Pencils on double coverings of curves

Abstract

Let X be a smooth curve of genus g. When π ≥ 3g and d ≥ π−2g+1 we show the existence of a double covering γ : C −→ X where C a smooth curve of genus π with a base-point-free pencil of degree d which is not the pull-back of a pencil on X.

Base point free pencils on multiple coverings of smooth curves

“…Thus X 2 0 = deg(R), X 2 1 = − deg(R) and X 0 ∩ X 1 = ∅. The case k = 2of the following lemma is just [1], Lemma 2.2. Unless otherwise stated X is a smooth and connected projective curve of genus q and S := P(O C ⊕ R), R ∈ Pic(X), e := deg(R) ≤ 0.…”

Linear systems on multiple coverings of a smooth curve

“…Let f : C → X be a double covering of smooth and connected projective curves. Here we propose a modification of [2], Question 3.4, concerning the existence of base point free linear systems on C not arising as the pull-back of a base point free linear system om X. Question 1.…”

“…Let X be a smooth and connected curve of genus g such that there is a degree d base point free linear system on C. Is there a double covering f : C → X such that C is a smooth genus π curve with a degree d base point free linear system not coming from C? Question 1 is just [2], Question 3.4, except that we require the existence of a degree d base point free linear system on X. The following proposition and examples show why this modification is necessary.…”

“…In the next 4 examples we consider a double covering f : C → X between smooth and connected projective curve such that g := p a (X) ≥ 2. For very nice existence theorems of base point free linear systems on C, see [2]. For older related results, see the papers quoted in the introduction of [2].…”

“…For very nice existence theorems of base point free linear systems on C, see [2]. For older related results, see the papers quoted in the introduction of [2].…”