For odd n, there exist at least 2^(n(n-1)) degree 2 Veronese n-folds through any (n+2 choose 2)+n+1 general points in complex projective space.
Interpolation and vector bundles on curves
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Interpolation for degree 2 Veroneses of odd dimension
For odd n, there exist at least 2^(n(n-1)) degree 2 Veronese n-folds through any (n+2 choose 2)+n+1 general points in complex projective space.