Algebraic Geometry Seminar
Friday, September 13, 2019, 3:15pm, Physics 119
Eric Larson (Stanford University, Mathematics)
The Maximal Rank Conjecture
Abstract:- Curves in projective space can be described by either parametric or
Cartesian equations. A natural question is how the parametric and
Cartesian descriptions of a curve relate to each other. We describe
the Maximal Rank Conjecture, formulated originally by Severi in 1915,
which prescribes a relationship between the "shape" of the parametric
and Cartesian equations --- i.e. 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 discuss the "interpolation problem" which asks how many
general points a curve of given type can pass through (for example a
line can pass through two general points but not three). We conclude
by sketching how recent results on the interpolation problem can be
used to prove the maximal rank conjecture.
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