Related papers: Totaro's Question on Zero-Cycles on Torsors

We prove that there exist smooth surfaces of degree d in projective 3-space such that the group of rational equivalence classes of decomposable 0-cycles has rank at least the integer part of (d-1)/3.

We show that given a smooth projective variety X over C with dim(X) > 2, an ample line bundle O(1) on X and an integer n > 1, any n distinct points on a generic hypersurface of degree d in X are linearly independent in CH_0(X) if d >> 0.…

The aim of this paper is to study the group of isomorphism classes of torsors of finite flat group schemes of rank 2 over a commutative ring $R$. This, in particular, generalises the group of quadratic algebras (free or projective), which…

We give a functorial construction of the genus zero chiral algebras of class $\mathcal{S}$, that is, the vertex algebras corresponding to the theory of class $\mathcal{S}$ associated with genus zero pointed Riemann surfaces via the 4d/2d…

Let $ Y \subseteq \Bbb P^N $ be a possibly singular projective variety, defined over the field of complex numbers. Let $X$ be the intersection of $Y$ with $h$ general hypersurfaces of sufficiently large degrees. Let $d>0$ be an integer, and…

The Ceresa cycle is an algebraic 1-cycle on the Jacobian of an algebraic curve. Although it is homologically trivial, Ceresa famously proved that for a very general complex curve of genus at least 3, it is non-trivial in the Chow group. In…

We prove that all points of a toroidal compactification lying over 0-dimensional cusps are rationally equivalent in the integral Chow group for most classical modular varieties (Siegel, Hilbert, orthogonal, Hermitian, quaternionic). This…

From the viewpoint of the division by zero $(0/0=1/0=z/0=0)$ and the division by zero calculus, we will show that in the very beautiful theorem by Descartes on three touching circles is valid for lines and points for circles except for one…

Finding necessary conditions for the geometry of flexible polyhedra is a classical problem that has received attention also in recent times. For flexible polyhedra with triangular faces, we showed in a previous work the existence of cycles…

This note is about an old conjecture of Voisin, which concerns zero--cycles on the self-product of surfaces of geometric genus one. We prove this conjecture for surfaces with $p_g=1$ and $q=2$.

We show that for each \ell\geq 4 every sufficiently large oriented graph G with \delta^+(G), \delta^-(G) \geq \lfloor |G|/3 \rfloor +1 contains an \ell-cycle. This is best possible for all those \ell\geq 4 which are not divisible by 3.…

Let $n_{1}$ and $n_{2}$ be two integers with $n_{1},n_{2}\geq3$ and $G$ a graph of order $n=n_{1}+n_{2}$. As a generalization of Ore's degree condition for the existence of Hamilton cycle in $G$, El-Zahar proved that if $\delta(G)\geq…

In this paper, we consider a general degree sum condition sufficient to imply the existence of $k$ vertex-disjoint chorded cycles in a graph $G$. Let $\sigma_t(G)$ be the minimum degree sum of $t$ independent vertices of $G$. We prove that…

Suppose $X$ is a smooth affine real variety and $\mathscr{E}$ is a vector bundle over $X$. We analyze the problem of splitting off a free rank one summand from $\mathscr{E}$ in corank $0$ and $1$. The problem in corank $0$ can be viewed as…

In these notes, we consider self-maps of degree > 1 on a weak del Pezzo surface X of degree < 8. We show that there are exactly 12 such X, modulo isomorphism. In particular, K_X^2 > 2, and if X has one self-map of degree > 1 then for every…

We propose a new definition of Koszulity for graded algebras where the degree zero part has finite global dimension, but is not necessarily semi-simple. The standard Koszul duality theorems hold in this setting. We give an application to…

For a vertex subset $X$ of a graph $G$, let $\Delta_{t}(X)$ be the maximum value of the degree sums of the subsets of $X$ of size $t$. In this paper, we prove the following result: Let $k$ be a positive integer, and let $G$ be an…

Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n \gg 1$ is solved in the affirmative.…

Comparing the module categories of an algebra and of the endomorphism algebra of a given support $\tau$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $\tau$-tilting theory. Afterwards…

A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic…