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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

14C20 Divisors, linear systems, invertible sheaves
14J10 Families, moduli, classification: algebraic theory
Full Text: DOI
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