Linear Series on 4 - Gonal Curves

Scroll codes over curves of higher genus: reducible and superstable vector bundles

“…The sequence γ t (C) is known for all hyperelliptic, trigonal and bielliptic curves (at least in characteristic = 2). For a general k-gonal curve a lot is known [2,3]. But the proofs in the papers [2] and [3] use some dimensional counts and deformation theory.…”

“…For a general k-gonal curve a lot is known [2,3]. But the proofs in the papers [2] and [3] use some dimensional counts and deformation theory. Hence we were unable to use them to study a fixed prescribed curve defined over F q .…”

On double coverings of hyperelliptic curves

“…[6,Remark 2.6]. In fact, the linear series |K − π * g 1 2 − p 1 − · · · − p g−6 | = g 2 g for generically chosen p 1 , .…”

The minimal degree of plane models of double covers of smooth curves

“…A general projection π : P r − P r−3 → P 3 maps φ(X) biregularly to a non-degenerate smooth curve X of degree d + 1. Then a general projection π : P 3 − {P} → P 2 with center P ∈ X maps X birationally onto a plane curve of degree d. Therefore X has a plane model of degree d for any d ≥ g x + 2, which shows that there is no maximum among degrees of plane models of X .…”

“…By the above discussion, s X (2) ≤ g x + 2. It has been classically known that s X (2) = [ 2(g x +4) 3 ] for a general curve X ; cf. Severi [1].…”