The Determination Of The Integral Closure Of A Quartic Extension And Geometric Applications

In number theory, algebraic function theory and algebraic geometry, it is a fundamental problem to compute the integral closure of a finite extension over a commutative ring. The corresponding problem is respectively to find the integral basis, the integral functions and the normalization of algebraic varieties.The computation of the integral closure of cyclic extension is well-known. While cubic extension is the first non-cyclic extension. For algebraic functions with one variable, Baur first gives the factorization of coefficients of a cubic function, and computes the integral functions. S. L. Tan determines the integral closure of a cubic extension over any Noetherian unique factorization domain. In2004, S. L. Tan and D. Q. Zhang extend this computation to any degree n Bring-Jerrard extension, i.e., an extension defined by zn+sz+t=0. In fact, they tried also the computation for general quartic extension by using base change technique.In this article, we first compute explicitly the integral closure of a quartic extension over a Noetherian UFD by using a system of syzygy equations. Then, we give a criterion to determine the non-normal locus of a extension defined by an equation of degree n.We apply our result to study quadruple covers over smooth algebraic va-rieties. We find the trace-free sheaf and the branch locus. We determine the Galois group of the cover by using the factorization of the coefficients of the quartic equation.In1996, by analyzing the ring structure of a quadruple cover; Hahn and Miranda prove that any flat quadruple cover on a smooth algebraic variety is determined by a rank3vector bundle ε and a totally decomposable section η∈H0(∧2S2ε*(?)∧3ε). In2004, M. Bhargava found that quartic rings are parameterized by pairs of ternary quadratic forms (A, B).In this article, by using our factorization of the coefficients of a quartic equation, we construct explicitly the totally decomposable section of a quadruple cover. On the other hand, we prove that the section comes locally from a pair of ternary quadratic forms. We find that the intersection forms of the corresponding plane conics determine the branch locus. This result answers a question of M. Bhargava about quartic rings.