Related papers: Multigraded Betti numbers of Veronese embeddings
We study $h$-vectors and graded Betti numbers of level modules up to multiplication by a rational number. Assuming a conjecture on the possible graded Betti numbers of Cohen-Macaulay modules we get a description of the possible $h$-vectors…
In this paper we study topology of the variety of closed planar polygons with given side lengths. We describe the Betti numbers of the moduli spaces as functions of the length vector. We also find sharp upper bounds on the sum of Betti…
We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR in the presence of a continuous real valued function. The refinement of Betti numbers consists of finite configurations of points…
We show that Kimura's necessary and sufficient condition for the non-vanishingness of multigraded Betti numbers of chordal graphs can be extended to the wider class of weakly chordal graphs.
We generalize a method by L. Ambrozio, A. Carlotto, and B. Sharp to study the Morse index of closed f-minimal hypersurfaces isometrically immersed in a general weighted manifold. The technique permits, in particular, to obtain a linear…
It has been shown that $1$-parameter persistence modules have a very simple classification, namely there is a discrete invariant called a barcode that completely characterizes $1$-parameter persistence modules up to isomorphism. In…
We formulate several conjectures which shed light on the structure of Veronese syzygies of projective spaces. Our conjectures are based on experimental data that we derived by developing a numerical linear algebra and distributed…
There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However…
In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector…
We study certain random simplicial complexes, called random quota complexes. A quota complex on $N+1$ weighted vertices is constructed by adding an $n$-simplex if the sum of the weights of the vertices is below a given quota, $q$. In this…
We compute the Betti numbers of the geometric spaces associated to nonrational simple convex polytopes and find that they depend on the combinatorial type of the polytope exactly as in the rational case. This shows that the combinatorial…
The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology…
We study bundles on projective spaces that have vanishing lower cohomologies using their short minimal free resolutions. We partition the moduli $\mathbf{M}$ according to the Hilbert function $H$ and classify all possible Hilbert functions…
We construct a nonminimal graded free resolution of Segre embeddings of $P^1\times P^1$, although we don't compute all maps. We use this to prove an explicit formula for certain nonzero entries in the graded Betti table, at the end of the…
We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the…
We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative…
We determine which simplicial complexes have the maximum or minimum sum of Betti numbers and sum of bigraded Betti numbers with a given number of vertices in each dimension.
We use the results by Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the…
We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each $p,d,r$, explicit bounds on the $p$-th Betti numbers of affine and projective subvarieties of $\mathrm{C}^k$,…
Let $\bar{M}_{0,n}(G(r,V), d)$ be the coarse moduli space of stable degree $d$ maps from $n$-pointed genus $0$ curves to a Grassmann variety $G(r,V)$. We provide a recursive method for the computation of the Hodge numbers and the Betti…