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We establish an algorithm for resolution of singularities of an idealistic filtration in dimension 3 (a local version) in positive characteristic, incorporating the method recently developed by Benito-Villamayor into our framework. Although (a global version of) our algorithm only implies embedded resolution of surfaces in a smooth ambient space of dimension 3, a classical result known before, we introduce some new invariant which effectively measures how much the singularities are improved in the process of our algorithm and which strictly decreases after each blow up. This is in contrast to the wellknown Abhyankar-Moh pathology of the increase of the residual order under blow up and the phenomenon of the "Kangaroo" points observed by Hauser.2010 Mathematics Subject Classification. 14E15.

This is the second of a series of four papers entitled "Toward resolution of singularities over a field of positive characteristic (the Idealistic Filtration Program)". The goal is to present the IFP, and to ultimately construct an explicit algorithm guided by the program.In the classical setting in characteristic zero, resolution of singularities was carried out by induction on dimension. We take a so-called "hypersurface of maximal contact" to reduce the dimension by one. In the algorithm, we construct the strand of invariants "inv classic " of the following form:where the unit (w, s) consists of the weak order w and the number s of the "old" components in the boundary. Going from one unit to the next, the dimension of the object which we use to extract the information to compute the invariants drops by one, manifesting the induction on dimension. We run the algorithm with the center of blowup determined as the maximal locus of "inv classic ".In our new setting in positive characteristic, we no longer have a hypersurface of maximal contact. However, we try to carry out the induction on "invariant σ", which indicates the behavior of "a Leading Generator System". The notion of an LGS plays the role of a collective substitute for a hypersurface of maximal contact in the IFP. Accordingly, in our new algorithm, we construct the strand of invariants "inv" of the following form:where the unit (σ, e µ, s) consists of the above mentioned σ, followed by e µ and s, which correspond to w and s in the classical setting, respectively. Going from one unit to the next, the invariant σ of the LGS of the object, namely an idealistic filtration, strictly drops, manifesting the induction on the invariant σ. We run the new algorithm with the center of blowup determined as the maximal locus of "inv".The main purpose of this paper, Part II of the series, is to study the basic properties of the invariants that appear in the strand of invariants "inv", establishing the upper semicontinuity of the pair (σ, e µ) among others.

In the category of free arrangements, inductively and recursively free arrangements are important. In particular, in the former, Terao's open problem asking whether freeness depends only on combinatorics is true. A long standing problem whether all free arrangements are recursively free or not was settled by the second author and Hoge very recently, by giving a free but non-recursively free plane arrangement consisting of 27 planes. In this paper, we construct free but non-recursively free plane arrangements consisting of 13 and 15 planes, and show that the example with 13 planes is the smallest in the sense of the cardinality of planes. In other words, all free plane arrangements consisting of at most 12 planes are recursively free. To show this, we completely classify all free plane arrangements in terms of inductive freeness and three exceptions when the number of planes is at most 12. Several properties of the 15 plane arrangement are proved by computer programs. Also, these two examples solve negatively a problem posed by Yoshinaga on the moduli spaces, (inductive) freeness and, rigidity of free arrangements. 1 Corollary 1.3 ). Freeness of arrangements depends only on their combinatorics for central arrangements A in C 3 with |A| ≤ 12.By investigating the structure of the classification in Theorem 1.1, we can say that almost all the free arrangements with small exponents are inductively free.then A is either inductively free or a dual Hesse arrangement appearing in Theorem 1.1.Another formulation of Problem 1.2 is using the moduli space of all arrangements whose intersection lattice is a given lattice, see Definition 2.16 for details:Open problem 1.5. Does a lattice L exist such that the moduli space V(L) contains a free and a non-free arrangement?Not much is known about this moduli space regarding freeness; Yuzvinsky [16] proved that the free arrangements form an open subset in V(L) (see also [15, Theorem 1.50] for the case of dimension three). The first intuition one gets when working with these notions is, that either the moduli space is very big and the free arrangements are inductively free, or the moduli space is zero dimensional and all its elements are Galois conjugates (let us call such an arrangement rigid). Yoshinaga proposed the following problem which is stronger than Terao's open problem since the property of being free is invariant under Galois automorphisms: NON-RECURSIVELY FREENESS AND NON-RIGIDITY 3 Problem 1.6 (Yoshinaga,[15, p. 20,(11)]). Is a free arrangement either inductively free, or rigid? In other words, does the following inclusion hold?{Free arrangements}⊂{Inductively free} ∪ {Rigid}.Beyond the fact that we find new small arrangements which are free but not recursively free, our examples with 13 and 15 planes provide a negative answer to Yoshinaga's problem 1.6, i.e., the following holds.Theorem 1.7. Let L be the intersection lattice of our 13 or 15 plane arrangements. Then the moduli spaces V(L) of them are one-dimensional (hence not rigid), and not inductively free, but free. I...

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