Related papers: A counterexample to the parity conjecture

Let $Hilb^d(A^3)$ be the Hilbert scheme of $d$ points in $A^3$, and let $T_z$ denote the tangent space to a point $z \in Hilb^d(A^3)$. Okounkov and Pandharipande have conjectured that $\dim T_z$ and $d$ have the same parity for every $z$.…

All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is $\sigma$-homogeneous. Inspired by this theorem, we obtain the following results: assuming $\mathsf{AD}$, every…

Inequalities among symmetric polynomial functions are fundamental questions in mathematics and have various applications in science and engineering. This paper investigates a beautiful and inspiring conjecture, proposed by Cuttler, Greene…

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the…

In this paper we study the tangent space to the Hilbert scheme $\mathrm{Hilb}^d \mathbf{P}^3$, motivated by Haiman's work on $\mathrm{Hilb}^d \mathbf{P}^2$ and by a long-standing conjecture of Brian\c{c}on and Iarrobino on the most singular…

In this short note we study the questions of (non-)L-equivalence of algebraic varieties, in particular, for abelian varieties and K3 surfaces. We disprove the original version of a conjecture of Huybrechts \cite[Conjecture 0.3]{H} stating…

We show that noncompact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions. The main ingredient in the proof…

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $\nu$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that…

In recent work of Monthoux and Pines~[1] and also in Rice et {\sl al.}'s work~[2], quasi-averages like $\langle c_{k \uparrow} c_{- k \downarrow} \rangle$ were considered even in the case of a dimension less or equal two. But it is well…

In 1989 Happel conjectured that for a finite-dimensional algebra $A$ over an algebraically closed field $k$, $\gl A< \infty$ if and only if $\hch A < \infty$. Recently Buchweitz-Green-Madsen-Solberg gave a counterexample to Happel's…

In this paper we complete the proof began by A. Polishchuk and E. Zaslow (math.AG/9801119) of a weak version of Kontsevich's homological mirror symmetry conjecture for elliptic curves. The main difference to the work of Polishchuk and…

We investigate some aspects of the geometry of two classical generalisations of the Hilbert schemes of points. Precisely, we show that parity conjecture for $\text{Quot}_r^d\mathbb{A}^3$ already fails for $d=8$ and $r=2$ and that lots of…

We prove that any two-dimensional real subspace of Schatten-3 can be linearly isometrically embedded into $L_{3}$. This resolves the case $p = 3$ of Hanner's inequality for Schatten classes, conjectured by Ball, Carlen and Lieb. We…

The paper concerns the affine varieties that are homogeneous with respect to a (non-standard) graduation over the group Z^m. Among the other properties it is shown that every such a variety can be embedded in its Zariski tangent space at…

We address a conjecture that $\pi_1$-surjective maps between closed aspherical 3-manifolds having the same rank on $\pi_1$ must be of non-zero degree. The conjecture is proved for Seifert manifolds, which is used in constructing the first…

We show that the Cancellation Conjecture does not hold for the affine space A^3_k over any field k of positive characteristic. We prove that an example of T. Asanuma provides a three-dimensional k-algebra A for which A is not isomorphic to…

Let k be an algebraically closed field. We study the cotangent space of a point t corresponding to a monomial ideal I of k[x_1, ..., x_r] in the Hilbert scheme of n points of affine r-space (so the k-dimension of k[x_1, ..., x_r]/I =…

Given a linear elliptic equation $\sum a_{ij} u_{ij} =0$ in $\mathbb{R}^3$, it is a classical problem to determine if its degree-one homogeneous solutions $u$ are linear. The answer is negative in general, by a construction of…

For each odd integer r greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise non-homeomorphic closed homogeneous spaces with fundamental group…

In 1989, D. Happel pointed out for a possible connection between the global dimension of a finite-dimensional algebra and its Hochschild cohomology: is it true that the vanishing of Hochschild cohomology higher groups is sufficient to…