Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

On projective Cohen-Macaulayness of a Del Pezzo surface embedded by a complete linear system

Abstract: Let k be an algebraically closed field. We understand by a Del Pezzo surface X over k a non-singular rational surface on which the anti-canonical sheaf ―wx is ample. We call the self-intersectionnumber d=a)x of wx the degree of X, then we get that 1^J^E9. It is well known that X is isomorphic to PlxP\ which has degree 8, or an image of P2 under a monoidal transformation with center the union of r―9―d points which satisfiesthe following conditions: (a) no three of them lie on a line; (b) no six of them lie on a conic; (c) there are no cubics which pass through seven of them and have a double point at the eighth point. Conversely any surface described above is a Del Pezzo surface of the corresponding degree ([8,in, Theorem 1]). It is also well known that ―o)x is very ample when d^3 and that ample divisors on X of degree 3, which is a cubic surface, are very ample too. In this paper we will get that ample divisorson X of degree d^3 are very ample and that ample divisors on X of degree 2 [resp. 1] other than ―o>x [resp. ―o)x nor ―2^x] ore very ample. A closed subscheme V in PN is said to be projectively Cohen-Macaulay if its affine cone is Cohen-Macaulay. It is equivalent to that H1(PN,Jv(m))=0 for every meZ and H\V, Ov(m))-0 for every meZ and 0

On nearly absolutely isolated hypersurface singularities of dimension 2

Abstract: A formula for the geometric genus of a nearly absolutely isolated hypersurface singularitiy of dimension 2 is found by using the canonical resolution. An upper bound for the fundamental cycle of such singularity is also given. Introduction. Let -r: M -V be a resolution of an isolated singularity of dimension 2. The number dim H?(V, R17r*Q(M)) is defined to be the geometric genus of the singularity. We study the case in which V can be embeded in C3. Our major result is a formula (Theorem 2) for the geometric genus when the singularity is nearly absolutely isolated (see ?2 for definition). Our proof is based on the canonical resolution of an m-tuple point, which is developed in the first section. The case m = 2 is well-known (cf. [2, pp. 47-48]). Finally we give a bound for the fundamental cycle of that kind of singularity as well as for its self-intersection number (Theorem 3). The base field is the complex number field C. A singular point p always means a hypersurface point of dimension 2. Sometimes we use the same notation for a line bundle and its corresponding divisor if it will not cause confusion. 1. m-tuple covering. Let m > 2. Let Y be a smooth surface covered by affine open sets {U i? , . Let C0, ... , Cm2 be effective divisors on Y locally defined in Ui by equations cs i = 0 (0 < s < m -2, i E I). Suppose there is a line bundle F over Y with transition function { fij } over { Ui n Uj ) such that (1) c5 = fiT7 Sc'j for all 0 < s < m 2. Let 0i be the fibre coordinates over Ui. Then the equations (2) 4im + C2_2 -2 + _ _ _ +Co,i 0 give rise to a surface X in F and the projection map from F to Y induces a finite morphism f: X -Yof degree m. DEFINITION. The surface X constructed as above is called the m-tuple cover of Y with branch locus data (CO, * * *, C. 2 ) Let Di be the discriminant of the equation (2) for i E I. Then { Di }i, I give rise to a divisor on Y, denoted by D. Obviously D is the branch locus of the map f. The map f is called totally ramified at a point p E Y if f '(p) consists of one point. Received by the editors September 12, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 14J17. ?1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

Finiteness of logarithmic crystalline representations II

Abstract: Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a simple normal crossings divisor and fix positive integers $r$ and $f$. We show that the set of absolutely irreducible representations $\pi_1(X_{\bar K})\rightarrow \mathrm{GL}_r(\mathbb{Z}_{p^f})$ that come from log crystalline $\mathbb Z_{p^f}$-local systems over $(X_K,S_K)$ of rank $r$ is finite. The proof uses $p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.

Computing the full quotient in bi-decomposition by approximation

Abstract: Bi-decomposition is a design technique widely used to realize logic functions by the composition of simpler components. It can be seen as a form of Boolean division, where a given function is split into a divisor and quotient (and a remainder, if needed). The key questions are how to find a good divisor and then how to compute the quotient. In this paper we choose as divisor an approximation of the given function, and characterize the incompletely specified function which describes the full flexibility for the quotient. We report at the end preliminary experiments for bi-decomposition based on two AND-like operators with a divisor approximation from 1 to 0, and discuss the impact of the approximation error rate on the final area of the components in the case of synthesis by three-level XOR-AND-OR forms.

Pro-l fundamental groups of generically ordinary semi-stable fibrations with low slope

Abstract: Let k be an algebraically closed field of characteristic p > 0 and l a prime that is distinct from p Let f : S \rightarrow C be a generically ordinary, semi-stable fibration of a projective smooth surface S to a projective smooth curve C over k Let F be a general fibre of f, which is a smooth curve of genus g \geq 2 We assume that f is generically strongly l-ordinary, by which we mean that every cyclic etale covering of degree l of the generic fibre of f is ordinary Suppose that f is not locally trivial and is relatively minimal Then deg f*\omegaS/C > 0, where \omegaS/C is the sheaf associated to the relative canonical divisor KS/C = KS − f*KC Hence the slope of f,\lambda( f ) = K2 S/C/deg f*\omegaS/C is well-defined Consider the push-out square \pi1(F) \rightarrow \pi1(S) \rightarrow \Pi(C) \rightarrow 1 \downarrow \Pi where \pi1 is the algebraic fundamental group and \pil1 is the pro-l fundamental group When f is non-hyperelliptic and \lambda(f) < 4, we show that the morphism \pil1(F)\rightarrow /alpha\Pi is trivial