is the ideal sheaf of B and that gives rise to Since an open subset of , there results in the map: Finally, when a basis of V is chosen, the above discussion becomes more down-to-earth (and that is the style used in Hartshorne, Algebraic Geometry). Linear system determined by a map to a projective space Each morphism from an algebraic variety to a projective space determines a base-point-free linear system on the variety; because of this, a base-point-free linear system and a map to a projective space are often used interchangeably. For a closed immersion of algebraic varieties there is a pullback of a linear system on to , defined as (page 158). O(1) on a projective variety A projective variety embedded in has a natural linear system determining a map to projective space from . This sends a point to its corresponding point . See also Brill–Noether theory Lefschetz pencil bundle of principal parts References Hartshorne, R. Algebraic Geometry, Springer-Verlag, 1977; corrected 6th printing, 1993. . Lazarsfeld, R., Positivity in Algebraic Geometry I, Springer-Verlag, 2004. . Geometry of divisors