# Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide

There are two definitions of intersection multiplicity of two complex algebraic plane curves. One of them is given in An introduction to algebraic curves by Griffiths. Let $$t \mapsto (t^k, y(t) )$$ be the local normalization of $$V_f$$ then $$(V_g \cdot V_f)_P = ord_P(g(t^k, y(t)))$$. The other one is given in Algebraic Geometry by Hartshorne. $$(V_g \cdot V_f)_P = \mathrm{length}(O_p/(f,g))$$ as $$O_P$$ module.

Do these definitions coincide?

There may be some approaches. Showing that completion preserves length may help cause you can thereby catch the normalization in complex analysis to algebraic one. Showing that $$\mathrm{length}(O_P/(g, f_1 f_2))= \mathrm{length}(O_P/(g, f_1))+\mathrm{length}(O_P/(g, f_2))$$ is also important.

• If your algebraic surface is smooth, a good (conceptual) way to compare the two is by comparing both to the cup product in cohomology. Jun 1 '20 at 3:48
• A proof (for the $n$-dimensional version) is given in Chapter 4 of arxiv.org/abs/1806.05346. Sorry for the self-promotion, but one reason for including this element in the (draft of the) book was that I could not locate a direct proof in the literature. Jun 1 '20 at 20:19