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Laurenţiu Păunescu

Publications and source records attributed to Laurenţiu Păunescu.

7 recordsLinked to original sources
arXiv2026

Bi-Lipschitz invariance of Newton polygons along gradient canyons

We study bi-Lipschitz right-equivalence of holomorphic function germs $f:(\mathbb{C}^2,0)\to(\mathbb{C},0)$ via polar arcs and gradient canyons. For a polar arc $γ$ we consider the Newton polygon of $f_x(X+γ(Y),Y)$ and define its augmentation by adjoining the point $(0,\operatorname{ord} f(γ(y),y)-1)$. We prove that the resulting augmented Newton polygon is constant along each gradient canyon of degree $>1$ and is invariant under bi-Lipschitz right-equivalence. Moreover, its compact edges decompose into a topological part and a Lipschitz part: the latter encodes, through simple intercept relations, the second-level Henry-Parusiński type invariants. As applications, we obtain two numerical bi-Lipschitz invariants attached to a canyon: its polar multiplicity and, via the Koike-Kuo-Păunescu curvature formula, the total asymptotic Gaussian curvature concentrated in it.

math.CV↗
arXiv2026

Zariski equisingularity of surface singularities in $\mathbb C^3$ by a local invariant

We associate to every analytic surface singularity $(V,0)$ in $(\mathbb C^3,0)$, not necessarily isolated, an invariant $mult^* (V)$ and show that an analytic family of such singularities $(V_t,0)$, $t\in (\mathbb C^l,0)$, is generically Zariski equisingular if and only if $mult^* (V_t)$ is constant. The invariant, that we call the multiplicity sequence of $V$, takes into account the multiplicities of the successive discriminants of $V$ by generic corank one projections.

math.AG↗
arXiv2022

Vanishing cohomology and Betti bounds for complex projective hypersurfaces

We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tibăr in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.

math.AG↗
arXiv2021

Clustering polar curves

This essay builds on the idea of grouping the polar curves of 2-variable function germs into polar clusters. In the topological category, one obtains a bijective correspondence between certain partitions of the polar quotients of two topologically equivalent function germs. We explain how this bijective correspondence may be refined in the Lipschitz category in terms of the associated gradient canyons.

math.CV↗
arXiv2021

The vanishing cohomology of non-isolated hypersurface singularities

We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower bound in dimension. Guided by geometric results, we alternately use the nearby and vanishing cycle functors to derive information about the Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead to the description of the reduced cohomology groups, except the top two, in terms of the vanishing cohomology of the nearby section. We use it to compute explicitly the lowest (possibly nontrivial) vanishing cohomology group of the Milnor fiber.

math.AG↗
arXiv2004

Closed manifolds coming from Artinian complete intersections

We reformulate the integrality property of the Poincaré inner product in the middle dimension, for an arbitrary Poincaré $\Q$-algebra, in classical terms (discriminant and local invariants). When the algebra is 1-connected, we show that this property is the only obstruction to realizing it by a closed manifold, up to dimension 11. We reinterpret a result of Eisenbud and Levine on finite map germs, relating the degree of the map germ to the signature of the associated local ring, to answer a question of Halperin on artinian weighted complete intersections.We analyse the homogeneous artinian complete intersections over $\Q$ realized by closed manifolds of dimensions 4 and 8, and their signatures.

math.AT↗