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

Sural Hypersensitivity after Nerve Transection depends on Anatomical Differences in the Distal Tibial Nerve of Mice and Rats

INTRODUCTION: Various mouse and rat models of neuropathic pain after nerve injury exist. Whilst some models involve a proximal nerve lesion or ligation of the sciatic trifurcation in mice and rats, others consists of a transection or ligation of distal nerves at the tibial bifurcation in mice or rats. The level of nerve cut directly affects the magnitude of hypersensitivity, and anatomical differences between mice and rats might therefore impact the development of hypersensitivity after distal tibial nerve transection as well. METHODS: The bifurcation of the distal tibial nerve into the medial and lateral plantar nerve (MPN and LPN), and the presence of anatomical differences in sural and tibial nerve distribution between mice and rat was evaluated. Sural mechanical sensitivity after transection of the MPN or whole tibial nerve was assessed using von Frey test until 8 weeks after surgery in 48 rats and 16 mice. RESULTS: The bifurcation of the tibial nerve into the MPN and LPN is situated proximal to the ankle in both mice and rats. The sural nerve joins the LPN in mice, but not in rats. A proximal communicating branch is present between the LPN and MPN in rats, but not in mice. MPN transection in mice caused hypersensitivity of the hindpaw innervated by the sural nerve, but not in rats. In rats, sural hypersensitivity only developed when both MPN and LPN were cut. CONCLUSION: Inter-species variation in nerve anatomy should be taken in consideration when performing surgery to induce plantar hypersensitivity in rodents