Linear system of divisors ($\mathbb{P}(V)$)

A linear system of divisors is a projective family of effective Weil divisors linearly equivalent to a fixed divisor. A complete linear system of a divisor contains every effective representative of that class; a projective subspace gives a subsystem. For a Cartier divisor, the family is obtained from global sections of its associated line bundle. The algebraic curve case recovers the usual moving family of effective divisors.

On a smooth variety over a field of characteristic zero, a general member of a basepoint-free linear system of divisors is smooth. More generally it is smooth outside the base locus. Irreducibility is a separate assertion and requires suitable hypotheses on the associated morphism; a pencil composed with a map to a curve can have disconnected general divisors.

On a smooth projective curve of genus two, $D=K+P+Q$ has three sections and no base points by the Riemann-Roch theorem. Its map to ${\mathbb{P}}^2$ is nondegenerate and has mapping degree times image degree four. The only nonbirational possibility is a double cover of a conic. Every degree-two divisor with two sections is canonical, again by Riemann-Roch theorem, so that possibility forces $D\sim 2K$, or $P+Q\sim K$. If this equivalence fails, the image is a quartic and the map is birational. The quartic is singular because a smooth plane quartic has genus three.

A linear system is base-point-free if its sections have no common zero. Their ratios then define an everywhere-defined morphism to projective space whose hyperplane pullbacks are members of the original system.

A very ample linear system defines an embedding into projective space. On a smooth projective curve this means it has no base points and separates both distinct points and tangent directions; the complete system of a very ample divisor is an example.

For a finite generating space $V\subseteq H^0(X,\mathcal{L})$ on a projective scheme, its map to projective space is a closed immersion exactly when, after algebraic closure of the ground field, the displayed evaluation is surjective for every length-two closed subscheme $E$. Two distinct points test point separation; a double point tests a direction in the Zariski tangent space. For the complete section space, this characterizes a very ample line bundle.