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.

  • 1
    $\begingroup$ If your algebraic surface is smooth, a good (conceptual) way to compare the two is by comparing both to the cup product in cohomology. $\endgroup$ – R. van Dobben de Bruyn Jun 1 '20 at 3:48
  • $\begingroup$ 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. $\endgroup$ – auniket Jun 1 '20 at 20:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.