Normality of secant varieties
-- Brooke Ullery, February 24, 2017

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If X is a smooth variety embedded in projective space, we can form a
new variety by looking at the closure of the union of all the lines
through 2 points on X. This is called the secant variety of
X. Similarly, the Hilbert scheme of 2 points on X parametrizes all
length 2 zero-dimensional subschemes. I will talk about how these two
constructions are related. More specifically, I will show how we can
use certain tautological vector bundles on the Hilbert scheme to help
us understand the geometry of the secant variety, leading to a proof
that for sufficiently positive embeddings of X, the secant variety is
a normal variety.