X and a global section s ∈ H^0(X, L^{⊗2}); Y is built as Spec(O_X ⊕ L^{−1} t) with t^2 = s. The branch locus is the divisor where s vanishes.

Examples and applications: Hyperelliptic curves are double covers of the projective line branched at 2g+2 points. The Riemann–Hurwitz formula relates genera: 2g_Y - 2 = 2(g_X - 1) + deg(R). Double covers encode Z/2-symmetry and are used in the study of spin structures and Galois covers.