From Pappus’s Theorem To Cayley-Bacharach Theorem

In this article,we will start from several important theorems in classical projective geometry,gradually advance to the Cayley-Bacharach theorem,and Finally study the so-called Cayley-Bacharach property of a nonsingular projective variety,as introduced by Sheng-Li Tan.A beautiful result due to him asserts the equivalence between the Cayley-Bacharach property and the k-very ampleness of certain adjoint linear system,which ties the classical Cayley-Bacharach theorem to the modern Fujita’s conjecture in complex algebraic geometry.Owing to the need for proof,we generalize the RiemannRoch theorem to a version for singular curves.