On deformations of quintic and septic hypersurfaces

An old question of Mori asks whether in dimension at least three, any smooth specialization of a hypersurface of prime degree is again a hypersurface. A positive answer to this question is only known in degrees two and three. In this paper, we settle the case of quintic hypersurfaces (in arbitrary dimension) as well as the case of septics in dimension three. Our results follow from numerical characterizations of the corresponding hypersurfaces. In the case of quintics, this extends famous work of Horikawa who analysed deformations of quintic surfaces.Comment: 23 pages, final version, to appear in Journal de Math\'ematiques Pures et Appliqu\'ee

Positivity of the diagonal

We study how the geometry of a projective variety $X$ is reflected in the positivity properties of the diagonal $\Delta_X$ considered as a cycle on $X \times X$. We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of $X$. For example, when the diagonal is big, we prove that the Hodge groups $H^{k,0}(X)$ vanish for $k>0$. We also classify varieties of low dimension where the diagonal is nef and big.Comment: 23 pages; v2: updated attributions and minor change

Crime and the Quality of Life in Wisconsin Counties

The impact of crime on the local quality of life of a region is examined. Using the methods suggested by Roback (1982) hedonic pricing analysis is used to examine the effects of eight categories of crime on property values and wages. The hedonic results are then used to calculate the implicit prices of the various types of crime. Prices are computed for both urban and rural areas reflecting differences in lifestyle and the corresponding impact of crime. As expected, crime has a measurable negative cost and lowers overall quality of life in a region and the level of impact varies significantly by type of crime.

Ample subvarieties and q-ample divisors

We introduce a notion of ampleness for subschemes of any codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti-Grauert vanishing theorem.This is the accepted manuscript. The final version is available from Elsevier at http://www.sciencedirect.com/science/article/pii/S0001870812000448