While what Brian wrote above is certainly correct, you can sometimes estimate the number of components without computing the normalization. For example, localize at the height one prime as described above. Then I think the multiplicity (with respect to the maximal ideal) of that non-normal 1-dimensional domain of will give you an upper bound on the number of divisors lying over your given divisor. This might be substantially harder than computing the resolution though, depending on what you know about your example.

If you know additional things about that 1-dimensional domain, like for example, it is Gorenstein and/or seminormal, then one should be able to make even more precise statements. For example, blowing up the conductor can under certain circumstances be shown to compute the normalization (see Greco-Traverso). Alternately, if the ring is seminormal, then the embedding dimension should give you a lower bound on the number of divisors lying over the point.

While what Brian wrote above is certainly correct, you can sometimes estimate the number of components without computing the normalization. For example, localize at the height one prime as described above. Then I think the multiplicity (with respect to the maximal ideal) of that non-normal 1-dimensional domain of will give you an upper bound on the number of divisors lying over your given divisor. This might be substantially harder than computing the resolution though, depending on what you know about your example.

If you know additional things about that 1-dimensional domain, like for example, it is Gorenstein and/or seminormal, then one should be able to make even more precise statements. For example, blowing up the conductor can under certain circumstances be shown to compute the normalization (see Greco-Traverso). Alternately, if the ring is seminormal, then the embedding dimension should give you an upper bound on the number of divisors lying over the point.

While what Brian wrote above is certainly correct, you can sometimes estimate the number of components without computing the normalization. For example, localize at the height one prime as described above. Then I think the multiplicity (with respect to the maximal ideal) of that non-normal 1-dimensional domain of will give you an upper bound on the number of divisors lying over your given divisor. This might be substantially harder than computing the resolution though, depending on what you know about your example.

If you know additional things about that 1-dimensional domain (like for example, it is Gorenstein and/or seminormal, then one should be able to make even more precise statements, for example, blowing up the conductor can under certain circumstances be shown to compute the normalization).

While what Brian wrote above is certainly correct, you can sometimes estimate the number of components without computing the normalization. For example, localize at the height one prime as described above. Then I think the multiplicity (with respect to the maximal ideal) of that non-normal 1-dimensional domain of will give you an upper bound on the number of divisors lying over your given divisor. This might be substantially harder than computing the resolution though, depending on what you know about your example.

If you know additional things about that 1-dimensional domain, like for example, it is Gorenstein and/or seminormal, then one should be able to make even more precise statements. For example, blowing up the conductor can under certain circumstances be shown to compute the normalization (see Greco-Traverso). Alternately, if the ring is seminormal, then the embedding dimension should give you a lower bound on the number of divisors lying over the point.