Integral closure of a cubic extension and applications (Q2716097)

The author computes explicitly the integral closure of a ring of the form \(A[x]/(x^3+ax+b)\), where \(A\) is a Noetherian unique factorization domain and \(p(x)=x^3+ax+b\) is an irreducible polynomial over \(A\). The main motivation and application of this result comes from the theory of triple covers, established by \textit{R. Miranda} [Am. J. Math. 107, 1123-1158 (1985; Zbl 0611.14011)], where it is shown that a finite flat map of degree 3, \(f: X\to Y\), is determined by a rank 2 vector bundle \(E\) and by a map \(S^3E\to \wedge^2 E\) [cf. also \textit{R. Pardini}, Ark. Mat 27, 319-341 (1989; Zbl 0707.14010) for the positive characteristic case]. Indeed, if \(f: X\to Y\) is a finite degree 3 map of normal varieties over an algebraically closed field of characteristic \(\neq 3\) with \(Y\) factorial, then \(f\) factorizes as \(X\to Z\to Y\) where \(X\to Z\) is a birational morphism and there exists a line bundle \({\mathcal L}\) on \(Y\) such that \(Z\subset V({\mathcal L})\) is defined by \(x^3+ax+b=0\), \(x\) being the tautological section and \(a\) and \(b\) sections of \({\mathcal L}^2\), \({\mathcal L}^3\), respectively. The map \(Z\to Y\) is of course the restriction of the bundle projection \(V({\mathcal L})\to Y\). The computation of the integral closure globalizes, so that it is possible to compute \(f_*{\mathcal O}_X\) and other invariants of \(f: X\to Y\) in terms of the triple \((a,b,{\mathcal L})\). In particular, when \(Y\) is a surface the map \(f\) is automatically flat and one recovers Miranda's description of \(f\).