arXiv subjects
Publications and source records attributed to Matthias Zach.
We improve a previously known theoretic method to compute A-resultants for suitable monomial support sets due to Weyman to the extent that it becomes computationally feasible and effective. This is achieved by introducing a new algorithm for the computation of direct images of complexes of coherent sheaves on toric varieties. The procedure does not rely on Gr\"obner basis computations at any stage.
Sheaf cohomology or, more generally, higher direct images of coherent sheaves along proper morphisms are central to modern algebraic geometry. However, the computation of these objects is a non-trivial and expensive task which easily challenges the capacities of modern computers. We describe an algorithm and its implementation to compute a spectral sequence converging to the higher direct images of a bounded complex of sheaves on a product of projective spaces $\mathbb P = \mathbb P^{r_1}\times \dots \times \mathbb P^{r_m}$ over an arbitrary affine base $\mathrm{Spec} R$. We assume the ring $R$ to be computable and the complex of sheaves to be represented by an actual complex of (multi-)graded modules.
We derive explicit equations for the Oguiso-Yu automorphism of minimum topological entropy on a complex Enriques surface. The approach is computer aided and makes use of elliptic fibrations.
We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$. An extension of Tib{\u a}r's Bouquet Theorem to this setup allows for a topological definition of Milnor numbers $\mu(\alpha; f)$ for each stratum $V^\alpha$ of $X$ and we prove several formulas which compute these numbers as (alternating) sums of certain ``homological indices''. The main technical result at work in the background is a local Riemann-Roch type theorem, relating a topological obstruction to holomorphic Euler characteristics.
We explain how to use the computer algebra system OSCAR to find all elliptic fibrations (up to automorphism) on a given surface and compute their Weierstrass models. This is illustrated for Vinberg's most algebraic K3 surface, the unique K3 surface of Picard rank 20 and discriminant 3.
We study the cohomology of the generic determinantal varieties $M_{m,n}^s = \{ φ\in \mathbb C^{m\times n} : \mathrm{rank} φ<s \}$, their polar multiplicities, their sections $D_k \cap M_{m,n}^s$ by generic hyperplanes $D_k$ of various dimension $k$, and the real and complex links of the spaces $(D_k\cap M_{m,n}^s,0)$. Such complex links were shown to provide the basic building blocks in a bouquet decomposition for the (determinantal) smoothings of smoothable isolated determinantal singularities. The detailed vanishing topology of such singularities was still not fully understood beyond isolated complete intersections and a few further special cases. Our results now allow to compute all distinct Betti numbers of any determinantal smoothing.
We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena such as for instance non-isolated singularities which are finitely determined, or smoothings with low connectivity; already the union of the coordinate axes in $(\mathbb{C}^3,0)$ is determinantal, but not a complete intersection. We start with the algebraic background and then continue by discussing the subtle interplay of unfoldings and deformations in this setting, including a survey of the case of determinantal hypersurfaces, Cohen-Macaulay codimension $2$ and Gorenstein codimension $3$ singularities, and determinantal rational surface singularities. We conclude with a discussion of essential smoothings and provide an appendix listing known classifications of simple determinantal singularities.
Let $f\colon (\mathbb C^n,S)\to (\mathbb C^p,0)$ be a finite map-germ with $n<p$ and $Y_δ$ the image of a small perturbation $f_δ$. We show that the reduced cohomology of $Y_δ$ is concentrated in a range of degrees determined by the dimension of the instability locus of $f$. In the case $n\geq p$ we obtain an analogous result, replacing finiteness by $\mathcal K$-finiteness and $Y_δ$ by the discriminant $Δ(f_δ)$. We also study the monodromy associated to the perturbation $f_δ$.
We describe a generalization of Milnor's formula for the Milnor number of an isolated hypersurface singularity to the case of a function $f$ whose restriction $f|(X,0)$ to an arbitrarily singular reduced complex analytic space $(X,0) \subset (\mathbb C^n,0)$ has an isolated singularity in the stratified sense. The corresponding analogue of the Milnor number, $μ_f(α;X,0)$, is the number of Morse critical points in a stratum $\mathscr S_α$ of $(X,0)$ in a morsification of $f|(X,0)$. Our formula expresses $μ_f(α;X,0)$ as a homological index based on the derived geometry of the Nash modification of the closure of the stratum $\mathscr S_α$. While most of the topological aspects in this setup were already understood, our considerations provide the corresponding analytic counterpart. We also describe how to compute the numbers $μ_f(α;X,0)$ by means of our formula in the case where the closure $\overline{ \mathscr S_α} \subset X$ of the stratum in question is a hypersurface.
We consider the possible disentanglements of holomorphic map germs $f \colon (\mathbb C^n,0) \to (\mathbb C^N,0)$, $n < N$, with nonisolated locus of instability $\operatorname{Inst}(f)$. The aim is to achieve lower bounds for their (homological) connectivity in terms of $\dim \operatorname{Inst}(f)$. Our methods apply in the case of corank $1$.
We provide a bouquet decomposition for the determinantal Milnor fiber of an essentially isolated determinantal singularity of arbitrary type $(m,n,t)$. The building blocks in the decomposition are (suspensions of) hyperplane sections of the associated generic determinantal variety $M_{m,n}^t$ in general position off the origin.
Isolated Cohen-Macaulay codimension 2 singularities share many common features with isolated complete intersection singularities, but they also exhibit some striking new behaviour. One such instance was recently observed by Damon and Pike in their study of the vanishing topology and Euler characteristic, where they took this class of singularities as examples. In this article, we explore their findings further by determining the Betti numbers explicitly and explain the new phenomena. An important tool here is the Tjurina modification relating a Cohen-Macaulay codimension 2 singularity to a finite number of complete intersection singularities.
We extend the results from the previous paper by A. Frühbis-Krüger and the author [arXiv:1501.01915] to the vanishing topology of those singularities in the title. Studying the case of possibly non-isolated singularities in the Tjurina- transform, we reveal that in dimension 3 and 2 there always is exactly one special vanishing cycle in degree 2 closely related to the determinantal structure of the singularity.