1.5. Hypersurfaces in toric varieties [04RU]

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1.5. Hypersurfaces in toric varieties

Let $f:(\mathbb{C}^{*})^{n+1}\to\mathbb{C}$ be a Laurent polynomial

f(z)=\sum\limits_{j}a_{j}z^{j},

where $z\in(\mathbb{C}^{*})^{n+1}$ and $j\in\mathbb{Z}^{n+1}$ is a multi-index.

We recall that the Newton polyhedron $\Delta$ of $f$ is the convex hull in $\mathbb{R}^{n+1}$ of the set of all indices $j\in\mathbb{Z}^{n+1}$ such that $a_{j}\neq 0$ . Since by assumption $f$ is a polynomial this set is finite and $\Delta$ is a bounded convex lattice polyhedron. We also call $\Delta$ the Newton polyhedron of the hypersurface ${V}^{\circ}=\{z\in(\mathbb{C}^{*})^{n+1}\ |\ f(z)=0\}$ . According to [4] we call the image $\operatorname{Log}({V}^{\circ})\subset\mathbb{R}^{n+1}$ the amoeba of ${V}^{\circ}$ .

For the rest of the paper we assume that $\Delta$ has a non-empty interior in $\mathbb{R}^{n+1}$ . Otherwise after a suitable (multiplicative) change of coordinates the polynomial $f$ can be transformed to a polynomial in smaller number of variables.

Let $\mathbb{C}T_{\Delta}$ be the complex toric variety (see e.g. [4]) associated to $\Delta$ . We define $V$ as the closure of the hypersurface ${V}^{\circ}=\{z\in(\mathbb{C}^{*})^{n+1}\ |\ f(z)=0\}$ in $\mathbb{C}T_{\Delta}$ . Taking the Newton polyhedron for $\Delta$ is a canonical choice. Of course, we can take such compactification for any convex lattice $(n+1)$ -polyhedron $\Delta$ , even if it was not the Newton polyhedron of ${V}^{\circ}$ . However the choice of the Newton polyhedron of ${V}^{\circ}$ as $\Delta$ produces the best results as the next proposition shows. Recall that in the toric construction there is a $k$ -dimensional complex toric subvariety $\mathbb{C}T_{\Delta^{\prime}}$ associated to any $k$ -dimensional face $\Delta^{\prime}\subset\Delta$ .

Proposition 1.18.

The hypersurface $V$ is disjoint from the points (i.e. the 0-dimensional toric varieties) corresponding to the vertices of $\Delta$ , but intersects all the tori corresponding to any positive-dimensional face of $\Delta$ .

Furthermore, this property characterizes $\mathbb{C}T_{\Delta}$ in the following sense. Let $\bar{\Delta}$ be a convex lattice polyhedron in $\mathbb{R}^{n+1}$ with a non-empty interior and $\bar{V}$ be the closure of ${V}^{\circ}$ in $\mathbb{C}T_{\bar{\Delta}}\supset(\mathbb{C}^{*})^{n+1}$ . If a hypersurface $\bar{V}$ is disjoint from the points corresponding to the vertices of $\bar{\Delta}$ but intersects all the tori corresponding to positive-dimensional faces of $\bar{\Delta}$ then $\mathbb{C}T_{\bar{\Delta}}=\mathbb{C}T_{\Delta}$ .

Remark 1.19.

Note that even though $\mathbb{C}T_{\Delta}$ is unique by this proposition, the polyhedron $\bar{\Delta}$ itself is not unique. The image of $\Delta$ by a homothety with an integer coefficient for $\bar{\Delta}$ corresponds to the same toric variety.

Proof.

Proposition 1.18 follows from the following Lemma. ∎

Lemma 1.20.

Let $\Delta^{\prime}\subset\Delta$ be a face. The intersection $V\cap\mathbb{C}T_{\Delta^{\prime}}$ coincides with the hypersurface cut on $\mathbb{C}T_{\Delta^{\prime}}$ by the closure of the zero set of the following $\Delta^{\prime}$ -truncation of the polynomial $f$

f_{\Delta^{\prime}}(z)=\sum\limits_{j\in\Delta^{\prime}}a_{j}z^{j}.

Proof.

To prove the lemma it suffices to note that the monomials from $\mathbb{Z}^{n+1}\cap\Delta^{\prime}$ have higher order of vanishing when $z\to\mathbb{C}\Delta^{\prime}$ . ∎

Remark 1.21.

The property of $V$ from Proposition 1.18 can be alternatively reformulated in terms of the moment map $\bar{\mu}_{\Delta}:\mathbb{C}T_{\Delta}\to\Delta$ , see 1.3. The image $\mu(V)$ is disjoint from the vertices of $\Delta$ but intersects every positive-dimensional face of $\Delta$ . According to [4] the image $\mu(V)$ is called the compactified amoeba of ${V}^{\circ}$ . This restatement is equivalent to the property from Proposition 1.18, since for any face $\Delta^{\prime}\subset\Delta$ we have $\mu(\mathbb{C}T_{\Delta^{\prime}})=\Delta^{\prime}$ .

Example 4.

Let $f(z,w)=zw+z+w-1$ . Then ${V}^{\circ}\subset(\mathbb{C}^{*})^{2}$ is a hyperbola. The Newton polygon $\Delta$ is a square $\{(x,y)\in\mathbb{R}^{2}\ |\ 0\leq x\leq 1,0\leq y\leq 1\}$ and the corresponding toric surface $\mathbb{C}T_{\Delta}$ is the hyperboloid ${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$ .

Take now $\bar{\Delta}=\{(x,y)\in\mathbb{R}^{2}\ |\ 0\leq x,0\leq y,x+y\leq 1\}$ . The corresponding toric surface is ${\mathbb{C}}{\mathbb{P}}^{2}\supset(\mathbb{C}^{*})^{2}$ . The images of ${V}^{\circ}$ under the associated moment maps are sketched on Figure 4.

Refer to caption — Figure 4. Images of the hyperbola $zw+z+w-1=0$ under the moment maps corresponding to its Newton polygon and another polygon.

The following example treats projective hypersurfaces.

Example 5.

Let $V\subset{\mathbb{C}}{\mathbb{P}}^{n+1}\supset(\mathbb{C}^{*})^{n+1}$ be a projective hypersurface of degree $d$ not passing through the points $[1:0:\dots:0],\dots,[0:\dots:0:1]$ . Then ${V}^{\circ}=V\cap(\mathbb{C}^{*})^{n+1}$ is given by a polynomial $f$ whose Newton polyhedron is

\Delta_{d}=\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\ |\ 0\leq x_{j},\sum\limits_{j}x_{j}\leq d\}.

Vice versa, $\mathbb{C}T_{\Delta}={\mathbb{C}}{\mathbb{P}}^{n+1}$ and the closure of ${V}^{\circ}$ in ${\mathbb{C}}{\mathbb{P}}^{n+1}$ is $V$ .