Varieties of low \(\Delta\)-genus
type: Item
instance of: scholarly article
title: Varieties of low \(\Delta\)-genus
zbMATH Open document ID: 0536.14021
DOI: 10.1215/S0012-7094-83-05031-7
author: Tony Horowitz
published in: Duke Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: Let V be a complex projective algebraic manifold of dimension r, L an ample line bundle on V and consider the Fujita \(\Delta\)-genus \(\Delta(V,L)=d+r-h^ 0(V,L),\) where \(d=c_ 1(L)^ r\). \textit{T. Fujita} classified pairs (V,L) with \(\Delta(V,L)=0\) [J. Fac. Sci., Univ. Tokyo, Sect. I A 22, 103-115 (1975; Zbl 0333.14004)] and those with \(\Delta(V,L)=1\) [J. Math. Soc. Japan 32, 709-725 (1980; Zbl 0474.14017) and 33, 415-434 (1981; Zbl 0474.14018)]. The author assumes that L is birationally very ample (i.e. \(| L|\) has no base locus and the associated map \(\rho_ L:V\to {\mathbb{P}}(H^ 0(V,L)\quad \check {\;})\) is birational) to study in great detail varieties of low \(\Delta\)-genus, i.e. pairs (V,L) satisfying \(\Delta(V,L)<h^ 0(V,L)-r,\) or equivalently, \(d>2\Delta\). The starting point is the following fact (holding also under weaker assumptions on L) which is well known to the experts. If V is a regular surface with \(\Delta\) (V,L)\(\geq 1\), then \(h^ 0(V,L)\leq 3\Delta +6,\) with the single exception of \(({\mathbb{P}}^ 2,{\mathcal O}(3))\), and equality characterizes geometrically ruled conic bundles. The first result is the following extension to irregular surfaces. Let \(r=2\), \((V,L)\neq({\mathbb{P}}^ 2,{\mathcal O}(3))\) and \(\Delta(V,L)<1/3 h^ 0(V,L)-2;\) then \(\rho_ L(V)\) is projectively ruled and \(\Delta(V,L)=rh^ 1({\mathcal O}_ V)-h^ 1(V,L.\) By the technique of ladders, this extends also to higher dimensions provided that L is very ample and \(2\leq \Delta(V,L)<1/3(h^ 0(V,L)-r-4).\) As to the next range of \(\Delta\), the author proves the following. Let \(r\geq 2\), \(2\leq \Delta(V,L)<frac{1}{2}(h^ 0(V,L)-r-6)\) and assume L very ample if \(r\geq 3\). Then \(\rho_ L(V)\) is ruled either by linear spaces or by quadrics. Moreover the author finds the minimum value of \(\Delta\) (V,L) for regular quadric bundles. The above results apply to the projective classification of irreducible varieties \(X\subset {\mathbb{P}}^ n\) of dimension r,…
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 3851266
zbMATH Keywords: polarized manifolds
zbMATH Keywords: Fujita delta genus
zbMATH Keywords: scrolls
zbMATH Keywords: quadric bundles
zbMATH Keywords: regular surface
zbMATH Keywords: geometrically ruled conic bundles
zbMATH Keywords: irregular surfaces
zbMATH Keywords: projective classification
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
cites work: Q4175362
cites work: Q4123438
cites work: Q4100195
cites work: On the structure of polarized manifolds with total deficiency one. I
cites work: On the structure of polarized manifolds with total deficiency one. II
cites work: Q4143433
cites work: Curves with high self-intersection on algebraic surfaces
cites work: FANO 3-FOLDS. I
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cites work: Hyperplane sections of projective surfaces. I: The adjunction mapping
full work available at URL: s0012-7094-83-05031-7
OpenAlex ID: W2009710683
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description: scientific article; zbMATH DE number 3851266