Moduli spaces of twisted K3 surfaces and cubic fourfolds

Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of untwisted polarized K3 surfaces as a quotient of a bounded symmetric domain.Comment: 23 page

Moduli spaces of K3 surfaces and cubic fourfolds

This thesis is concerned with the Hodge-theoretic relation between polarized K3 surfaces of degree d and special cubic fourfolds of discriminant d, as introduced by Hassett. For half of the d, K3 surfaces associated to cubic fourfolds come naturally in pairs. As our first main result, we prove that if (S,L) and (St,Lt) form such a pair of polarized K3 surfaces, then St is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3,L,d/6). We also explain for which d the Hilbert schemes Hilbn(S) and Hilbn(St) are birational. Next, we study the more general concept of associated twisted K3 surfaces. Our main contribution here is the construction of moduli spaces of polarized twisted K3 surfaces of fixed degree and order. We strengthen a theorem of Huybrechts about the existence of associated twisted K3 surfaces. We show that like in the untwisted situation, half of the time, associated twisted K3 surfaces come in pairs, and we explain how the elements of such a pair are related to each other

Kodaira dimension of moduli spaces of hyperk\"ahler varieties

We study the Kodaira dimension of moduli spaces of polarized hyperk\"ahler varieties deformation equivalent to the Hilbert scheme of points on a K3 surface or O'Grady's ten dimensional variety. This question was studied by Gritsenko-Hulek-Sankaran in the cases of $K3^{[2]}$ and OG10 type when the divisibility of the polarization is one. We generalize their results to higher dimension and divisibility. As a main result, for almost all dimensions $2n$ we provide a lower bound on the degree such that for all higher degrees, every component of the moduli space of polarized hyperk\"ahler varieties of $K3^{[n]}$ type is of general type.Comment: 56 pages. Comments are welcome

Numerical Solutions of 2-D Steady Incompressible Flow in a Driven Skewed Cavity

The benchmark test case for non-orthogonal grid mesh, the "driven skewed cavity flow", first introduced by Demirdzic et al. (1992, IJNMF, 15, 329) for skew angles of alpha=30 and alpha=45, is reintroduced with a more variety of skew angles. The benchmark problem has non-orthogonal, skewed grid mesh with skew angle (alpha). The governing 2-D steady incompressible Navier-Stokes equations in general curvilinear coordinates are solved for the solution of driven skewed cavity flow with non-orthogonal grid mesh using a numerical method which is efficient and stable even at extreme skew angles. Highly accurate numerical solutions of the driven skewed cavity flow, solved using a fine grid (512x512) mesh, are presented for Reynolds number of 100 and 1000 for skew angles ranging between 15<alpha<165

Melanocortin Receptor 4 Deficiency Affects Body Weight Regulation, Grooming Behavior, and Substrate Preference in the Rat

Obesity is caused by an imbalance between energy intake and expenditure and has become a major health-care problem in western society. The central melanocortin system plays a crucial role in the regulation of feeding and energy expenditure, and functional loss of melanocortin receptor 4 (MC4R) is the most common genetic cause of human obesity. In this study, we present the first functional Mc4r knockout model in the rat, resulting from an N-ethyl-N-nitrosourea mutagenesis–induced point mutation. In vitro observations revealed impaired membrane-binding and subsequent nonfunctionality of the receptor, whereas in vivo observations showed that functional loss of MC4R increased body weight, food intake, white adipose mass, and changed substrate preference. In addition, intracerebroventricular (ICV) administration of Agouti-Related Protein79–129 (AgRP79–129), an MC4R inverse agonist, or Melanotan-II (MTII), an MC4R agonist, did affect feeding behavior in wild-type rats but not in homozygous mutant rats, confirming complete loss of MC4R function in vivo. Finally, ICV administration of MTII induced excessive grooming behavior in wild-type rats, whereas this effect was absent in homozygous mutant rats, indicating that MTII-induced grooming behavior is exclusively regulated via MC4R pathways. Taken together, we expect that the MC4R rat model described here will be a valuable tool for studying monogenic obesity in humans. More specifically, the relative big size and increased cognitive capacity of rats as compared to mice will facilitate complex behavioral studies and detailed mechanistic studies regarding central function of MC4R, both of which ultimately may help to further understand the specific mechanisms that induce obesity during loss of MC4R function