The Chord Construction

Let F be a smooth plane curve of degree 3. Let β be an element in Pic(F )[2]− {0}. Let us define F ′ := {p(p+ β)|p ∈ F} ⊂ (P2)∗. In this note we show that F ′ is a smooth embedding of F/β. Moreover, let β′ be the generator of Pic(F )/β, and let p ∈ F be a flex, then p(p+ β) + β′ is a flex on F ′. We present two proofs. 1. Notation. Let F be a smooth curve of genus 1, β ∈ Pic(F )[2]. Let us denote F ′ := F/β. Let π be the quotient map F → F ′ and let β be the image of the non-zero element in Pic(F )[2]/β in Pic(F ). Let H be an element in Pic(F ), and i : F → P2 the map associated to H. This notation will hold in the next definition and couple of lemmas. 2. Definition (The chord construction (c.c.)). Let us define i : F/β → P∗ {p, p + β} 7→ i(p)i(p + β) 3. Theorem. The c.c is an embedding. Proof. Let us identify F with its image i(F ). The c.c. is 1-1: Let H be the line corresponding to a point {p, β+p} ∈ F/β. Then F ∩H consists of three points: {p, p + β,H − (2p+ β)}. If (H − (2p + β)) − β 6= p, p + β, then i(H) consist of the unique point {p, p+β}, so it is 1-1. If (H − (2p+β))− β = p then H − (2p+β) = p+ β, so i is 1-1 here too. The remaining case is similar. (in the last two cases p (respectively p+ β) is a flex). For p ∈ P2, let p be the line it defines in P2∗ If F does not have a CM then for a generic p ∈ F , p is transversal to F : Suppose it is not. By upper-semicontinuity, it is never transversal, so F is the dual curve of i(F ). Since i(F ) is not a line (it is an immersion of F ), this induces a degree 1 map i(F ) → F , and therefor a degree 1 map F ′ → F . This means that F has a CM. Date: February 1, 2008. 1991 Mathematics Subject Classification. 14N15,14H52.