We construct the moduli of stable supermaps as an algebraic superstack with superschematic and separated diagonal. We prove that its bosonic reduction is an affine linear scheme over the stack of stable spin maps, so that it is not proper except in a few particular cases. We also compute the virtual dimension of the moduli superstack of stable supermaps, and prove that it coincides with...
In this talk, I shall survey three papers devoted to the study of moduli spaces of framed sheaves on Hirzebruch surfaces, two of which were written in collaboration with Ugo Bruzzo. These contributions pursue a common objective: the construction of a quiver-theoretic description of such moduli spaces, starting from the monadic description previously established by Bartocci, Bruzzo, and Rava....
We consider a Higgs bundle $(E, \phi)$. Its Higgs Grassmiannans are subschemes of the usual Grassmannian bundles of $E$ that parameterise Higgs quotients of $(E, \phi)$. We recall how to define them, present some results about their structure, and explain how they can be used to prove some results about Higgs bundles satisfying a strong semistability condition.
In this talk we realize the nested Hilbert scheme of points on affine spaces and varieties as quiver varieties. In addition, we provide a schematic construction to a set-theoretical result concerning the nested Hilbert schemes of points on $\mathbb A^2$ with quotients supported on curves, provided by Santos; we compute interesting examples and an explicit formula for the tangent space to these...
We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers $\mathbb{Z}[t]/(t^2)$. For a compact quantum space, we give sufficient conditions that guarantee there is a morphism of abelian groups from $K_0 \to \mathbb{Z}[t]/(t^2)$, compatible with the tensor product of bimodules.
Applications include the standard quantum sphere $S^2_q$ and a quantum...
In 2012, Bruzzo and Grassi proved a Noether-Lefschetz theorem for toric varieties, which claims that for a (2k+1)-dimensional projective toric orbifold with suitable conditions on a very general quasi-smooth hypersurface $X$, each $(k,k)$-cohomology class on $X$ comes from the ambient toric variety. The Noether-Lefschetz locus is the locus of quasi-smooth hypersurfaces with the same degree...
In this talk, I will address the following question, attributed to Gizatullin: ``Which automorphisms of a smooth quartic surface in projective 3-space are restrictions of Cremona transformations of the ambient space?'' Corti and Kaloghiros have introduced a general framework that is extremely useful for approaching this problem, namely, a special version of the Sarkisov program for Calabi-Yau...
We study $G$-graded Artinian algebras having Poincaré duality and their Lefschetz properties. We prove the equivalence between the toric setup and the $G$-graded one. We prove a Hessian criterion in the $G$-graded setup. We provide an application to toric geometry.
Enumerative geometry, as formulated in Gromov--Witten theory, encodes curve-counting information on smooth projective varieties. Such data can be organized in different ways, giving rise to rich geometric structures and invariants, including quantum cohomology and quantum spectra. In the work \emph{G.~Cotti, ``Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution...
In this talk, we will introduce basic concepts of superalgebraic geometry and explore why certain classical foundational results, such as the Birkhoff-Grothendieck splitting criterion, do not extend naturally to the supergeometric setting. We then present a splitting criterion for supervector bundles and give some examples of supervector bundles with vanishing cohomology that do not...
In this talk, we briefly introduce stability conditions, which we apply to the category of coherent systems on an integral curve $C$.
We define Bridgeland stability conditions on its derived category. We also study the semistability of certain objects with respect to these conditions. We use some results we got to address the problem of finding bounds for the dimension of the space of global...
The Hilbert scheme of points on a quasi-projective variety is a classical object in algebraic geometry. However, its geometry is nowadays still not completely accessible. On the other hand, the motive of a variety $X$ is an invariant attached to $X$ carrying a lot of information about its geometry, and it is considered as a universal Euler characteristic. In a joint project with Monavari,...
In 2020, Sagan and Tirrell introduced Lucas atoms, which are irreducible factors of Lucas polynomials. Their main goal was to investigate when certain combinatorial rational functions are actually polynomials. In a joint work with Miska, Murru, and Romeo, we present Lucas atoms in a more natural way than the original definition, providing straightforward proofs of their main properties....
Let $G$ be a finite abelian subgroup of $\mathsf{SL}(n, \mathbb{C})$, and suppose there exists a toric crepant resolution $ \phi: X \longrightarrow \mathbb{C}^n / G$ of the quotient variety $\mathbb{C}^n / G$. Let $\mathsf{Exc}(\phi) = E_1 \cup \dots \cup E_s$ be the decomposition of the exceptional set of $\phi$ into irreducible components. In this seminar, I will show that for every $i$ ...
In 1996, David Cox proved that for a given projective toric variety of dimension $n$, its homogeneous coordinate ring modulo $n+1$ forms with the same ample degree, that do not vanish simultaneously, must have dimension one in the component of the critical degree of the forms.
This result, known as the Codimension One Theorem, was generalized by Cattani-Cox-Dickenstein and even further by...
Being Kähler imposes severe constraints on the cohomology of compact complex manifolds such as the Hard Lefschetz property, and the question of how far this generalises beyond the class of Kähler manifolds has been of great interest for a while. In this talk, I shall report on ongoing joint work with Mario García Fernández and Raúl González Molina that abstracts out the definition of a...
Let $X$ be a smooth, irreducible, projective surface. A coherent system on $X$ is a pair $(E, V)$ where $E$ is a coherent sheaf on $X$ and $V$ is a finite-dimensional vector space. Associated to coherent systems there is a notion of stability that depends on a parameter $\alpha \in \mathbb{Q}[m]$. In this talk, we describe the moduli space of coherent systems for $\alpha \gg 0$, present...
will review some results on Donagi-Markman cubics (infinitesimal period maps) for the pure and generalised Hitchin system. I will discuss how these fit into the context of special Kaehler geometry, and also will discuss some work in progress. Joint with Ugo Bruzzo.
