Let \(k\) be an infinite field and \(R\) a graded \(k\)-algebra generated by finitely many \(1\)-forms. The Hilbert function \(H(R,d)\) is an important invariant of \(R\) and the projective variety \(\text{Proj } R\). In the present paper, the authors study the Hilbert function of the general hypersurface section of \(\text{Proj } R\). They give a function \(f(d,i,n)\) satisfying the following property:
Let \(h \in R\) be a general \(s\)-form. If \(\text{ch} k = 0\), then \(H(R/hR, d)\leq \sum_{i=0}^{s-1} f(d,i, H(R,d))\) for all \(d \geq s\).
When \(s = 1\), the inequality above is already known [M. Green in: Algebraic curves and projective geometry, Proc. Conf., Trento 1988, Lect. Notes Math. 1389, 76-86 (1989; Zbl 0717.14002)] even if \(\text{ch } k > 0\). The authors pose a conjecture: the unfortunate assumption that \(\text{ch } k = 0\) is superfluous.