arXiv subjects
Publications and source records attributed to Julius Giesler.
We study degenerations (along an edge) of surfaces in (fake) weighted projective $3$-space. We introduce the Hodge theoretic data and show that the Hodge loci of vanishing cycles both might be proper or not. This illustrates results of Green and Voisin.
In this article we deal with jacobian rings and identify a mixed Hodge component of a nondegenerate hypersurface in the torus with a lattice geometric quotient vector space. We introduce a period map, study its differential and compute the kernel of the differential much explicitly via certain Laurent polynomials. As a main application we deal with the infinitesimal Torelli theorem (ITT) for such explicit deformations. We study the kernel of the cohomological map for explicit deformations and complete the ITT by dealing with the remaining part $\coker(\kappa_{\mathbb{P},f})$ (cokernel of the Kodaira-Spencer map) in dimensions $n \geq 4$.
In this article we study infinitesimal deformations of toric hypersurfaces. We introduce a Kodaira-Spencer map and compute its kernel. By introducing some new Laurent polynomials we make our computation as explicit as possible. This widely generalizes results of Griffiths for projective hypersurfaces.
In this article we present a formula for the plurigenera of minimal models of nondegenerate toric hypersurfaces, which is valid in arbitrary dimension and which expresses these invariants through lattice points on the Fine interior. From this formula we derive a formula for the volume of the canonical divisor of the toric hypersurface. We also study the pluricanonical mappings and show under some restrictions how to construct a canonical model of the toric hypersurface.
In the first part of this article we show for some examples of surfaces of general type in toric 3-folds how to construct minimal and canonical models by toric methods explicitly. The examples we study turn out to be surfaces of general type, namely so called Kanev surfaces and Todorov surfaces. We show how properties of our examples of surfaces could be derived directly from properties of some polytopes and we compute the singularities of their canonical models.