Properties of extreme rays

Question

Let $X$ be a projective variety over $\mathbb{C}$, then the effective curves module the numerical equivalence form a cone. For my understanding, if the extreme ray $[C]$ has the property that $[C]\cdot K_X < 0$, then one can contract this extreme ray to get a morphism $f: X \to Y$.

I want to know if there is any restriction for the exceptional divisor $Exc(f)$ of this morphism. For example, could $Exc(f)$ be a Calabi-Yau variety (i.e. the canonical divisor is trivial)?

My knowledge about birational geometry is very limited, the "counter-example" I can think of is a a cone (this is $Y$ in the question) over a Calabi-Yau variety $D$ and $X$ is a blowup of $Y$ at the vertex. I think the exceptional divisor of $X \to Y$ is $D$, but I don't know if this is the contraction of extreme ray.

Any suggestions/references are very welcome!

Answer 1

Let $X$ be a smooth projective variety, and let $L(R)$ be the locus of a $K_X$-nagative extremal ray $R$. Any irreducible component $Z$ of $L(R)$ is uniruled. If the contraction associated to $R$ is divisorial then $L(R)$ is irreducible and uniruled. You can find this, for instance, in Proposition $6.10$ of Debarre's book "Higher dimensional algebraic geometry". In particular the exceptional divisor has negative Kodaira dimension. Hence, it can not be Calabi-Yau.

If the locus of $R$ is $X$ then $X$ itself is uniruled and the contraction corresponding to $R$ gives a Mori fiber space i.e. the general fiber is Fano.

If the locus of $R$ is a divisor the image $Y$ of the contraction may be singular but some multiple of $K_Y$ is Cartier. In this case $Y$ has at most terminal singularites. This is why your example does not work. The vertex of a cone over a Calabi-Yau is not terminal. For instance think about the cone over an elliptic curve $E$. Even the vertex of a quadric cone surface is not terminal (for surfaces terminal = smooth ). On the other hand the vertex of a quadric cone is a canonical singularity. For surfaces the canonical singularities are precisely finite rational quotient singularities. In your example the vertex is not even a rational singularity becuase $dim(H^{1}(E,\mathcal{O}_{E})) = 1$ and $R^1f_{*}\mathcal{O}_{X}\neq 0$.

For the singularities of the minimal model program you can look either at Kollar's book "Singularities of the Minimal Model Program" or at these notes http://www-fourier.ujf-grenoble.fr/sites/ifmaquette.ujf-grenoble.fr/files/mella.pdf.

If the contracted locus is in codimension greater or equal that two the image $Y$ is not locally factorial. The canonical divisor $K_Y$ is not $\mathbb{Q}$-Cartier. Therefore to continue the MMP one must perform a flip.