On morphisms onto quadrics (Q2473738)

Let \(f:X\to Y\) be a surjective morphism of smooth projective varieties. It is natural to ask under what conditions, on both \(Y\) and \(X\), one can bound the degree of such an \(f\), see [\textit{J.-M. Hwang, N. Mok}, Math. Z. 238, No. 1, 89--100 (2001; Zbl 1076.14021)] and [\textit{C. Schuhmann}, Math. Ann. 306, No. 3, 571--581 (1996; Zbl 0873.14035)]. The aim of this note is to improve on Schuhmann's paper and prove that if \(\text{rk Pic}(X)=1\) and \(Y\) is a quadric then the degree of \(f\) is bounded in terms of discrete invariants of \(X\). The nice proof relies on ``infinitesimal Noether--Lefschetz'' type result, see [\textit{L. Ein, R. Lazarsfeld}, Invent. Math. 111, No. 1, 51--67 (1993; Zbl 0814.14040)] and [\textit{J. Carlson, M. Green, P. Griffiths, J. Harris}, Compos. Math. 50, 109--205 (1983; Zbl 0531.14006)].