Regular del Pezzo surfaces with irregularity

We construct the first examples of regular del Pezzo surfaces for which the irregularity (i.e. the dimension of the first cohomology group of the structure sheaf) is nonzero. We also find a restriction on the integer pairs that are possible as the anti-canonical degree and irregularity of such a surface.Comment: 23 pages, 1 figure (on page 2

Level Sets of the Takagi Function: Generic Level Sets

The Takagi function {\tau} : [0, 1] \rightarrow [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. This paper studies the level sets L(y) = {x : {\tau}(x) = y} of the Takagi function {\tau}(x). It shows that for a full Lebesgue measure set of ordinates y, these level sets are finite sets, but whose expected number of points is infinite. Complementing this, it shows that the set of ordinates y whose level set has positive Hausdorff dimension is itself a set of full Hausdorff dimension 1 (but Lebesgue measure zero). Finally it shows that the level sets have a nontrivial Hausdorff dimension spectrum. The results are obtained using a notion of "local level set" introduced in a previous paper, along with a singular measure parameterizing such sets.Comment: Comments welcome. 23 pages, 2 figures. Latest version is an extensive rewrite of earlier version

A BOUND ON EMBEDDING DIMENSIONS OF GEOMETRIC GENERIC FIBERS

The author finds a limit on the singularities that arise in geometric generic fibers of morphisms between smooth varieties of positive characteristic by studying changes in embedding dimension under inseparable field extensions. This result is then used in the context of the minimal model program to rule out the existence of smooth varieties fibered by certain nonnormal del Pezzo surfaces over bases of small dimension.</jats:p

Level Sets of the Takagi Function: Local Level Sets

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.Comment: 32 pages, 2 figures, 1 table. Latest version has updated equation numbering. The final publication will soon be available at springerlink.co

Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory

This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anti-canonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semi-stable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry

Regular del Pezzo surfaces with irregularity

We construct the first examples of regular del Pezzo surfaces X X for which ℎ¹(𝒪 _{𝒳})&gt;0. We also find a restriction on the integer pairs that are possible as the anti-canonical degree K X 2 K_X^2 and irregularity h 1 ( O X ) h^1(\mathcal {O}_X) of such a surface. Our method of proof is by generalizing results of Ekedahl on foliations to the setting of regular varieties.</p

Is Pornography Consumption Related to Risky Behaviors During Friends with Benefits Relationships?

Abstract Background Friends with benefits encounters are a relatively new pattern of relating among emerging adults where risky sexual behavior may occur. Aim To understand whether pornography consumption is associated with riskier behaviors during friends with benefits encounters. Methods Cross-sectional study of 2 samples of emerging adults who have engaged in friends with benefits relationships (study 1, N = 411; study 2, N = 394). For binary outcomes, we used logistic regression and report odds ratios. For ordinal outcomes, we used ordered logistic regression and reported odds ratios. We tested for moderation by biological sex. Results Men who consumed pornography more frequently were more likely to engage in risky sexual behaviors during their friends with benefits encounters. More frequent pornography consumption was associated with increased likelihood and amount of intoxication for both the respondent and his partner, less frequent condom use, and a higher probability of having penetrative friends with benefits encounters while intoxicated and not using a condom. For each of these outcomes, our parameter estimates from study 2 fell within the 95% confidence intervals from study 1. These associations persisted when controlling for the effects of binge drinking frequency, broader patterns of problematic alcohol use, trait self-control, openness to experience, and permissive attitudes toward casual sex. The findings of this study may inform interventions to reduce risky behaviors among emerging adults. Limitations Our cross-sectional studies examined only emerging adults in college with measurement that was exclusively self-reported. Conclusions These results are discussed in terms of sexual script theory, and several implications for intervention are outlined. </jats:sec