Abstract
In June 2010, we have the honor of seeing Prof Hironaka at Kyoto. He conjectures that the number \(d_2\) is bounded under permissible blow-ups (see below). The next step is certainly to show that the number \(d_2\) will drop under permissible blow-ups. Implicitly, the further conjectures about \(d_3,d_4,\ldots \) will clarify the types of singularities under permissible blow-ups and produce a local version of resolution of singularities. We were unable to prove the Conjecture in its entirety. What we can prove is that in a sequence of permissible blow-ups, the number \(d_2\) may have a bounded increase, if the jumping reach the bound, then it may be stationary for a few steps and then must have a decrease. However, it can not be eliminated by our result that if the increase does not reach the bound or after the decrease, the resulting number \(d_2\) will not again increase beyond the old bound. Finally, we have no counter-example to Hironaka’s conjecture.
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