Let U be a unirational variety over a field of characteristic 0. I have read that its canonical divisor cannot be ample, but I don't know why.


Any references will be most helpful.



Unirational and ampleness
**Smooth varieties** 


Let $X$ be a smooth uniruled variety. Then there is a free rational curve $f:\mathbb{P}^1\rightarrow X$, that is $H^1(\mathbb{P}^1,f^\*T\_X\otimes\mathcal{O}\_{\mathbb{P}^1}(-1)) = 0$. Now $f^{\*}T\_X$ is a rank $n = dim(X)$ vector bundle on $\mathbb{P}^1$. Therefore we can write
$$f^\*T\_X = \mathcal{O}\_{\mathbb{P}^1}(a\_1)\oplus...\oplus\mathcal{O}\_{\mathbb{P}^1}(a\_n).$$
Now, $f^{\*}T\_X$ contains $T\_{\mathbb{P}^1}\cong\mathcal{O}\_{\mathbb{P}^1}(2)$. We have $K\_X\cdot f\_\*\mathbb{P}^1 = -\sum\_{i=1}^{n}a\_i\leq -2$.
Therefore, $X$ uniruled implies that there is a free rational curve on $X$ which implies that $K\_X$ is not nef. So, if $K\_X$ is ample (in particular $K\_X$ is nef) then $X$ is not uniruled. 


Now, $X$ unirational $\Rightarrow$ $X$ rationally connected $\Rightarrow$ $X$ uniruled $\Rightarrow$ $K\_X$ is not nef (in particular $K\_X$ is not ample).


In other terms. If $X$ is uniruled then $X$ has nagative Kodaira dimension. If the canonical bundle of $X$ is ample (it is enough big) then $X$ has Kodaira dimension $dim(X)$. 


**Singular varieties**


There are singular uniruled varieties with ample canonical bundle. In particular there are many examples of rational singular surfaces with ample canonical bundle. The following example is due to Kollar. You can find the details and many other interesting constructions here <http://arxiv.org/abs/1007.1936>


Let us consider the surface
$$Y = Y(a\_1, a\_2, a\_3, a\_4) := (x\_1^{a\_1}x\_2 + x\_2^{a\_2}x\_3 + x\_3^{a\_3}x\_4 + x\_4^{a\_4}x\_1 = 0)$$
in the weighted projective space $\mathbb{P}(w\_1, w\_2, w\_3, w\_4)$, such that
$$a\_1w\_1+w\_2 = a\_2w\_2 + w\_3 = a\_3w\_3 + w\_4 = a\_4w\_4+w\_1 = d$$
and
$$\begin{array}{ll} w\_1 = \frac{1}{w^\*}(a\_2 a\_3 a\_4-a\_3 a\_4+a\_4-1),
& w\_2 = \frac{1}{w^\*}(a\_1 a\_3 a\_4-a\_1 a\_4+a\_1-1),\\
 w\_3 =
\frac{1}{w^\*}(a\_1 a\_2 a\_4-a\_1 a\_2 + a\_2-1), & w\_4 =
\frac{1}{w^\*}(a\_1 a\_2 a\_3- a\_2 a\_3 + a\_3-1),
\end{array}$$
$$d = \frac{1}{w^\*}(a\_1 a\_2 a\_3 a\_4-1)$$ where $w^\* = \gcd(w\_1, w\_2,
w\_3, w\_4).$ If $w^\* = 1$, then $Y$
is a rational surface with $4$ cyclic singularities and $H\_2(Y, \mathbb{Q}) \cong
\mathbb{Q}^3$. The rational curves
$$C\_1 := (x\_1 = x\_3 = 0),\quad C\_2 := (x\_2 = x\_4 = 0)$$ are extremal rays
with respect the $K\_Y + (1- \epsilon)(C\_1 + C\_2)$ minimal model program for
$0 < \epsilon \ll 1$. Therefore $C\_1$ and $C\_2$ are contractible
to quotient singularities and we get a rational surface of Picard
number 1,
$$\pi : Y=Y(a\_1,a\_2,a\_3,a\_4) \rightarrow X = X(a\_1, a\_2, a\_3, a\_4).$$
If $a\_1, a\_2, a\_3, a\_4 \geq 4$, then $K\_{X}$ is ample.


For examples of big surfaces with big cotangent bundle you could take a look at this <http://www.latp.univ-mrs.fr/~eroussea/big-surfaces.pdf>


For a classification of complex compact Gorenstein surfaces with negative canonical bundle you can read this <http://link.springer.com/article/10.1007%2FBF01421952#page-2>. A classical example of a rational surface with negative canonical bundle is the quadric cone in $\mathbb{P}^3$. The canonical divisor $K\_S$ is $-2H\_{|S}$, where $H$ is a plane in $\mathbb{P}^3$ through the vertex.


