Reducible and Multiple Algebraic Curves

a₁ additive divisors affine space algebraic curve algebraic family algebraic map algebraic sheaf algebraic varieties algébrique bijective called canonical CHEVALLEY closed set composition sequence constructed corollary D₁ definition deg(D denote dimension divisor classes divisor variety divisorial sheaf exact sequence exists a neighbourhood family of divisors finite formula GROTHENDIECK group of divisors group variety H¹(C H¹(W H¹(X hence imbedded points induces infinitesimally injective irreducible isomorphism Jacobian variety lemma is proved linear local rings Ly(D Math morphism Ni(P nilpotent elements nonsingular divisor nonsingular points notations open set Ox(P p-primary parametrized Pic(X Picard variety presheaf primary ideals principal divisor proposition proves the lemma R₁ rational function rational homomorphism reduced curve Riemann-Roch theorem ring of quotients ringed space ROSENLICHT SERRE sheaves singular points subset subsheaf Suppose surjective topology total ring U₁ variety and let write X₁ Xred Y₁