Let S be a rational projective surface given by means of parametrization whose base locus satisfies mild assumption. In this paper we present an algorithm that provides three maps f,g,h:A2⇢S⊂Pn such the union images covers S. As consequence, second generates two f,g˜:A2⇢S, its affine S∩An. case, number involved in cover is general optimal.

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This paper characterizes the injective and projective objects in the category of directed multigraphs, or quivers. Further, the injective envelope and projective cover of any quiver in this category is constructed.

Let X → S be an abelian scheme over a finite field. We show that for a set of rational primes l of density one, if q ≡ 1 mod l is sufficiently large, then the proportion of s ∈ S(Fq) for which l| ∣Xs(Fq) ∣∣ is at least 1/l−O(1/l2). We apply this to study class numbers of cyclic covers of the projective line.

In this paper, we study the relationship between two main categories of S-acts for a monoid S with zero from viewpoint existence projective covers. particular, prove that condition all acts have cover holds in category if and only it pointed acts. Furthermore, connected are cyclic they satisfy ascending chain on subacts.