The Maximal Rank Conjecture
-- Eric Larson, November 30, 2018

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Curves in projective space can be described in either
parametric or Cartesian equations. We begin by describing the Maximal
Rank Conjecture, formulated originally by Severi in 1915, which
prescribes a relationship between the "shape" of the parametric and
Cartesian equations --- that is, which gives the Hilbert function of a
general curve of genus g, embedded in P^r via a general linear series
of degree d. We then explain how recent results on the interpolation
problem can be used to prove this conjecture.