Given any s points P1,…, Ps in the projective plane and s positive integers m1,…, ms, let Sn be the linear system of plane curves of degree n through Pi with multiplicity at least mi (1 ⩽ i ⩽ s). We give numerical bounds for the regularity of Sn in the following cases (a) the points Pi are non-singular points of an integral curve of degree d; (b) the Pi's are in general position; (c) the Pi's are in uniform position; (d) the Pi's are generic points of the projective plane. We also study the sharpness of such bounds.
Linear systems of plane curves through fixed "fat'' points of P^2
CATALISANO, MARIA VIRGINIA
1991-01-01
Abstract
Given any s points P1,…, Ps in the projective plane and s positive integers m1,…, ms, let Sn be the linear system of plane curves of degree n through Pi with multiplicity at least mi (1 ⩽ i ⩽ s). We give numerical bounds for the regularity of Sn in the following cases (a) the points Pi are non-singular points of an integral curve of degree d; (b) the Pi's are in general position; (c) the Pi's are in uniform position; (d) the Pi's are generic points of the projective plane. We also study the sharpness of such bounds.
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