Sectional invariants of hyperquadric fibrations over a smooth projective curve
The paper under review continues a series of papers devoted to the study of \textit{sectional invariants} associated to a polarized manifold \((X,L)\) [see \textit{Y. Fukuma}, Rend. Semin. Mat. Univ. Padova 121, 93--119 (2009; Zbl 1173.14040), Adv. Geom. 8, No. 4, 591--614 (2008; Zbl 1156.14007) and J. Pure Appl. Algebra 209, No. 1, 99--117 (2007; Zbl 1106.14003)]. The idea for these sectional invariants comes from the case in which \(L\) is globally generated. In fact, as a consequence of Bertini's Theorem one can construct a chain of smooth subvarieties of \(X\), say \(X_{n-1} \subset \cdots\subset X_1 \subset X_0:=X\) such that dim\((X_i)=n-i\) and \(X_i \in |L|_{X_{i-1}}|\). Hence one can associate to \((X,L)\) a collection of sectional invariants defined on each section \(X_i\). For example the \(i\)-sectional Euler number \(e_i(X,L)=e(X_{n-i})\) and the \(i\)-sectional Betti number \(b_i(X,L)=b_i(X_{n-i})\) appear. These notions extend properly to the case in which \(L\) is not necessarily globally generated. In this paper the authors calculate the \(i\)-sectional Euler numbers \(e_i(X,L)\) and lower bounds for the \(i\)-sectional Betti numbers \(b_i(X,L)\) (\(i \leq 4\)) in the case of hyperquadric fibrations over a smooth curve \(C\) (see Def. 2.3 for a proper definition). The formulae for \(e_i(X,L)\) are given in Theorem 3.1 and involve the genus of \(C\), the degree of \(f_*L\) (\(f:X \to C\) giving the structure of fibration) and the degree of \(B\), encoding \(B \in \)Pic\((C)\) information of the natural embedding of \(X\) in the projective bundle \(P_C(f_*L)\). The lower bounds for \(b_i(X,L)\) (\(\geq 3\) for \(i=2\) and \(n \geq 3\), \(\geq 0\) for \(i=3\) and \(n \geq 4\), \(\geq 3\) for \(i=4\) and \(n \geq 5\)) are provided in Theorem 4.1 where a classification of the extremal cases is also given.
scientific article; zbMATH DE number 5776098
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