Punctual Hilbert schemes for Kleinian singularities as quiver varieties

For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/\Gamma$ as a Nakajima quiver variety for the framed McKay quiver of $\Gamma$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of stability parameter.Comment: 24 pages. v3: Definition of cornered algebras simplified. To be published in Algebraic Geometr

Putative Biomarkers for Acute Pulmonary Embolism in Exhaled Breath Condensate

Current diagnostic markers for pulmonary embolism (PE) are unspecific. We investigated the proteome of the exhaled breath condensate (EBC) in a porcine model of acute PE in order to identify putative diagnostic markers for PE. EBC was collected at baseline and after the induction of autologous intermediate-risk PE in 14 pigs, plus four negative control pigs. The protein profiles of the EBC were analyzed using label-free quantitative nano liquid chromatography–tandem mass spectrometry. A total of 897 proteins were identified in the EBCs from the pigs. Alterations were found in the levels of 145 different proteins after PE compared with the baseline and negative controls: albumin was among the most upregulated proteins, with 14-fold higher levels 2.5 h after PE (p-value: 0.02). The levels of 49 other proteins were between 1.3- and 17.1-fold higher after PE. The levels of 95 proteins were lower after PE. Neutrophil gelatinase-associated lipocalin (fold change 0.3, p-value < 0.01) was among the most reduced proteins 2.5 h after PE. A prediction model based on penalized regression identified five proteins including albumin and neutrophil gelatinase-associated lipocalin. The model was capable of discriminating baseline samples from EBC samples collected 2.5 h after PE correctly in 22 out of 27 samples. In conclusion, the EBC from pigs with acute PE contained several putative diagnostic markers of PE

Cubic hypersurfaces, their Fano schemes, and special subvarieties

This thesis investigates cubic hypersurfaces and their Fano schemes. After introducing the Fano schemes through low-dimensional examples, we move on to investigate cubic fourfolds. For the cubic fourfolds, we give complete proofs of some statements of Beauville-Donagi and Amerik. Following Beauville-Donagi, we introduce the Abel-Jacobi map. Together with its transpose, we use this to investigate cubic fourfolds and their varieties of lines. This is used to -among other things- to prove the Hodge conjecture for cubic fourfolds. For some cubics, we are able to prove the Integral Hodge conjecture. We also investigate linear subspaces of varieties. Here we generalize the techniques of Clemens and Griffiths, which leads to characterizations of linear spaces tangent to hypersurfaces. We continue by investigating the Eckardt points on cubic hypersurfaces. Studying these points is not a new idea, but our approach focusing on the lines through an Eckardt point is, -as far as we know- novel. We give several other characterizations of these points, and show that they influence whether a cubic fourfold is rational. Following this, we investigate some highly special cubic fourfolds, such as the Fermat cubic. We prove the Hodge conjecture for their Fano schemes. The second-to-last chapter introduces special cubic fourfolds, following the classification of Hassett . We describe some of the divisors in the moduli space of cubic fourfolds explicitly. These investigations lead us to answer a question raised by Nuer on the existence of smooth rational surfaces in cubic fourfolds. The chapter continues by discussing the effective and nef cones of 2-cycles on special cubic fourfolds. We give a new and complete description of their cones for fourfolds containing a plane. Some conjectures of Hassett and Tschinkel lead us to investigate the cones of nef cycles on their Fano schemes, and we fill in some details of their paper. The final chapter deviates from the theme. It is a vast generalization of our analysis of cubic fourfolds containing a plane. We give a complete description of the cones of effective and nef m-cycles for hypersurfaces of dimension 2m of sufficiently large degree. In certain cases, toric geometry leads to improved results. This result is, as far as we know, new

Quot schemes for Kleinian orbifolds

For a finite subgroup $\Gamma\subset {\mathrm{SL}}(2,\mathbb{C})$, we identify fine moduli spaces of certain cornered quiver algebras, defined in earlier work, with orbifold Quot schemes for the Kleinian orbifold $[\mathbb{C}^2/\Gamma]$. We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of $\Gamma$, taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. Our results generalise our previous work on the Hilbert scheme of points on $\mathbb{C}^2/\Gamma$; we present arguments that completely bypass the ADE classification.Comment: Inaccuracy correcte