ac.commutative algebra - Isolated hypersurface singularities, Chow groups and D-branes

Assume $k= mathbf C$ and $W$ homogeneous. Let $X=Proj (k[x_1,cdots,x_n]/(W))$. $X$ is then a smooth hypersuraface in $mathbb P_{n-1}$.

Assume $n=2d$ is even. Corollary 3.10 of the paper you quoted says that $theta=0$ for all pairs iff the homological Chow group $CH^{d-1}_{hom}$ modulo $[h]^{d-1}$ is not torsion (here $[h]$ is the class of the hyperplane section). So your question, in this case, is equivalent to
($l=d-1$):

Examples of smooth hypersurfaces of dimension $2l$ such that $CH^{l}_{hom}/([h]^{l})$ is not torsion ?

(By the way, I think if you phrased your question this way, it probably would become more popular, consider how many geometry-inclined people visit this site! So if you want more and better answers, consider changing the title.)

Now, a cheap way to get examples you want is to take $W= x_1x_{d+1} + cdots + x_dx_{2d}$. Then the cycle defined by $(x_1,...,x_d)$ will not be a multiple of a power of the hyperplane section. Why? Because, I am waving my hand a bit here, if it is then the intersection with the cycle $(x_{d+1},cdots, x_{2d})$ would be positive. But they are disjointed in $X$!

The same trick works for generalized quadrics, i.e. if $W = f_1g_1 +cdots +f_dg_d$.

EDIT: Let me give more details here. In this situation you can easily make $W$ non-homogeneous as you desire. But the trouble is you can't use my argument above as there is no longer a projective variety $X$. But one can get around this. Let $S=k[x_1,cdots,x_{2d}]_{m}$

here $m$ is the irrelevant ideal. Suppose $W = f_1g_1 +cdots +f_dg_d$ and assume that $(f_1,cdots, f_d, g_1,cdots, g_d)$ is a full system of parameters in $S$. Let $R=S/(W)$, $P=(f_1,cdots,f_d)$ and $Q=(g_1,cdots,g_d)$. I claim that $theta^R(R/P,R/Q) neq 0$.

The reason is that $theta^R(R/P,R/Q) = chi^S(S/P,S/Q)$, the Serre's intersection multiplicity (see Hochster's original paper). Because $dim S/P + dim S/Q = d+d =dim S$, we must have $chi^S(S/P,S/Q)>0$ by Positivity, which is known in this case.

More exotic examples should be abound, and I am sure people who know more intersection theory can provide some, once they are aware of what this question is about. I would be interested in hearing more answers along that line.

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