Eight-secant conics for space curves (Q1204305)

Let \(C\) be a generic complete intersection of two surfaces of fixed degrees at least 15 or a generic rational curve of fixed degree at least 15 in \(\mathbb{P}^ 3\) over an algebraically closed field of characteristic zero. We prove that for such curves the classical formula by Severi for the number of eight-secant conics as a polynomial in the degree and genus of \(C\) has a straightforward interpretation in the sense that it gives the number of irreducible conics that intersect \(C\) exactly 8 times. None of these conics is tangent to \(C\), and no conic intersects \(C\) in 9 points. In addition there is no other conic \(Q\), irreducible or reducible (line pair), reduced or non-reduced (double line), such that there is a finite scheme \(S\), of length 8, simultaneously contained in \(Q\) and in \(C\).