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A linear series      ( V , L ) (V,\mathcal {L})       on a curve      X X       has an      N N      -fold point along a divisor      D D       of degree      N N       if      dim ⁡  ( V ∩  H 0 ( X , L ( −  D ) ) ) ≥  dim V −  1 \dim (V \cap {H^0}\;(X,\mathcal {L}\,(- D))) \geq \dim \;V - 1      . The dimensions of the families of linear series with an      N N      -fold point are determined for general curves.

Linear series with an 𝑁-fold point on a general curve