arXiv subjects

Dennis Presotto

Publications and source records attributed to Dennis Presotto.

7 recordsLinked to original sources

arXiv2018

Construction of noncommutative surfaces with exceptional collections of length 4

Recently de Thanhoffer de Völcsey and Van den Bergh classified the Euler forms on a free abelian group of rank 4 having the properties of the Euler form of a smooth projective surface. There are two types of solutions: one corresponding to $\mathbb{P}^1\times\mathbb{P}^1$ (and noncommutative quadrics), and an infinite family indexed by the natural numbers. For $m=0,1$ there are commutative and noncommutative surfaces having this Euler form, whilst for $m\geq 2$ there are no commutative surfaces. In this paper we construct sheaves of maximal orders on surfaces having these Euler forms, giving a geometric construction for their numerical blowups.

math.AG↗

arXiv2018

Comparison of two constructions of noncommutative surfaces with exceptional collections of length 4

Recently the Euler forms on numerical Grothendieck groups of rank 4 whose properties mimick that of the Euler form of a smooth projective surface have been classified. This classification depends on a natural number $m$, and suggests the existence of noncommutative surfaces which up to that point had not been considered for $m\geq 2$. These have been constructed for $m=2$ using noncommutative $\mathbb{P}^1$-bundles, and for all $m\geq 2$ by a different construction using maximal orders on $\mathrm{Bl}_x\mathbb{P}^2$. In this article we compare the constructions for $m=2$, i.e. we compare the categories arising from half-ruled del Pezzo quaternion orders on $\mathbb{F}_1$ with noncommutative $\mathbb{P}^1$-bundles on $\mathbb{P}^1$. This can be seen as a noncommutative instance of the classical isomorphism $\mathbb{F}_1\cong\mathrm{Bl}_x\mathbb{P}^2$.

math.AG↗

arXiv2017

$\mathbb{Z}^2$-algebras as noncommutative blow-ups

The goal of this note is to first prove that for a well behaved $\mathbb{Z}^2$-algebra $R$, the category $QGr(R) := Gr(R)/Tors(R)$ is equivalent to $QGr(R_Δ)$ where $R_Δ$ is a diagonal-like sub-$\mathbb{Z}$-algebra of $R$. Afterwards we use this result to prove that the $\mathbb{Z}^2$-algebras as introduced in [ArXiV:1607.08383] are QGr-equivalent to a diagonal-like sub-$\mathbb{Z}$-algebra which is a simultaneous noncommutative blow-up of a quadratic and a cubic Sklyanin algebra. As such we link the noncommutative birational transformation and the associated $\mathbb{Z}^2$-algebras as appearing in the work of Van den Bergh and Presotto with the noncommutative blowups appearing in the work of Rogalski, Sierra and Stafford.

math.AG↗

arXiv2016

Symmetric noncommutative birational transformations

In a previous paper (arXiv:1410.5207) certain birational transformations were constructed between the noncommutative schemes associated to quadratic and cubic three dimensional Sklyanin algebras. In the current paper we consider the inverse birational transformations and show that they are of the same type. Moreover we extend everything to the $\mathbb{Z}$-algebras context, which allows us to incorporate the noncommutative quadrics introduced by Van den Bergh.

math.AG↗

arXiv2016

Homological properties of a certain noncommutative Del Pezzo surface

Recently, de Thanhoffer de Volcsey and Van den Bergh showed that Grothendieck groups of "noncommutative Del Pezzo surfaces" with an exceptional sequence of length 4 are isomorphic to one of three types, the third one not coming from a commutative Del Pezzo surface. In this paper, we adapt the theory of noncommutative $\mathbb{P}^1$-bundles as appearing in the work of Van den Bergh and Nyman to produce a sheaf $\mathbb{Z}$-algebra whose associated Proj has an exceptional sequence of length 4 for which the Gram matrix is of this third type. We show that this noncommutative scheme is noetherian and describe its local structure through the use of our generalized preprojective algebras.

math.AG↗

arXiv2015

Some generalizations of Preprojective algebras and their properties

In this note we consider a notion of relative Frobenius pairs of commutative rings $S/R$. To such a pair, we associate an $\mathbb{N}$-graded $R$-algebra $Π_R(S)$ which has a simple description and coincides with the preprojective algebra of a quiver with a single central node and several outgoing edges in the split case. If the rank of $S$ over $R$ is 4 and $R$ is noetherian, we prove that $Π_R(S)$ is itself noetherian and finite over its center and that each $Π_R(S)_d$ is finitely generated projective. We also prove that $Π_R(S)$ is of finite global dimension if $R$ and $S$ are regular.

math.RA↗

arXiv2015

Noncommutative versions of some classical birational transformations

In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct and analogue of the Cremona transform for non-commutative projective planes.

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