# zbMATH — the first resource for mathematics

Remarks on quasi-polarized varieties. (English) Zbl 0699.14002
Let V be an irreducible reduced n-dimensional projective scheme over an algebraically closed field k of any characteristic, and let L be a nef and big line bundle on V. The pair $$(V,L)$$ is called a quasi-polarized variety. Let $$\Delta(V,L)= n+L^ n-h^ 0(V,L)$$ be the $$\Delta$$-genus of $$(V,L)$$. - The author shows that $$\Delta\geq 0$$ for any quasi-polarized variety (V,L) and describes the structure of (V,L) with $$\Delta =0$$ precisely. In particular the sectional genus $$g=g(L)$$ is zero in this case. It is also conjectured the converse:
Conjecture. $$g\geq 0$$ for any quasi-polarized variety. Moreover $$g=0$$ implies $$\Delta =0$$ if V is normal.
Let now $$char(k)=0$$. Then a characterization of $${\mathbb{P}}^ n$$ and hyperquadrics is given. This establishes the above conjecture for $$L^ n\leq 2$$. Then the author shows that the conjecture follows from the flip conjecture and hence it is true when $$n\leq 3$$ by virtue of Mori’s results. - In the case when L is spanned by global sections (by the way, in this case the conjecture is easily verified) the main result is the following nefness ascent theorem which slightly improves a result by Sommese.
Theorem. Let (V,L) be a polarized variety and let A be an irreducible reduced member of $$| L|$$. Suppose that
(1) the double dual $$\omega^ r$$ of the r-th tensor product of the canonical sheaf $$\omega$$ of V is invertible for some $$r>0$$, and $$\omega^ m$$ is invertible in a neighborhood of A for a possible smaller positive integer m,
(2) $$U=V-Y$$ has only log-terminal singularities for some finite set Y, and
(3) $$(\omega +tL)_ A$$ is a nef $${\mathbb{Q}}$$-bundle on A for some rational number t with $$t\geq 2-m^{-1}.$$
Then $$\omega +tL$$ is nef on V unless $$n=\dim (V)=2$$ and (V,L) is a scroll over a curve isomorphic to A.
Reviewer: M.Beltrametti

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14J10 Families, moduli, classification: algebraic theory
Full Text:
##### References:
 [1] Lecture Notes in Math. 439 (1975) [2] J. reine angew. Math. 335 pp 1– (1982) [3] DOI: 10.1007/BF01434964 · Zbl 0306.14006 · doi:10.1007/BF01434964 [4] DOI: 10.1070/IM1986v026n03ABEH001160 · Zbl 0605.14006 · doi:10.1070/IM1986v026n03ABEH001160 [5] J. reine angew. Math. 366 pp 121– (1986) [6] Sijthoff & Noordhoff, Alphen aan den Rijn pp 273– (1980) [7] J. of AMS 1 pp 117– (1988) [8] DOI: 10.2307/2007050 · Zbl 0557.14021 · doi:10.2307/2007050 [9] Algebraic Geometry Sendai 1985 pp 283– (1987) [10] pp 149– (1985) [11] DOI: 10.1007/BF01456407 · Zbl 0476.14007 · doi:10.1007/BF01456407 [12] DOI: 10.2969/jmsj/03810019 · Zbl 0627.14031 · doi:10.2969/jmsj/03810019 [13] DOI: 10.1017/S0305004100064409 · Zbl 0619.14004 · doi:10.1017/S0305004100064409 [14] J. Fac. Sci. Univ. of Tokyo 30 pp 353– (1983) [15] DOI: 10.2307/1970486 · Zbl 0122.38603 · doi:10.2307/1970486 [16] DOI: 10.2969/jmsj/03440709 · Zbl 0489.14001 · doi:10.2969/jmsj/03440709 [17] Proc. Japan Acad. 64 pp 88– (1988) [18] DOI: 10.2748/tmj/1178229197 · Zbl 0489.14002 · doi:10.2748/tmj/1178229197 [19] pp 167– (1987) [20] J. Fac. Sci. Univ. of Tokyo 28 pp 215– (1981) [21] DOI: 10.2969/jmsj/03240709 · Zbl 0474.14017 · doi:10.2969/jmsj/03240709 [22] DOI: 10.2969/jmsj/03210153 · doi:10.2969/jmsj/03210153 [23] J. Fac. Sci. Univ. of Tokyo 22 pp 103– (1975) [24] Complex Analysis and Algebraic Geometry pp 175– (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.