Do projective hypersurfaces contain projective toric varieties?

Question

Is there an example of a smooth projective hypersurface in $\mathbb{P}^n_k$ ($k=\overline{k}$) that does not contain any projective toric varieties (edit: of positive dimension)? Or is it the case that every such hypersurface will contain a projective toric variety (edit: of positive dimension)?

Answer 1

As Alexander Woo said in a comment, toric varieties are rational. Now, it turns out that projective hypersurfaces have strong hyperbolicity-type properties. This properties have been established by several authors in the last decades.

First, in 1986 Clemens showed that if $X$ is a generic hypersurface of degree $d \ge 2$ in $\mathbb P^{n+1}$, then $X$ does not admit an irreducible family $f\colon\mathcal C\to X$ of immersed curves of genus $g$ and fixed immersion degree $\deg f$ which cover a variety of codimension less than $D = ((2 -2g)/ \deg f) + d - (n + 2)$. As an immediate consequence, one gets, for example, that there are no rational curves on generic hypersurfaces $X$ of degree $d \ge2n + 1$ in $\mathbb P^{n+1}$.

Two years later, Ein studied the Hilbert scheme of $X \subset G$, a generic complete intersection of type $(m_1,\dots,m_k)$ in the Grassmann variety $G = G(r,n+2)$. As a remarkable corollary one gets that any smooth projective subvariety of $X$ is of general type if $m_1 + m_2 +\cdots+ m_k \ge\dim X + n + 2$. It is also proved that the Hilbert scheme of $X$ is smooth at points corresponding to smooth rational curves of "low" degree.

In 1996, Voisin had the idea of regarding the hypersurfaces in family and to use the positivity property of the tangent bundle of the family itself. Her main result is the following theorem which improves Ein's result in the case of hypersurfaces:

Let $X\subset\mathbb P^{n+1}$ be a hypersurface of degree $d$. If $d\ge 2n-\ell+ 1$, $1 \le\ell\le n - 2$, then any $\ell$-dimensional subvariety $Y$ of $X$ has a desingularization $\widetilde Y$ with an effective canonical bundle. Moreover, if the inequality is strict, then the sections of $K_{\widetilde Y}$ separate generic points of $\widetilde Y$.

The bound is now sharp and, in particular, the theorem implies that generic hypersurfaces in $\mathbb P^{n+1}$ of degree $d\ge 2n$, $n\ge 3$, contain no rational curves. The method also gives an improvement of a result of Xu as well as a simplied proof of Ein's original result.

Lastly, let me cite a result by Pacienza in 2004: this paper gives the sharp bound $d\ge 2n$ for a general projective hypersurface $X$ of degree $d$ in $\mathbb P^{n+1}$ containing only subvarieties of general type, for $n\ge 6$. This result improves the aforesaid results of Voisin and Ein.

Answer 2

It was shown by H. Clemens (1986) that a general hypersurface of degree $d$ in $\mathbb P^n$ does not contain any rational curves if $d$ is sufficiently large, specifically for $d \geq 2n-1$ for $n \geq 3$. Such a hypersurface can never contain a toric subvariety since any such subvariety must contain a rational curve.

Answer 3

If $X$ is an abelian variety then it contains no rational curves. Indeed, $\Omega_X$ is generated by global sections, hence if $f:P^1 \to X$ is a map then the image of the morphism $f^*\Omega_X \to \Omega_{P^1}$ is generated by global sections as well. But $\Omega_{P^1}$ has no global sections at all, so the image is $0$, which means that the map $f$ is constant.