## Final Report Summary - CODIM1SING (Codimension-one properties of singularities)

Our scientific modelling of nature is essentially based on various types of systems of equations (linear, polynomial, analytic, differential, etc.). Solving such equations allow us to make predictions. In many cases where explicit solutions can not be found abstract considerations can yield at least qualitative insights. Here the understanding of so-called singularities of equations is crucial. These are points of local instability that can have tremendous impact on the global behaviour of solutions. In algebraic geometry and singularity theory one considers solutions of polynomial or analytic equations as geometric objects, so called varieties, and studies them based on an interplay of geometry, topology and algebra. Here the singular locus consists of points where the geometric solution space is, intuitively speaking, not smooth but has edges or corners. In elementary situations one knows such a phenomenon to be related to the derivative of a function being zero. Derivations and differential forms give an abstract algebraic framework that captures this idea in general. By the very nature of a singularity it is determined by what happens close to it. Codim1Sing follows this idea by considering certain properties in codimension one.

A main tool to study singularities is resolution of singularities which relates complicated singularities to simpler ones, so-called normal crossing divisors. The complicatedness of the singularity is encoded in a map from the simpler singularity to the original one.

Normalization is a step towards resolution that removes singularities in codimension one. More general than normal crossing divisors are so-called free divisors whose singular locus is purely one-codimensional. These objects play a prominent role in singularity theory for instance as discriminants in deformation theory. They are subject to long standing conjectures such as Terao's conjecture that freeness of hyperplane arrangements is a combinatorial proerty. Codim1Sing aims to understand the geometry of free divisors, in particular, the relation between free divisors and normal crossing divisors. Free divisors stem from Kyoji Saito's theory of logarithmic derivations and logarithmic differential forms. While this theory deals only with hypersurfaces, that is, zero sets of a single function, Codim1Sing aims at extending it to classes of more general varieties. This concerns in particular dualities and algebraic properties of modules of derivations and differential forms. The approach proposed by Codim1Sing is to relate Saito's theory to that of regular differential forms by Kersken and Kunz/Waldi and that of multilogarithmic differential forms by Aleksandrov. The long term goal of Codim1Sing was a deeper insight into the nature of free divisors and substantial generalizations of existing results on hypersurface singularities. As a side-product Codim1Sing contributed functionality to the computer algebra system SINGULAR supporting research in the area.

Codim1Sing made the following major achievements:

* A generalized notion of Saito freeness for Gorenstein singularities was introduced.

* For a general class of singularities, the equality of 0th regular differential forms on the singularity and its normalization was shown to correspond to the singularity being normal crossing in codimension one.

This generalizes an earlier result by Granger and the PI for hypersurface singularities that gave an algebraic companion of the classical Lê-Saito theorem.

* Faber's conjecture was confirmed for quasihomogeneous divisors: A quasihomogeneous free divisor which is normal crossing in codimension one is a normal crossing divisor. Notably the proof makes use of the Lie algebra structure of the module of logarithmic derivations.

* A combinatorial duality on so-called good semigroup ideals was established. It mirrors the Cohen-Macaulay duality on fractional ideals on a general class of curve singularities. The correspondence from algebra to combinatorics is given by taking value semigroup ideals with respect to the normalization.

A main tool to study singularities is resolution of singularities which relates complicated singularities to simpler ones, so-called normal crossing divisors. The complicatedness of the singularity is encoded in a map from the simpler singularity to the original one.

Normalization is a step towards resolution that removes singularities in codimension one. More general than normal crossing divisors are so-called free divisors whose singular locus is purely one-codimensional. These objects play a prominent role in singularity theory for instance as discriminants in deformation theory. They are subject to long standing conjectures such as Terao's conjecture that freeness of hyperplane arrangements is a combinatorial proerty. Codim1Sing aims to understand the geometry of free divisors, in particular, the relation between free divisors and normal crossing divisors. Free divisors stem from Kyoji Saito's theory of logarithmic derivations and logarithmic differential forms. While this theory deals only with hypersurfaces, that is, zero sets of a single function, Codim1Sing aims at extending it to classes of more general varieties. This concerns in particular dualities and algebraic properties of modules of derivations and differential forms. The approach proposed by Codim1Sing is to relate Saito's theory to that of regular differential forms by Kersken and Kunz/Waldi and that of multilogarithmic differential forms by Aleksandrov. The long term goal of Codim1Sing was a deeper insight into the nature of free divisors and substantial generalizations of existing results on hypersurface singularities. As a side-product Codim1Sing contributed functionality to the computer algebra system SINGULAR supporting research in the area.

Codim1Sing made the following major achievements:

* A generalized notion of Saito freeness for Gorenstein singularities was introduced.

* For a general class of singularities, the equality of 0th regular differential forms on the singularity and its normalization was shown to correspond to the singularity being normal crossing in codimension one.

This generalizes an earlier result by Granger and the PI for hypersurface singularities that gave an algebraic companion of the classical Lê-Saito theorem.

* Faber's conjecture was confirmed for quasihomogeneous divisors: A quasihomogeneous free divisor which is normal crossing in codimension one is a normal crossing divisor. Notably the proof makes use of the Lie algebra structure of the module of logarithmic derivations.

* A combinatorial duality on so-called good semigroup ideals was established. It mirrors the Cohen-Macaulay duality on fractional ideals on a general class of curve singularities. The correspondence from algebra to combinatorics is given by taking value semigroup ideals with respect to the normalization.