Inspired by A. Landesman's undergraduate thesis, we prove that del Pezzo surfaces satisfy interpolation with respect to incidence conditions with linear spaces. The case of 3-Veronese surfaces merits mentioning, because it involves an amusing use of A. Coble's theory of association, now called Gale duality. In particular, we show that through thirteen general points in P^9, there exist a finite, positive, number of 3-Veronese surfaces.

Quite generally, every Veronese variety (i.e. embeddings of projective spaces by complete linear series) comes equipped with a similar enumerative problem, asking how many interpolate through the appropriate number of points. The case of Veronese curves, i.e. rational normal curves is ultra-classical: through n+3 general points in P^n there exists a unique rational normal curve. In approximately 1920, A. Coble proved that through nine general points in P^5 there are precisely four 2-Veronese surfaces. No other Veronese counts are currently known. The curious geometry described in our paper offers a very promising pathway for the case of 3-Veronese surfaces.