Computing jumping numbers in higher dimensions

Abstract

The aim of this paper is to generalize the algorithm to compute jumping numbers on rational surfaces described in [AAD14] to varieties of dimension at least 3. Therefore, we introduce the notion of π-antieffective divisors, generalizing antinef divisors. Using these divisors, we present a way to find a small subset of the 'classical' candidate jumping numbers of an ideal, containing all the jumping numbers. Moreover, many of these numbers are automatically jumping numbers, and in many other cases, it can be easily checked.Proposition-Definition 2.5. Let a be an ideal sheaf on X. There exists an increasing sequence of rational numbers). The numbers λ i , i > 0, are called the jumping numbers of a.

Contribution of jumping numbers by exceptional divisors

“…Since E 1 is the only divisor for which 6 7 is a candidate jumping number, we can even conclude that 6 7 is not a jumping number. Using for example the algorithm of [BD16], we find that the complete list of jumping numbers in (0, 1] is 3 7 , 1 2 , 5 6 and 1, which also yields the result. 4.2.…”

“…However, since D 5 is a log canonical singularity, it has no jumping numbers in (0, 1). (This can also be verified using the algorithm of [BD16].) We can conclude that contribution of jumping numbers by an exceptional divisor cannot be decided by only looking at the intersection configuration.…”

“…Using Proposition 4.3, one can show that they are not contributed by E 0 (similarly as in Example 4.2). With for example the algorithm of [BD16], one can compute that the jumping numbers are in fact the numbers in the set 7 10 , 9 10 , 1 + Z ≥0 , and then the statement for 3 4 also follows. We show that E 0 is not contracted in the log canonical model using Lemma 3.12.…”