We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal compactifications of the moduli space.

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We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs (X, R) over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.

BackgroundIt is important to generate biofuels and society must be weaned from its dependency on fossil fuels. In order to produce biofuels, lignocellulose is pretreated and the resulting cellulose is hydrolyzed by cellulases such as cellobiohydrolases (CBH) and endoglucanases (EG). Until now, the biofuel industry has usually applied impractical celluloses to screen for cellulases capable of degrading naturally occurring, insoluble cellulose. This study investigates how these cellulases adsorb and hydrolyze insoluble α-cellulose − considered to be a more practical substrate which mimics the alkaline-pretreated biomass used in biorefineries. Moreover, this study investigates how hydrodynamics affects cellulase adsorption and activity onto α-cellulose.ResultsFirst, the cellulases CBH I, CBH II, EG I and EG II were purified from Trichoderma reesei and CBH I and EG I were utilized in order to study and model the adsorption isotherms (Langmuir) and kinetics (pseudo-first-order). Second, the adsorption kinetics and cellulase activities were studied under different hydrodynamic conditions, including liquid mixing and particle suspension. Third, in order to compare α-cellulose with three typically used celluloses, the exact cellulase activities towards all four substrates were measured.It was found that, using α-cellulose, the adsorption models fitted to the experimental data and yielded parameters comparable to those for filter paper. Moreover, it was determined that higher shaking frequencies clearly improved the adsorption of cellulases onto α-cellulose and thus bolstered their activity. Complete suspension of α-cellulose particles was the optimal operating condition in order to ensure efficient cellulase adsorption and activity. Finally, all four purified cellulases displayed comparable activities only on insoluble α-cellulose.Conclusionsα-Cellulose is an excellent substrate to screen for CBHs and EGs. This current investigation shows in detail, for the first time, the adsorption of purified cellulases onto α-cellulose, the effect of hydrodynamics on cellulase adsorption and the correlation between the adsorption and the activity of cellulases at different hydrodynamic conditions. Complete suspension of the substrate has to be ensured in order to optimize the cellulase attack. In the future, screenings should be conducted with α-cellulose so that proper cellulases are selected to best hydrolyze the real alkaline-pretreated biomass used in biorefineries.

Let F be a moduli space of lattice-polarized K3 surfaces and suppose that one has chosen a canonical effective ample divisor on a general fiber. We call this divisor "recognizable" if its flat limit on Kulikov surfaces is well defined. We prove that the normalization of the stable pair compactification F R for a recognizable divisor R is semitoroidal.We also prove that for polarized K3 surfaces (X, L) of degree 2d the sum R of rational curves in the linear system |L| is a recognizable divisor, giving a modular semitoroidal compactification of F 2d for all d.