Related papers: Conics in the Grothendieck ring

Polytopes in R^n with integral vertices form a monoid under the Minkowski sum, and the Grothendieck construction gives rise to a group. We show that every symmetric polytope is a norm in this group for every n.

We compute the Grothendieck group K_0 of non-commutative analogues of quantum projective space bundles. Our results specialize to give the Grothendieck groups of non-commutative analogues of projective spaces, and specialize to recover the…

Though the Chow group of 0-cycles on a K3 surface is quite large, we observe that the subgroup generated by product of divisors is cyclic.

We prove the conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of smooth cubic surfaces. The proof is based on a toric…

Recent discoveries make it possible to compute the K-theory of certain rings from their cyclic homology and certain versions of their cdh-cohomology. We extend the work of G. Corti\~nas et al. who calculated the K-theory of, in addition to…

We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a…

In this expository article, we prove a birational classification of smooth projective models of surfaces with negative Kodaira dimension over $\mathbb{Z}$ and over more general rings of integers $\mathcal{O}_K$, depending on their…

This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable $\infty$-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups…

It is a question by C.Sormani that whether there exists a $k \in \mathbb N$, such that any compact, smooth and simply connected manifold has a 1/k-geodesic. We prove in this paper that this is not true by showing for each $k$, there exists…

We give a direct proof of the following known result: the Grothendieck group of a triangulated category with a silting subcategory is isomorphic to the split Grothendieck group of the silting subcategory. Moreover, we obtain its…

The Cox ring provides a coordinate system on a toric variety analogous to the homogeneous coordinate ring of projective space. Rational maps between projective spaces are described using polynomials in the coordinate ring, and we generalise…

In this note, we work out a simple inductive proof showing that every polyhedral cone K is the conic hull of a finite set X of vectors. The base cases of the induction are linear subspaces and linear halfspaces of linear subspaces. The…

Let $G$ be a finite group, $X$ be a smooth complex projective variety with a faithful $G$-action, and $Y$ be a resolution of singularities of $X/G$. Larsen and Lunts asked whether $[X/G]-[Y]$ is divisible by $[\mathbb{A}^1]$ in the…

In this article we prove several important results on graded rings, especially monoid-rings, that are motivated and inspired by Kaplansky's zero-divisor, unit and idempotents conjectures. Among the main results, we first generalize…

We establish an analogue of the Grothendieck inequality where the rectangular matrix is replaced by a symmetric/Hermitian matrix and the bilinear form by a quadratic form. We call this the symmetric Grothendieck inequality; despite its…

If $G$ is a finite group, the Grothendieck group ${\mathbf{K}}\_G(G)$ of the category of $G$-equivariant ${\mathbb{C}}$-vector bundles on $G$ (for the action of $G$ on itself by conjugation) is endowed with a structure of (commutative)…

In this paper we use families of finite subgroups to study Grothendieck rings associated to certain discrete groups, such as the arithmetic ones.

We compute the integral Grothendieck rings of the moduli stacks, $\mathcal{M}_2$, $\overline{\mathcal{M}}_2$ of smooth and stable curves of genus two respectively. We compute $K_0(\mathcal{M}_2)$ by using the presentation of $\mathcal{M}_2$…

A conjecture by Corti, Filip and Petracci, inspired by mirror symmetry, states that smoothing types of affine Gorenstein toric 3-folds correspond to zero mutable Laurent polynomials. We propose a method to prove this conjecture via log…

We consider correspondences on smooth quasiprojective varieties $U$. An algebraic cycle inducing the K\"unneth projector onto $H^1(U)$ is constructed. Assuming normal crossings at infinity, the existence of relative motivic cohomology is…