Addenda and Errata: on Nondegeneracy of Curves

(1) Beginning of Section 5: We write that every genus g hyperelliptic curve over a perfect field k is birationally equivalent (over k) to a curve of the form y + q(x)y = p(x) where p(x), q(x) ∈ k[x] satisfy 2 deg q(x) ≤ deg p(x) and deg p(x) ∈ {2g + 1, 2g + 2}. This is false for (and only for) k = F2. Namely, this will fail for any hyperelliptic curve C over k = F2 for which the degree 2 morphism π : C → P splits completely over k, meaning that above each point 0, 1,∞ ∈ P(k) there are two distinct k-rational points of C. For any other perfect field, the statement is true. This is easily deduced from a result of Enge [5, Theorem 7]. In particular, since we assume #k ≥ 17 in this context anyway, this erratum has no effect on any further statement. (2) Section 6 (Curve of genus 4, hyperboloidal case),“Then Q ∼= Pk × Pk and V can be projected”: Should be “Then Q ∼= Pk × Pk”. (3) Proof of Lemma 10.5, “The dual loop P∨ walks through the normal vectors of ∆”: In fact it walks through the direction vectors of the edges of ∆. The same conclusion follows. (4) Poof of Theorem 12.1, “More generally, let k, l ∈ Z≥2 satisfy k ≤ l, let ∆ be the trapezium