Classification Of Points In Special Position With Respect To Low Degree Curves On Projective Plane

It is a significant problem in algebraic geometry to classify the sets of points in special position with respect to curves of given degree d in projective plan. We know that the set ofequations of plane curves of degree d is a vector space of dimension m(d)=(d+1)(d+2)/2For a given set of N points in the plane, it gives N linear conditions on the vector space if we require the curves pass through these points. If the N conditions are linearly dependent, then we call that the set of N points is in a special position with respect to curves of degree d. When the number of pints is bigger than m(d), then the conditions are always linearly dependent. When the number N is exactly m(d), then these points are in special points if and only if they lie in a curve of degree d. Therefore, we always rule out these trivial cases, namely, we assume that N < m(d).It is well-known that N < 6 points are in special position with respect to conics if and only if 4 points are collinear. N < 10 points are in special position with respect to cubics if and only if either 9 points lie in two different cubics, or 7 points lie in a conic, are 5 points are collinear.Sheng-Li Tan [4,5] gives an effective method to deal with this problem base on Bogomolov's inequality [3, 10] for semistable rank two vector bundles on an algebraic surface, namely a method to classify those special sets with N≤(d+3)~2/4 points. In particular, he give a complete list of the sets of points in a special position with respect to curves of degree 4≤d≤8.We try to give a different method to deal with the problem for the case when (d+3)~2/4N < m(d). The main purpose is to classify the sets of points in a special position with respect to curves of degree d = 4,5 or 6. (Theorems A, B, C). Our method is based on Noether's Fundamental Theorem and Cayley-Bacharach Theorem.