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Search: MSC category 14C20 ( Divisors, linear systems, invertible sheaves )

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1. CMB Online first

Miranda-Neto, Cleto Brasileiro
 A module-theoretic characterization of algebraic hypersurfaces In this note we prove the following surprising characterization: if $X\subset {\mathbb A}^n$ is an (embedded, non-empty, proper) algebraic variety defined over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ of logarithmic vector fields of $X$ is a reflexive ${\mathcal O}_{{\mathbb A}^n}$-module. As a consequence of this result, we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a free ${\mathcal O}_{{\mathbb A}^n}$-module, which is shown to be equivalent to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily $X$ is a Saito free divisor. Keywords:hypersurface, logarithmic vector field, logarithmic derivation, free divisorCategories:14J70, 13N15, 32S22, 13C05, 13C10, 14N20, , , , , 14C20, 32M25

2. CMB 2013 (vol 57 pp. 562)

Kaveh, Kiumars; Khovanskii, A. G.
 Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety $X$ over $\mathbf{k} = \mathbb{C}$. In this short note, we first extend this intersection theory to an arbitrary algebraically closed ground field $\mathbf{k}$. Secondly we give an isomorphism between the group of Cartier $b$-divisors on the birational class of $X$ and the Grothendieck group of the semigroup of subspaces of rational functions on $X$. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative point of view on Cartier $b$-divisors and their intersection theory. Keywords:intersection number, Cartier divisor, Cartier b-divisor, Grothendieck groupCategories:14C20, 14Exx

3. CMB 2001 (vol 44 pp. 452)

Ishihara, Hironobu
 Some Adjunction Properties of Ample Vector Bundles Let $\ce$ be an ample vector bundle of rank $r$ on a projective variety $X$ with only log-terminal singularities. We consider the nefness of adjoint divisors $K_X + (t-r) \det \ce$ when $t \ge \dim X$ and $t>r$. As an application, we classify pairs $(X,\ce)$ with $c_r$-sectional genus zero. Keywords:ample vector bundle, adjunction, sectional genusCategories:14J60, 14C20, 14F05, 14J40

4. CMB 1998 (vol 41 pp. 267)

Fukuma, Yoshiaki
 On the nonemptiness of the adjoint linear system of polarized manifold Let $(X,L)$ be a polarized manifold over the complex number field with $\dim X=n$. In this paper, we consider a conjecture of M.~C.~Beltrametti and A.~J.~Sommese and we obtain that this conjecture is true if $n=3$ and $h^{0}(L)\geq 2$, or $\dim \Bs |L|\leq 0$ for any $n\geq 3$. Moreover we can generalize the result of Sommese. Keywords:Polarized manifold, adjoint bundleCategories:14C20, 14J99
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