A Jacobian module for disentanglements and applications to Mond's conjecture

Given a germ of holomorphic map $f$ from $\mathbb C^n$ to $\mathbb C^{n+1}$, we define a module $M(f)$ whose dimension over $\mathbb C$ is an upper bound for the $\mathscr A$-codimension of $f$, with equality if $f$ is weighted homogeneous. We also define a relative version $M_y(F)$ of the module, for unfoldings $F$ of $f$. The main result is that if $(n,n+1)$ are nice dimensions, then the dimension of $M(f)$ over $\mathbb C$ is an upper bound of the image Milnor number of $f$, with equality if and only if the relative module $M_y(F)$ is Cohen-Macaulay for some stable unfolding $F$. In particular, if $M_y(F)$ is Cohen-Macaulay, then we have Mond's conjecture for $f$. Furthermore, if $f$ is quasi-homogeneous, then Mond's conjecture for $f$ is equivalent to the fact that $M_y(F)$ is Cohen-Macaulay. Finally, we observe that to prove Mond's conjecture, it suffices to prove it in a suitable family of examples.Comment: 19 page

Isolated singularities of binary differential equations of degree n

We study isolated singularities of binary differential equations of degree n which are totally real. This means that at any regular point, the associated algebraic equation of degree n has exactly n different real roots (this generalizes the so called positive quadratic differential forms when n = 2). We introduce the concept of index for isolated singularities and generalize Poincar'e-Hopf theorem and Bendixson formula. Moreover, we give a classification of phase portraits of the n-web around a generic singular point. We show that there are only three types, which generalize the Darbouxian umbilics D1, D2 and D3

Disentangling mappings defined on ICIS

We study germs of hypersurfaces $(Y,0)\subset (\mathbb C^{n+1},0)$ that can be described as the image of $\mathscr A$-finite mappings $f:(X,S)\rightarrow (\mathbb C^{n+1},0)$ defined on an ICIS $(X,S)$ of dimension $n$. We extend the definition of the Jacobian module given by Fern\'andez de Bobadilla, Nu\~no-Ballesteros and Pe\~nafort-Sanchis when $X=\mathbb C^n$, which controls the image Milnor number $\mu_I(X,f)$. We apply these results to prove the case $n=2$ of the generalised Mond conjecture, which states that $\mu_I(X,f)\geq codim_{\mathscr A_e} (X,f)$, with equality if $(Y,0)$ is weighted homogeneous.Comment: 19 page

Isolated singularities of binary differential equations of degree nn

We study isolated singularities of binary differential equations of degree n which are totally real. This means that at any regular point, the associated algebraic equation of degree n has exactly n different real roots (this generalizes the so called positive quadratic differential forms when n = 2). We introduce the concept of index for isolated singularities and generalize Poincar'e-Hopf theorem and Bendixson formula. Moreover, we give a classification of phase portraits of the n-web around a generic singular point. We show that there are only three types, which generalize the Darbouxian umbilics D1, D2 and D3