The input to the NP-hard POINT LINE COVER problem (PLC) consists of a set P of n points on the plane and a positive integer k, and the question is whether there exists a set of at most k lines which pass through all points in P. By straightforward reduction rules one can efficiently reduce any input to one with at most k2 points. We show that this easy reduction is already essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, for any ε> 0, there is no polynomial-time algorithm that reduces every instance (P, k) of PLC to an equivalent instance with O(k2−ε) points. This answers, in the negative, an open problem posed by Lokshtanov (PhD Thesis, 2009). Our proof uses the notion of a kernel from parameterized complexity, and the machinery for deriving lower bounds on the size of kernels developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients: We first show, by reduction from VERTEX COVER, that— unless the polynomial hierarchy collapses—PLC has no kernel of total size O(k2−ε) bits. This does not directly imply the claimed lower bound on the number of points, since the best known polynomial-time encoding of a PLC instance with n points requires ω(n2) bits. To get