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Question 1. Given a Laurent polynomial $f(z_1,\cdots,z_n)$, such that the corresponding zero locus $Z$ of $f$ in $(\mathbb{C}^*)^n$ is smooth, can we find a smooth toric variety $\bar{X}$ (together with a line bundle $L$ and an extension of $f$) such that:

a) the compactified hypersurface $\bar{Z}$ is smooth.

b) let $D$ be the toric divisors, $\bar{Z} \cap D$ is smooth normal crossings.

More precise question: What conditions are needed to ensure that $\bar{Z} \cap D$ is ample? What is the algorithm for producing these compactifications?

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  • $\begingroup$ Aren't these the subvarieties that Jenia Tevelev calls "sch"on"? $\endgroup$
    – Jason Starr
    Commented Feb 26, 2014 at 10:23
  • $\begingroup$ @JasonStarr Indeed! Thanks for the key word. $\endgroup$
    – user36931
    Commented Feb 26, 2014 at 12:49

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An obvious thing to try is to consider the Newton polytope $\Delta$ of $f$ and take $\rm Proj$ of the corresponding semigroup algebra $$ P=\rm{Proj}\oplus_{k\geq 0}\mathbb C[k\Delta]. $$ Then $Z$ is $\rm{Proj}\oplus_{k\geq 0}\mathbb C[k\Delta]/\langle f\rangle$.

You are not a priori guaranteed that the intersection of $Z$ with the smaller dimensional strata are transversal (although that would be the case if you wiggled the coefficients a bit). The condition for that is smoothness of the hypersurfaces given by a restriction to a face of $\Delta$. See Batyrev's paper "Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori" for this notion of non-degeneracy.

You are definitely not guaranteed that that $P$ is smooth, but perhaps this is what you are trying to look for. Attempts at toric desingularization will destroy ampleness (but you will still have base point free and big properties).

If $\Delta$ is simple (the number of facets through each vertex is equal to the dimension) then $P$ has only abelian quotient singularities. In that case you can consider the corresponding smooth toric stack, and you would still have ampleness.

If you really insist on the best case scenario, i.e. $P$ is a smooth variety, then you want to require the simplicity of $\Delta$ as well as unimodularity, which means that the aforementioned facets through each vertex come from pairing with an \emph{integer} basis of the dual lattice.

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  • $\begingroup$ Thank you for your answer. If I may ask for one clarification: are you giving a condition such that Z above is smooth or such that Z is smooth and the induced compactification is simple normal crossings? $\endgroup$
    – user36931
    Commented Feb 26, 2014 at 12:53
  • $\begingroup$ If $P$ is smooth and $Z$ intersects all strata transversely, then $Z$ is smooth and the simple normal crossing infinity divisor on $P$ induces one on $Z$. Otherwise, it is a bit unclear what you require. For example, if I have a conic in $\mathbb P^2$ which is tangent to a coordinate line, is this good for you or not (the set-theoretic intersection with $D$ is snc, but scheme-theoretic one is not). $\endgroup$
    – Lev Borisov
    Commented Feb 26, 2014 at 13:01
  • $\begingroup$ I think the type of compactification you mention might be fine for my purposes. However in one higher dimension you could imagine the intersection of your hyperplane H with Z is reduced but is a singular curve and that would pose a problem. $\endgroup$
    – user36931
    Commented Feb 26, 2014 at 13:18
  • $\begingroup$ Right, this is why Batyrev puts the condition of smoothness on the restrictions to strata. Take a look at his paper -- it has a nice algebraic reformulation of the criterion in terms of regularity of the sequence of logarithmic partial derivatives. $\endgroup$
    – Lev Borisov
    Commented Feb 26, 2014 at 13:44

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