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Given a plane affine curve $\sum_{i,j}a_{i,j}X^iY^j = 0$, its genus can be calculated as the number of integral points of the interior of the convex hull of $\{(i,j) \mid a_{i,j} \neq 0\}$. (claimed here: http://lamington.wordpress.com/2009/09/23/how-to-see-the-genus/)

How can this be proved?

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    $\begingroup$ It is not true without a genericity assumption. $\endgroup$
    – Petya
    Commented Feb 27, 2010 at 15:58
  • $\begingroup$ I added a comment to Bjorn Poonen's answer giving a reference for the general statement about the genus. $\endgroup$
    – François Brunault
    Commented Jul 24, 2019 at 12:26

1 Answer 1

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Here are the references I know concerning this:

H. F. Baker, Examples of applications of Newton's polygon to the theory of singular points of algebraic functions, Trans. Cambridge Phil. Soc. 15 (1893), 403-450.

A. G. Khovanskii, Newton polyhedra and the genus of complete intersections, Funct. Anal. i ego pril. English translation: Functional Anal. Appl., 12 (1978), 38-46.

V. I. Danilov and A. G. Khovanskii, Newton polyhedra and an algorithm for computing Hodge-Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 925-945; English translation: Math. USSR-Izv. 29 (1987), 279-298.

P. Beelen and R. Pellikaan, The Newton polygon of plane curves with many rational points, Designs, Codes and Cryptography 21 (2000), 41-67. (See Theorem 4.2.)

I think the statement should really be, given an irreducible curve in $\mathbf{G}_m^2$, a formula for the arithmetic genus of its closure in the 2-dimensional projective toric variety corresponding to the polygon. This way one should not need to impose genericity hypotheses or restrictions on the characteristic. The references above don't quite do all of this, however, so there is still room for a better reference or proof, I think.

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  • $\begingroup$ I learned of these references from Dan Bernstein, Fernando Rodriguez Villegas, and John Voight. $\endgroup$
    – Bjorn Poonen
    Commented Feb 27, 2010 at 16:20
  • $\begingroup$ Here is a reference for the genus formula you mention at the end: T. Beck, J. Schicho, Parametrization of algebraic curves defined by sparse equations, Appl. Algebra Engrg. Comm. Comput. 18 (2007), no. 1-2, 127-150. $\endgroup$
    – François Brunault
    Commented Jul 24, 2019 at 12:25
  • $\begingroup$ Thank you for this. It seems that Beck and Schicho, at the end of the proof of their Proposition 9, are just quoting the result from page 91 of Fulton, Introduction to toric varieties, so perhaps Fulton's book is the best reference. I haven't checked all the hypotheses, but I think Fulton's proof, which is partly based on work of Danilov, does prove the general result. $\endgroup$
    – Bjorn Poonen
    Commented Jul 25, 2019 at 13:41

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