The Vandermonde determinant plays a role in the proof by Kempf and Kleiman-Laksov of the existence portion of the famous Brill-Noether theorem (formulated, but not proved, by Brill and Noether): a general, genus $g$ projective curve has an algebraic line bundle of degree $d$ and $(r+1)$-dimensional space of global sections if and only if the "naive parameter count" for the dimension of such, $\rho(g,r,d) = g-(r+1)(g-r+d)$, is nonnegative. The point is that they set up the enumerative formula to count the number of such line bundles (satisfying some appropriate additional conditions). Miraculously the formula comes out to a Vandermonde determinant which can be explicitly evaluated as being nonzero (as opposed to many similar enumerative problems in algebraic geometry which have no such closed formula).
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