Models of triple covers (Q2334918)

This short note studies the problem that whether a generical finite morphism between projective varieties has a ``good'' biratioanl model, in the sense that the birational model is a finite cover with smooth branching divisor. It is well-known that such a good model exists for generically double covers over fields of characteristic 0. As the main result of this note, such a good model exists for generically triple covers over perfect fields of characteristic not \(2\) or \(3\) under a natural assumption. To be more precise, if \(T\to S\) is a generically finite morphism of degree 3 between projective varieties, over a perfect field \(k\) with \(\text{char}\,k\neq 2, 3\), suppose that we have an embedded resolution of singularities of divisors on \(S\) by iterated blowing ups with smooth centers, then a good model exists.