This article aims to study the behavior of certain types of singularities in a universally equidimensional morphism (i.e., open with constant pure-dimensional fibers). These singularities are those of reduced complex spaces of pure dimension m for which the sheaf LZm (whose sections are meromorphic forms that extend analytically over any desingularization of Z) has depth at least two and are called weakly-1-rational and denoted BR1; for spaces whose singular locus is of codimension at least two, they are called weakly-2-rational and denoted BR2. Our study focuses on the possibility of transferring this type of singularities from the total space to the base, from the base and the fibers to the total space, and from the latter to the fibers.
This is the second of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to connections with the topic of singularities of curves on surfaces. In the prior paper we studied those invariants of Goursat distributions akin to those of curves on surfaces, which we call structural invariants. In this paper we study invariants arising from the small growth sequence of a Goursat distribution, and relate them to the the structural invariants.
We extend the computation of the invariant $η(ω,C,a)$ defined in arXiv:2409.01751 to special points on the line at infinity and show that, as in the affine case, its value is determined purely by the geometry of the integral curve C. By incorporating points at infinity, the invariant $η$ yields effective geometric criteria that certify Darboux integrability in cases not covered by affine data alone. As an application we construct six new codimension-11 components of the degree-3 center variety
Starting from the Weierstrass elliptic function, we study the associated Frobenius structure, incorporating the perspective of derived categories, particularly that of homological mirror symmetry. Given a deformation of the Weierstrass elliptic function, we construct a primitive form normalized to be compatible with the period map for integral cycles, and obtain a Frobenius structure whose Frobenius potential is defined over the rational numbers. We also construct a Frobenius structure using elliptic Weyl group invariants (as opposed to Jacobi group invariants), and establish an isomorphism between these two Frobenius structures. We further examine the relationship between the degree of the Lyashko–Looijenga map modulo the modular group and the number of full exceptional collections up to the braid group action and translations, as well as the associated Gamma-integral structure.
In this article we extend the theory of the binary codes (the strict code 𝒦 and the extended code 𝒦'), associated to a projective nodal surface, to a coding theory for normal surfaces, with special consideration of the surfaces with ADE (Rational Double Points) singularities. We define a new theory of generalized labeled codes, establish in the geometric case basic restrictions for the weights of these codes, and some basic inequality. A crucial method that we establish is the extension of the concept of `code shortening' to the case of generalized codes: this is the algebraic counterpart of the geometric notion of a partial smoothing of the singular points, and leads to the concept of ancestors, which we illustrate through several examples.
Isolated hypersurface singularities come equipped with distinguished bases of their Milnor lattices and with upper triangular integral matrices, which are called here distinguished matrices. These matrices form an orbit of a braid group and a sign change group. This paper proposes to characterize the distinguished matrices of singularities within all upper triangular integral matrices in terms of the variance of certain spectral numbers. It succeeds in the positive definite and the positive semidefinite cases. The ADE root lattices are crucial. In the semidefinite cases, results on non-reduced presentations of Weyl group elements are used.
. The study of evolutes of plane curves goes back at least to Huygens, and was continued and extended to space curves by Monge, Darboux, and others. Salmon studied projective curves and surfaces and their evolutes and gave many enumerative formulas for their degrees and number of singularities. We define envelopes of families of linear spaces in projective space. In order to define evolutes we impose a notion of perpendicularity, which allows us to consider the normal spaces to a projective variety. The evolute of a projective hypersurface is the envelope of the family of normal lines. For a variety of dimension r in n-space, the evolute is defined as the (n - r)th "iterated" cuspidal locus of the map from the total space of the normal spaces to projective space. Thus the envelope can be interpreted as a (n - r)th order Thom-Boardman singularity. Further higher order Thom-Boardman singularities correspond, for a curve in the plane or in 3-space, to classical objects like the vertices of the curve; for a surface in 3-space, they give the cuspidal curve - and its cusps - on the evolute. Using known formulas for Thom polynomials we are able to verify and generalize many of Salmon's formulas and find new ones.
Saito theory associates to an isolated singularity rich structure that plays an important role in mirror symmetry. In this note we construct Saito theory for A and D type Landau–Ginzburg orbifolds. Namely, for the pairs (f,G), where f defines an isolated singularity of A and D type and G is a group of symmetries of f. In total we consider five families of such pairs. In particular, we construct the orbifold versions of Brieskorn lattice and the Gauss–Manin connection computing them explicitly for the A and D type Landau–Ginzburg orbifolds.
For curves singularities all smoothing components of the deformation space have the same dimension, but there can be components of different dimensions. We are interested in the question of what the generic singularities are that appear in the fibre over a component. To this end we revisit the known examples of non-smoothable singularities and study their deformations. There are two general methods available to show that a curve is not smoothable. In the first method one exhibits a family of singularities of a certain type and then uses a dimension count to prove that the family cannot lie in the closure of the space of smooth curves. The other method uses the semicontinuity of a certain invariant, related to the Dedekind different. This invariant vanishes for Gorenstein singularities, so in particular for smooth curves. With these methods and computations with computer algebra systems we study monomial curves and cones over point sets in projective space. We also give new explicit examples of non-smoothable singularities. In particular, we find non-smoothable Gorenstein curve singularities. The cone over a general self-associated point set in Pg-2 is not smoothable if g is at least 11, as then the point set can not be a hyperplane section of a canonical curve of genus g.
We discuss the universal orbifold Euler characteristic and generalized orbifold Euler characteristics corresponding to finitely generated groups $A$ (the $A$-Euler characteristics). We show that the collection of all $A$-Euler characteristics for $A$ of the form $A'\times Z$ ($Z$ is the group of integers) with finite $A'$ determine the universal orbifold Euler characteristic.
We describe a system of plane algebraic curves defined over , attached naturally to the exponential function. On of these is a remarkable curve of degree 6 that has genus equal to 1. As the sectic curve has rational points, it is an elliptic curveand can be tranformed over into the curve 1584.j1 of the LMFDB. One is left to wonder what the number 11, appearing in the factorisation 1584=2^4.3^2.11 has to do with the exponential function.
In this paper we introduce and study divisorial (i) classes for the blow up of projective space in several points for i=-1,0 and 1. We generalize Noether's inequality, and we prove that all divisorial (i) classes are in bijective correspondence with the orbit of the Weyl group action on one exceptional divisor following Nagata's original approach. Moreover, we prove that the irreducibility condition from the definition of divisorial (i) classes can be replaced by the numerical condition of having positive intersection with all divisorial (-1) classes of smaller degree via the Mukai pairing.
We construct random Morse functions on surfaces by random walk and compute related distributions. We study the space of Morse functions through these random variables. We consider subspaces characterized by the surfaces with boundary obtained by cutting the closed domain surface of the Morse function at the levels of regular values. We consider Morse functions having a bounded number of critical points and one single local minimum. We find a small set of Morse functions which are close enough to any other Morse function in the sense that they share the same characterizing surfaces with boundary.
We construct some version of the trace morphism between the Du Bois complexes, with applications towards the behavior of the local cohomological dimension and some Hodge theoretic aspects of singularities under finite morphisms.
The purpose of this paper is to understand generic behavior of constraint functions in optimization problems relying on singularity theory of smooth mappings. To this end, we will focus on a subgroup of the Mather's contact group, whose action to constraint map-germs preserves the corresponding feasible set-germs (i.e. the set consisting of points satisfying the constraints). We will classify map-germs with small stratum extended-co dimensions with respect to the subgroup we introduce, and calculate the co dimensions of the orbits by the subgroup of jets represented by germs in the classification lists and those of the complements of these orbits. Applying these results and a variant of the transversality theorem, we will show that families of constraint mappings whose germ at any point in the corresponding feasible set is equivalent to one of the normal forms in the classification list compose a residual set in the entire space of constraint mappings with at most four parameters. These results enable us to quantify genericity of given constraint mappings, and thus evaluate to what extent known test suites are generic.
