arXiv subjects

Amal Mattoo

Publications and source records attributed to Amal Mattoo.

5 recordsLinked to original sources

arXiv2025

Objects of a Phantom on a Rational Surface

Johannes Krah showed that the blowup of $\mathbf{P}^{2}$ in $10$ general points admits a phantom subcategory. We construct three types of objects in such a phantom: a strong generator, projections of skyscraper sheaves, and a family of objects with two nonzero cohomology sheaves. We study the deformation theory of these objects to show that the phantom contains rich geometry, such as encoding the blowdown map to $\mathbf{P}^{2}$. We also show that there exists a co-connective dg-algebra whose derived category is a phantom.

math.AG↗

arXiv2024

Non-smoothness of Moduli Spaces of Higher Genus Curves on Low Degree Hypersurfaces

We show that the moduli space of degree $e$ maps from smooth genus $g \ge 1$ curves to an arbitrary low degree smooth hypersurface is singular when $e$ is large compared to $g$. We also give a lower bound for the dimension of the singular locus.

math.AG↗

arXiv2023

A Counterexample to a Question on Grothendieck Groups of Schemes

If an element of the Grothendieck group of the derived category of a scheme is locally represented by perfect complexes, then can the original element be represented by a perfect complex? We provide a counterexample on a projective variety of dimension 2, as well as a counterexample on a thickening of a Dedekind domain.

math.AG↗

arXiv2021

Saturation of Newton polytopes of type A and D cluster variables

We study Newton polytopes for cluster variables in cluster algebras $\mathcal{A}(Σ)$ of types A and D. A famous property of cluster algebras is the Laurent phenomenon: each cluster variable can be written as a Laurent polynomial in the cluster variables of the initial seed $Σ$. The cluster variable Newton polytopes are the Newton polytopes of these Laurent polynomials. We show that if $Σ$ has principal coefficients or boundary frozen variables, then all cluster variable Newton polytopes are saturated. We also characterize when these Newton polytopes are \emph{empty}; that is, when they have no non-vertex lattice points.

math.CO↗

arXiv2020

Virtual Complete Intersections in $\mathbb{P}^1 \times \mathbb{P}^1$

The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$, we investigate which sets of points have a virtual resolution that is a Koszul complex on a regular sequence. This paper provides conditions on sets of points; some of which guarantee the points have this property, and some of which guarantee the points do not have this property.

math.AG↗