In the book "Introduction to toric varieties" by William Fulton (p. 100) there is a formula for the intersection product of two subvarieties. Let $\tau,\sigma$ be cones in the fan $\Delta$, we have $$ V(\sigma) \cap V(\tau) = \begin{cases}V(\gamma) \text{ if $\sigma,\tau$ span the cone $\gamma$} \\ \emptyset \text{ otherwise}\end{cases} $$

However, knowing the intersection is not enough, we also need the corresponding multiplicity to determine the intersection product. Fulton defines the product as follows: If $\sigma,\tau$ span $\gamma$ with $\dim \gamma = \dim \tau + \dim \sigma$ we have $$ V(\sigma) \cdot V(\tau) = \frac{\text{mult}(\sigma)\cdot \text{mult}(\tau)}{\text{mult}(\gamma)} \cdot V(\gamma), $$ if the span of $\sigma,\tau$ is no cone in $\Delta$ the product is $V(\sigma)\cdot V(\tau) = 0$.

My question is now, how are these multiplicities $\text{mult}(\sigma)$ defined? For simplicial cones we can find the following definition: If $u_1,...,u_d$ are the generators of $\sigma$, $N_\sigma = \text{Span}(\sigma)\cap N$ we define the multiplicity of $\sigma$ as $$ \text{mult}(\sigma) = [N_\sigma: \mathbb{Z}u_1+...+\mathbb{Z}u_d]. $$

But this definition is only given for simplicial cones, the above formula for the intersection product is given for arbitrary cones. How can the multiplicity of arbitrary cones be calculated?

Another approach would be the transformation of arbitrary cones into simplicial ones. I found some proposition that says, by refining $\Delta$, arbitrary fans can be converted into simplicial ones. However, this corresponds to blow-ups and, thus, changes the original variety and the intersection product should change as well, correct?

Any help resolving this dilemma is appreciated!

  • $\begingroup$ I found out that for intersection theory varieties should be smooth (for singular varieties one can work with Euler characteristic, but that's not as intuitive as intersecting divisors directly) and that smooth varieties are closely related to simplicial fans. But how close is this relation? Is this equivalence relation or are there smooth varieties that are not simplicial?? $\endgroup$
    – LegNaiB
    Jul 15, 2021 at 18:19


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