I’d like to make something more explicit than I have. An effective Cartier divisor on a scheme is a closed subscheme locally cut out by one function, and that function is not a zero-divisor. (Translation: the zero-set does not contain any associated points.) PicX = group of line bundles = Cartier divisors modulo linear equivalence = Cartier divisors modulo principal Cartier divisors. We get a m...

Let S be a finite, nonempty set of positive integers. Then, the divisor graph ) (S D of S has S as its vertex set, and vertices i and j are adjacent if and only if either j i or i j . This paper investigates the vertex-chromatic number, the clique number, the clique cover number, and the independence number of ]) ([n D and its complement, where } , 1 : { ] [ Ν ∈ ≤ ≤ = n n i i n . Besides, we di...

For a commutative semigroup S with 0, the zero-divisor graph of S denoted by Γ(S) is the graph whose vertices are nonzero zero-divisor of S, and two vertices x, y are adjacent in case xy = 0 in S. In this paper we study median and center of this graph. Also we show that if Ass(S) has more than two elements, then the girth of Γ(S) is three.