Singularity Theory and Algebraic Geometryhttp://hdl.handle.net/20.500.11824/182021-09-06T05:38:44Z2021-09-06T05:38:44ZImage Milnor Number Formulas for Weighted-Homogeneous Map-GermsPallarés, IrmaPeñafort Sanchis, Guillermohttp://hdl.handle.net/20.500.11824/13172021-08-10T01:00:30Z2021-07-05T00:00:00ZImage Milnor Number Formulas for Weighted-Homogeneous Map-Germs
Pallarés, Irma; Peñafort Sanchis, Guillermo
We give formulas for the image Milnor number of a weighted-homogeneous map-germ $(\mathbb{C}^n,0)\to(\mathbb{C}^{n+1},0)$, for $n=4$ and $5$, in terms of weights and degrees. Our expressions are obtained by a purely interpolative method, applied to a result by Ohmoto. We use our approach to recover the formulas for $n=2$ and $3$ due to Mond and Ohmoto, respectively. For $n\geq 6$, the method is valid as long as certain multi-singularity conjecture holds.
2021-07-05T00:00:00ZClassification of smooth factorial affine surfaces of Kodaira dimension zero with trivial unitsPełka, T.Raźny, P.http://hdl.handle.net/20.500.11824/13142021-08-05T01:00:35Z2021-07-31T00:00:00ZClassification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units
Pełka, T.; Raźny, P.
We give a corrected statement of (Gurjar and Miyanishi 1988, Theorem 2), which classifies smooth affine surfaces of Kodaira dimension zero, whose coordinate ring is factorial and has trivial units. Denote the class of such surfaces by $\mathcal{S}_0$. An infinite series of surfaces in $\mathcal{S}_0$, not listed in loc. cit., was recently obtained by Freudenburg, Kojima and Nagamine (2019) as affine modifications of the plane. We complete their list to a series containing arbitrarily high-dimensional families of pairwise nonisomorphic surfaces in $\mathcal{S}_0$. Moreover, we classify them up to a diffeomorphism, showing that each occurs as an interior of a 4-manifold whose boundary is an exceptional surgery on a 2-bridge knot. In particular, we show that $\mathcal{S}_0$ contains countably many pairwise nonhomeomorphic surfaces.
2021-07-31T00:00:00ZSome contributions to the theory of singularities and their characteristic classesPallarés Torres, I.http://hdl.handle.net/20.500.11824/12942021-07-08T07:02:46Z2021-06-02T00:00:00ZSome contributions to the theory of singularities and their characteristic classes
Pallarés Torres, I.
In this Ph.D. thesis, we give some contributions to the theory of singularities, as well as to the theory of characteristic classes of singular spaces. The first part of this thesis is devoted to the theory of singularities of mappings. We obtain formulas for an important analytical invariant, the image Milnor number of a map-germ. The first contribution is a version of the Lê-Greuel formula for the image Milnor number for corank 1 map-germs, which gives a recursive method to compute it. This work is in collaboration with my Ph.D. supervisor J. J. Nuño. In the second work of this thesis, we obtain two formulas for the image Milnor number for weighted-homogeneous map-germs for dimensions four and five. These formulas are obtained by combining the theory of characteristic classes of singular spaces with a recursive method through examples of singularities. This work is in collaboration with Prof. G. Peñafort. The last part which composes the main work of this Ph.D. thesis is the proof for projective varieties of the Brasselet-Schürmann-Yokura conjecture. This conjecture is within the theory of characteristic classes of singular spaces. Characteristic classes of singular varieties are homology classes generalizing the classical cohomological characteristic classes of manifolds. The conjecture states that two different characteristic classes coincide for compact complex algebraic varieties that are rational homology manifolds. This work is in collaboration with my Ph.D. supervisor J. Fernández de Bobadilla. This conjecture is the characteristic class version for rational homology manifolds of the famous Hodge Index Theorem which computes the signature of a compact complex manifold in terms of Hodge numbers.
2021-06-02T00:00:00ZThe abel map for surface singularities II. Generic analytic structureNagy, J.Nemethi, A.http://hdl.handle.net/20.500.11824/12152021-07-08T07:02:49Z2019-01-01T00:00:00ZThe abel map for surface singularities II. Generic analytic structure
Nagy, J.; Nemethi, A.
We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants include: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincar ́e series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle.
The first part contains the definition of ‘generic structure’ based on the work of Laufer [La73]. The second technical ingredient is the Abel map developed in [NN18].
The results can be compared with certain parallel statements from the Brill–Noether theory and from the theory of Abel map associated with projective smooth curves, though the tools and machineries are very different.
2019-01-01T00:00:00Z