Question

(This is a generalization of a question I posted a week ago.)

I'm looking at a variety sitting inside the algebraic torus $(\mathbb{C}\setminus 0)^n$ generated by the ideal $I = (*x_1^{\alpha_1} + \cdots + *x_n^{\alpha_1}, \dots, *x_1^{\alpha_k} + \cdots + *x_n^{\alpha_k})$, where the $*$ are generic complex numbers who may be zero. Each of the $k\leq n$ polynomials in the ideal belongs to $\mathbb{C}[x_1,\dots,x_n]$ and is homogeneous with no cross terms.

I want to compute the Euler characteristic (i.e. the alternating sum of Betti numbers) of this variety. My question is the following:

How, if at all, is the Euler characteristic of the variety generated by $I$ related to the Euler characteristic of the linear variety generated by $I' := (*x_1 + \cdots + *x_n, \dots, *x_1 + \cdots + *x_n)$?

My hope is that these two numbers are either equal or differ only by a factor of the $\alpha_i$.

I'm coming to this problem from a combinatorics standpoint, and so I have a hope that this Euler characteristic is actually just counting the number of regions of a certain hyperplane arrangement.

Thanks for anything!

Answer 1

Such a variety always has Euler number zero. This is because:

1. Due to being cut out by homogenous equations, it is invariant under the $\mathbf{C}^*$ action $t.(x_1,\ldots,x_n) \to (tx_1, \ldots, t x_n)$.

2. This action on $(\mathbf{C} \setminus 0)^n$ has no fixed points.

3. The Euler number of a variety equipped with a $\mathbf{C}^*$ action is equal to the sum of the Euler numbers of the fixed loci.