Extremal trigonal fibrations on rational surfaces

Abstract

For a relatively minimal fibration 𝑓 : 𝑋 → ℙ¹ of non-hyperelliptic curves of genus 𝑔, we know the Picard number 𝜌(𝑋) ≤ 3𝑔 + 8. We study the case where 𝜌(𝑋) = 3𝑔 + 8 and the Mordell–Weil group of 𝑓 is trivial. Such an 𝑓 occurs only if 𝑔 ≡ 0 or 1 (mod 3), and we describe such 𝑓 : 𝑋 → ℙ¹ explicitly.