Interpolation for Curves in Projective Space with Bounded Error. (31st July 2019) - Larson, Eric
Abstract: Given n general points p_1, p_2, …, p_n P^r it is natural to ask whether there is a curve of given degree d and genus g passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if equation*n ≤ (r + 1)d - (r - 3)(g - 1)r - 1floor.equation* The case of curves with nonspecial hyperplane section was recently studied in [2 ], where the above conjecture was shown to hold with exactly three exceptions. In this paper, we prove a "bounded-error analog" for special linear series on general curves; more precisely we show that existence of such a curve subject to the stronger inequality equation*n ≤ (r + 1)d - (r - 3)(g - 1)r - 1floor - 3.equation* Note that the -3 cannot be replaced with -2 without introducing exceptions (as a canonical curve in P^3 can only pass through nine general points, while a naive dimension count predicts twelve). We also use the same technique to prove that the twist of the normal bundle N_C(-1) satisfies interpolation for curves whose degree is sufficiently large relative to their genus, and deduce from this that the number of general points contained in the hyperplane section of a general curve is at least equation*(d, (r - 1)^2 d - (r - 2)^2 g - (2r^2 - 5r + 12)(r - 2)^2).equation* As explained in [7 ], these results play a key role in the author's proof of the maximal rank conjecture [9 ].