arXiv subjects
Publications and source records attributed to Mihai Tibăr.
We prove a formula for the polar degree of projective hypersurfaces in terms of the Milnor data of the singularities, extending to 1-dimensional singularities the Dimca-Papadima result for isolated singularities. We discuss the semi-continuity of the polar degree in deformations, and we classify the homaloidal cubic surfaces with 1-dimensional singular locus. Some open questions are pointed out along the way.
We address the question "when the local image of a map is well defined" and answer it in case of holomorphic map germs with target $(\bC^{2}, 0)$. We prove a criterion for holomorphic map germs $(X, x)\to (Y, y)$ to be locally open, solving a conjecture by Huckleberry in all dimensions.
We show that the variation of the topology at infinity of a two-variable polynomial function is localisable at a finite number of "atypical points" at infinity. We construct an effective algorithm with low complexity in order to detect sharply the bifurcation values of the polynomial function.
For a map germ $G$ with target $(\mathbb C^{p}, 0)$ or $(\mathbb R^{p}, 0)$ with $p\ge 2$, we address two phenomena which do not occur when $p=1$: the image of $G$ may be not well-defined as a set germ, and a local fibration near the origin may not exist. We show how these two phenomena are related, and how they can be characterised.
We define local fibration structures for real map germs with strictly positive dimensional discriminant: a local fibration structure over the complement of the discriminant, and a complete local fibration structure which includes the stratified discriminant into the picture. We provide new classes of map germs endowed with such local fibration structures.
We present a general criterion for the existence of open book structures defined by real map germs $(\bR^m, 0) \to (\bR^p, 0)$, where $m> p \ge 2$, with isolated critical point. We show that this is satisfied by weighted-homogeneous maps. We also derive sufficient conditions in case of map germs with isolated critical value.
We compare the Euler-Poincaré characteristic to the global Euler obstruction, in case of singular affine varieties, and point out a certain duality among their expressions in terms of strata of a Whitney stratification.