Is there a good reference/book on Intersection theory on schemes with Gorenstein singularities? Does the construction of the intersection of cycles discussed in Fulton's book also hold for schemes with Gorenstein singularities?
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2$\begingroup$ What kind of results are looking for, exactly? Note that, even for the simplest type of Gorenstein singularities (namely, ordinary double points of surfaces) one cannot hope for an integer-valued intersection theory. For instance, if $L$ is a line in the ruling of a quadric cone in $\mathbb{P}^3$, since $2L$ is a linearly equivalent to a smooth hyperplane section we have $(2L)^2=2$, hence $L^2=1/2$. $\endgroup$– Francesco PolizziCommented Jan 9, 2023 at 13:34
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2$\begingroup$ Note that the previous example is $\mathbb{Q}$-factorial, so a rational intersection theory is defined. But there are also example of Gorenstein varieties that are not $\mathbb{Q}$-factorial: for instance, take a $3$-fold $X \subset \mathbb{P}^4$ containing a plane $H$. In this case, no multiple of $H$ is a line bundle and I am not aware of any kind of intersection theory in this situation (but I am not an expert on the subject). $\endgroup$– Francesco PolizziCommented Jan 9, 2023 at 13:45
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$\begingroup$ @Francesco Polizzi I understand that integer-valued intersection theory is not possible in general. But I was curious whether there is a theory which gives positive intersection numbers and algebraic equivalence implies numerical equivalence. $\endgroup$– user497552Commented Jan 9, 2023 at 13:49
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