Conjectures on Stably Newton Degenerate Singularities

We discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

NOTES ON VANISHING CYCLES AND APPLICATIONS

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.

Homology supported in Lagrangian submanifolds in mirror quintic threefolds

In this note, we study homology classes in the mirror quintic Calabi–Yau threefold that can be realized by special Lagrangian submanifolds. We have used Picard–Lefschetz theory to establish the monodromy action and to study the orbit of Lagrangian vanishing cycles. For many prime numbers $p,$ we can compute the orbit modulo p. We conjecture that the orbit in homology with coefficients in $\mathbb {Z}$ can be determined by these orbits with coefficients in $\mathbb {Z}_p$ .

Image Milnor number and 𝒜e-codimension for maps between weighted homogeneous irreducible curves

Let (X, 0) ⊂ (ℂn, 0) be an irreducible weighted homogeneous singularity curve and let f : (X, 0) → (ℂ2, 0) be a finite map germ, one-to-one and weighted homogeneous with the same weights of (X, 0). We show that 𝒜e-codim(X, f) = μI(f), where the 𝒜e-codimension 𝒜e-codim(X, f) is the minimum number of parameters in a versal deformation and μI(f) is the image Milnor number, i.e. the number of vanishing cycles in the image of a stabilization of f.