Let $X$ be a variety over some algebraically closed field $k$. In order to define the intersection product of the Chow ring one usually demands $X$ to be smooth. This is for example well explained by Fulton in his book Intersection Theory.
I was wondering whether it is possible to weaken this assumption. I would like it to work on normal varieties but I guess that is too much to hope for. Generalizations that only work on small dimensional varieties might also be helpful for me.
In my specific situation I also have a birational map $f:X \to Y$ where $Y$ is a smooth variety. However, I'm unsure whether this helps at all for defining the product.
Best Answer
One thing that works under pretty general conditions is intersecting curves with $\mathbb Q$-Cartier divisors. This covers for instance the example mentioned by David.
Just in case:
Definition Let $X$ be a normal variety and $D$ a Weil divisor on $X$. Then $D$ is called $\mathbb Q$-Cartier if there exists an $m\in\mathbb N$ such that $mD$ is Cartier. A normal variety $X$ is called $\mathbb Q$-factorial if all Weil divisors are $\mathbb Q$-Cartier.
Now if $X$ is a normal variety, $D$ is a $\mathbb Q$-Cartier divisor on $X$ and $C\subset X$ is a proper curve, then one can define $$ C\cdot D := \frac {C\cdot mD}m $$ for any $m$ such that $mD$ is a Cartier divisor and of course $C\cdot mD$ is defined the usual way: If $L$ is a Cartier divisor and $\gamma:\widetilde C\to C$ is the normalization of $C$, then $$ C\cdot L:= \deg_{\widetilde C} \ \gamma^*\left(\mathscr O_X(L)|_C\right). $$
In David's example a line through the singular point is not Cartier, but twice the line is, because it is linearly equivalent to the hyperplane section. The intersection of a hyperplane section and a line is clearly $1$, so the intersection of two lines through the singular point has to be $\dfrac 12$.
It turns out that $\mathbb Q$-factorial varieties exist in large numbers. For instance the minimal model program (this is currently in char $0$) produces $\mathbb Q$-factorial varieties and there is even a process of $\mathbb Q$-factorialization, which is a partial resolution and on not too bad singularities it is actually a small resolution, so the divisors are not effected.
In the particular case of a normal surface this means that if it is $\mathbb Q$-factorial, then you can intersect any two $1$-cycles, but of course the intersection numbers may be rational numbers and not integers even between integral curves. Although the intersection between a Cartier divisor and an integral curve will still always be an integer.
For a concrete example one can mention that rational surface singularities are always $\mathbb Q$-factorial. The example in David's answer falls in this category. (This is not true in higher dimensions).
As far as trying to use the resolution, the problem is that pulling back cycles does not respect linear equivalence. The only reasonable way to do it is that Cartier divisors can be pulled back as line bundles and then you can extend this numerically to $\mathbb Q$-Cartier divisors. However, then you're back to the same restriction as above.
Alternatively, one can try numerical pull-back as in Sakai's paper mentioned by Francesco. I am not sure how well that works in higher dimensions. I think Batyrev had some work on that and more recently Araujo. I don't remember off the top of my head.
For problems with trying to intersect with non-$\mathbb Q$-Cartier divisors see Tom G's answer.