Varieties of low \(\Delta\)-genus
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, whose degree is less than 3(n-r-1)/2.
scientific article; zbMATH DE number 3851266
scientific article