In this largely-expository note, we describe a class of divisors on elliptic curves that index the inflection points of linear series arising (as subspaces of holomorphic sections) from line bundles on $\mathbb{ P}^1$ via pullback along the canonical 2-to-1 projection. Associated to each inflection divisor on an elliptic curve $E_{ \lambda}: y^2= x(x-1)(x-\lambda)$, there is an associated { \it inflectionary curve} in (the projective compactification of) the affine plane in coordinates $x$ and $\lambda$. These inflectionary curves have remarkable features; among other things, they lead directly to an explicit conjecture for the number of { \it real} inflection points of linear series on $E_{ \lambda}$ whenever the Legendre parameter $\lambda$ is real.
Categories: math.AG