Covers of algebraic varieties III. The discriminant of a cover of degree 4 and the trigonal construction

Abstract

Abstract. For each Gorenstein cover : X → Y of degree 4 we define a scheme ∆(X) and a generically finite map ∆( ): ∆(X) → Y of degree 3 called the discriminant of . Using this construction we deal with smooth degree 4 covers : X → P n C with n ≥ 5. Moreover we also generalize the trigonal construction of S. Recillas.

Lifting low-gonal curves for use in Tuitman’s algorithm

“…In other words, the dehomogenized polynomials Q 1 (y 1 , y 2 , 1) and Q 2 (y 1 , y 2 , 1) are supported on the polytopes from Figure 2. 5 Alternatively, the reader can check that res…”

“…The central tool in Bhargava's correspondence is the fundamental resolvent map, which is the bilinear alternating form (1) + α (2) α (3) + α (6) α (4) + α (5) β (1) + β (2) β (3) + β (6) β (4) + β (5) .…”

“…Namely, we will not be computing the fundamental resolvent map with respect to our reduced bases directly, but rather compute this for certain naive bases and apply change of basis. Concretely, consider the naive bases 1, y, y 2 , y 3 , y 4 for ‫ކ‬ q (C) and 1, y, y 2 , y 3 , y 4 , y 5 for ‫ކ‬ q (C res ), along with the slightly altered fundamental resolvent map g : ‫ކ‬ q (C res ) × ‫ކ‬ q (C res ) → ‫ކ‬ q (C) : (α, β) → disc f • 1 1 1 α (1) + α (2) α (3) + α (6) α (4) + α (5) β (1) + β (2) β (3) + β (6) β (4) + β (5) where disc f = det((y i ) ( j) ) 0≤i≤4,1≤ j≤5 . We compute the M (r ) i j ∈ ‫ކ‬ q [x] for which…”

“…Then the Maroni invariants of its cubic resolvent are b 1 + 2, b 2 + 2.Proof. This result is due to Casnati[5, Definition 6.4], although he formulated it in terms of Recillas' trigonal construction, which is the geometric counterpart of Lagrange's cubic resolvent, as pointed out in[19, Section 8.6].…”

Stacks of Ramified Covers Under Diagonalizable Group Schemes

“…In general, however, there is no comparable description of G-covers. Very little is known when G is not abelian, beyond the cases G = S d with d = 3, 4, 5: see [6] for the case G = S 3 and [4,5,9,15] for the non-Galois case.…”