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1. The integral cohomology of the Hilbert scheme of points on a surface

   https://doi.org/10.1017/fms.2020.35  

   Totaro, Burt (January 2020, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $$X^{[n]}$$ has torsion-free cohomology for every natural number n . This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces. 

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2. On birational boundedness of foliated surfaces

   https://doi.org/10.1515/crelle-2020-0009  

   Hacon, Christopher D.; Langer, Adrian (January 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}} , then there exists an integer {N>0} such that if {(X,{\mathcal{F}})} is a canonical or nef model of a foliation of general type with Hilbert polynomial {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all {m\in\mathbb{Z}_{\geq 0}} , then {|mK_{\mathcal{F}}|} defines a birational map for all {m\geq N} . On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities. 

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3. Vapor-phase grafting of functional silanes on atomic layer deposited Al2O3

   https://doi.org/10.1116/6.0002364  

   Rozyyev, Vepa; Shevate, Rahul; Pathak, Rajesh; Murphy, Julia G.; Mane, Anil U.; Sibener, S. J.; Elam, Jeffrey W. (May 2023, Journal of Vacuum Science & Technology A)

   Fundamental studies are needed to advance our understanding of selective adsorption in aqueous environments and develop more effective sorbents and filters for water treatment. Vapor-phase grafting of functional silanes is an effective method to prepare well-defined surfaces to study selective adsorption. In this investigation, we perform vapor phase grafting of five different silane compounds on aluminum oxide (Al2O3) surfaces prepared by atomic layer deposition. These silane compounds have the general formula L3Si–C3H6–X where the ligand, L, controls the reactivity with the hydroxylated Al2O3 surface and the functional moiety, X, dictates the surface properties of the grafted layer. We study the grafting process using in situ Fourier transform infrared spectroscopy and ex situ x-ray photoelectron spectroscopy measurements, and we characterize the surfaces using scanning electron microscopy, atomic force microscopy, and water contact angle measurements. We found that the structure and density of grafted aminosilanes are influenced by their chemical reactivity and steric constraints around the silicon atom as well as by the nature of the anchoring functional groups. Methyl substituted aminosilanes yielded more hydrophobic surfaces with a higher surface density at higher grafting temperatures. Thiol and nitrile terminated silanes were also studied and compared to the aminosilane terminated surfaces. Uniform monolayer coatings were observed for ethoxy-based silanes, but chlorosilanes exhibited nonuniform coatings as verified by atomic force microscopy measurements. 

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4. On the arithmetic of a family of degree - two K3 surfaces

   https://doi.org/10.1017/S0305004118000087  

   BOUYER, FLORIAN; COSTA, EDGAR; FESTI, DINO; NICHOLLS, CHRISTOPHER; WEST, MCKENZIE (May 2019, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x , y , z and w ; let $$\mathcal{X}$$ be the generic element of the family of surfaces in ℙ given by \begin{equation*}X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2.\end{equation*} The surface $$\mathcal{X}$$ is a K3 surface over the function field ℚ( t ). In this paper, we explicitly compute the geometric Picard lattice of $$\mathcal{X}$$ , together with its Galois module structure, as well as derive more results on the arithmetic of $$\mathcal{X}$$ and other elements of the family X . 

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5. The Flex Divisor of a K3 Surface

   https://doi.org/10.1093/imrn/rnac089  

   Alexeev, Valery; Engel, Philip (May 2022, International Mathematics Research Notices)

   Abstract The flex divisor$$R_{\textrm flex}$$ of a primitively polarized K3 surface $(X,L)$ is, generically, the set of all points $$x\in X$$ for which there exists a pencil $$V\subset |L|$$ whose base locus is $$\{x\}$$. We show that if $L^2=2d$ then $$R_{\textrm flex}\in |n_dL|$$ with $$ \begin{align*} &n_d= \frac{(2d)!(2d+1)!}{d!^2(d+1)!^2} =(2d+1)C(d)^2,\end{align*}$$where $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $$F_{2d}$$ of polarized K3 surfaces. 

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