Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I take to be algebraically closed).

It is classical that classifying normal affine toric varieties is the same thing as classifying rational polyhedral cones. The latter is a classical part of discrete and combinatorial geometry. (Apologies for overusing the word "classical".)

Question. Is there some combinatorial data which classifies non-normal affine toric varieties?

Background and what I am not looking for. I am not an algebraic geometer. I work a lot with non-commutative semigroups. So often people asked me what is the classification of finitely generated cancellative commutative monoids, expecting an answer like the group case. The category of affine monoids (=finitely generated submonoids of free abelian groups) is dual to the category of affine toric varieties, hence my question. The normal affine toric varieties correspond to saturated or normal affine monoids (also called integral polyhedral cones by some). For this reason I don't consider affine monoids as an answer to the question.

  • $\begingroup$ Take a (rational polyhedral) cone $\sigma$ in $\mathbb{R}^n$ and remove a finite number of non-invertible elements of $P:=\sigma\cap \mathbb{Z}^n$. If the resulting $P'$ is a monoid (and if there are no invertible elements, you can make $P'$ a monoid by removing finitely many more elements), it gives you a non-normal affine toric variety, and all not necessarily normal affine toric varieties arise this way. I guess that pairs (cone, finite subset of integral points) is not what you're looking for. What answer would you like to have in case $P\subseteq \mathbb{N}$? $\endgroup$ Mar 25, 2014 at 16:28
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    $\begingroup$ (cont.) Note that any finite set of positive integers generates such a $P\subseteq \mathbb{N}$, but it's quite difficult to see whether two such sets generate the same monoid (see "Frobenius problem"). $\endgroup$ Mar 25, 2014 at 16:31
  • $\begingroup$ There's even a game exploiting this difficulty: en.wikipedia.org/wiki/Sylver_coinage $\endgroup$ Mar 25, 2014 at 16:41
  • $\begingroup$ The answer maybe there is no answer. I was hoping for an answer not using monoids. $\endgroup$ Mar 25, 2014 at 21:34

2 Answers 2


Bernd Sturmfels and others have studied varieties defined by binomial ideals. This is what you are looking for.

  • $\begingroup$ Is this very different than saying it reduces to looking at monoids or do they have combinatorial descriptions? $\endgroup$ Apr 14, 2016 at 12:27
  • $\begingroup$ I am afraid not. $\endgroup$ Apr 14, 2016 at 18:43

There is paper of Patrick Popescu-Pampu and Dmitry Stepanov called Local tropicalization, where they propose a different notion of a fan that pretends to encode nonnormal toric varieties, namely a fan of semigroups, defined as a lattice fan $\Delta$ endowed with an affine subsemigroup (affine semigroups are pointed, commutative cancellative, finite type and torsion free) $\Gamma_{\alpha}$ of $\Gamma$ for each cone $\sigma\in\Delta$ with:

i) For each cone $\sigma\in \Delta$, $\sigma(\Gamma_{\alpha})=\alpha$ and $M(\Gamma_{\alpha})=M$.

ii) $\beta$ is a face $\alpha\in \Delta$, then $\Gamma_{\beta}=\Gamma_{\alpha}+M(\beta,\Gamma_{\alpha})$.

I think this can help

  • $\begingroup$ Thanks. I'll take a look. $\endgroup$ Apr 9, 2021 at 18:02

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