Matrix factorizations for domestic triangle singularities

2017

The Alexander polynomial of a plane curve is an important invariant in global theories on curves. However, it seems that this invariant and even a much stronger one as the fundamental group of the complement of a plane curve may not distinguish non-reduced curves. In this article, we consider a general problem which concerns a hypersurface of the complex projective space $\mathbb P^n$ defined by an arbitrary homogeneous polynomial $f$. The singularity of $f$ at the origin of $\mathbb C^{n+1}$ is studied, by means of the characteristic polynomials $\Delta_l(t)$ of the monodromy, and via the relation between the monodromy zeta function and the Hodge spectrum. Especially, we go further with $\Delta_1(t)$ in the case $n=2$ and aim to regard it as an alternative object of the Alexander polynomial for $f$ non-reduced. This work is based on knowledge of multiplier ideals and local systems.

Lecture Notes in Pure and Applied Mathematics, 2005

Advances in Mathematics, 2016

We provide geometric constructions of modules over the graded Cherednik algebra H gr ν and the rational Cherednik algebra H rat ν attached to a simple algebraic group G together with a pinned automorphism θ. These modules are realized on the cohomology of affine Springer fibers (of finite type) that admit C * -actions. In the rational Cherednik algebra case, the standard grading on these modules is derived from the perverse filtration on the cohomology of affine Springer fibers coming from its global analog: Hitchin fibers. When θ is trivial, we show that our construction gives the irreducible finite-dimensional spherical modules Lν (triv) of H gr ν and of H rat ν . We give a formula for the dimension of Lν (triv) and give a geometric interpretation of its Frobenius algebra structure. The rank two cases are studied in further details. Part 3. Representations 49 7. Geometric modules of the graded Cherednik algebra 49 7.1. The H gr -action on the cohomology of homogeneous affine Springer fibers 49 7.2. The polynomial representation of H gr 51 7.3. The global sheaf-theoretic action of H gr 52 7.4. Local-global compatibility 54 8. Geometric modules of the rational Cherednik algebra 55 8.1. The polynomial representation of H rat 55 8.2. The H rat -action on the cohomology of homogeneous affine Springer fibers 56 8.3. The perverse filtration 57 8.4. The global sheaf-theoretic action of H rat 60 8.5. Proof of Theorem 8.2.3(1) 61 8.6. Frobenius algebra structure and proof of Theorem 8.2.3(2) 62 8.7. Langlands duality and Fourier transform 65 9. Examples 66 9.1. Algebraic theory 67 9.2. Type 2 A 2 , m 1 = 2 67 9.

Journal of Algebra, 2010

Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We then propose the definition of a categorical resolution of singularities. Our main examples are concerned with a categorical resolution of the derived category of quasi-coherent sheaves on a scheme. We propose two kinds of such resolutions. 2 VALERY A. LUNTS 7. Quasi-coherent sheaves on poset schemes 33 7.1. Operations with quasi-coherent sheaves on poset schemes 34 7.2. Summary of functors and their properties 37 7.3. Cohomological dimension of poset schemes 37 8. Derived categories of poset schemes 38 8.1. Summary of functors and their properties 39 8.2. Semi-orthogonal decompositions 40 9. Compact objects and perfect complexes on poset schemes 42 9.1. Existence of a compact generator 43 10. Smoothness of poset schemes 43 10.1. A spectral sequence 45 11. Categorical resolutions by poset schemes 46 11.1. Categorical resolution of reducible schemes with smooth components 48 11.2. Categorical resolution of the cone over a plane cubic 48 12. Appendix 50 References 53

A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley-Reisner ring $R/I_\Lambda$ and the inverse system algebra $R/I_\Delta$. We give a careful study of both of these algebras. Our main results are a full description of the graded Betti numbers of both algebras in the more general setting of linear spaces (giving the result for the projective planes as a special case), and a classification of the characteristics in which the inverse system algebra associated to a finite projective plane has the Weak or Strong Lefschetz Property.

Mathematische Zeitschrift, 2019

We continue, generalize and expand our study of linear degenerations of flag varieties from [5]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel-Weil theorem for the flat irreducible locus.

Advances in Mathematics, 2010

We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a form of simplification of the singularities, known as the reduction to the monomial case. Contents Part 3. Elimination 7. Elimination via universal invariants 8. A smooth local projection and the elimination algebra 9. Elimination algebras and permissible monoidal transformations Part 4. Main Theorem and inductive invariants 10. Main Theorem 10.1 11. Proof of Theorem 10.1 12. The non-simple case 13. Stratification of the singular locus by smooth strata Part 5. Epilogue and example References Key words and phrases. Resolution of singularities. Rees algebras. 2000 Mathematics subject classification. 14E15.

Communications in Algebra, 2019

The Hodge spectrum is an important analytic invariant of singularities encoding the Hodge filtration and the monodromy of the Milnor fiber. However, explicit formulas exist in only a few cases. In this article the main result is a combinatorial formula for the Hodge spectrum of any homogeneous polynomials in three variables whose zero locus is a projective curve arrangement having only ordinary multiple points.

Asian Journal of Mathematics, 2011

Given a variety X over a field k one wants to find a desingularization, which is a proper and birational morphism X ′ → X, where X ′ is a regular variety and the morphism is an isomorphism over the regular points of X. If X is embedded in a regular variety W , there is a notion of embedded desingularization and related to this is the notion of log-resolution of ideals in O W . When the field k has characteristic zero it is well known that the problem of resolution is solved. The first proof of the existence of resolution of singularities is due to H. Hironaka in his monumental work [Hir64] (see also ). If characteristic of k is positive the problem of resolution in arbitrary dimension is still open. See [Hau10] for recent advances and obstructions (see also [Hau03]). The proof by Hironaka is existential. There are constructive proofs, always in characteristic zero case, see for instance [VU89], [VU92], [BM97], we refer to [Hau03] for a complete list of references. Those constructive proofs give rise to algorithmic resolution of singularities, that allows to perform implementation at the computer [BS00], [FKP04].

Journal of Pure and Applied Algebra, 1983