The Exceptional Locus in the Bertini Irreducibility Theorem for a Morphism

We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $\phi \colon X \to{\mathbb{P}}^n$ such that $X$ is geometrically irreducible and the nonempty fibers of $\phi $ all have the same dimension, the locus of hyperplanes $H$ such that $\phi ^{-1} H$ is not geometrically irreducible has dimension at most ${\operatorname{codim}}\ \phi (X)+1$. We give an application to monodromy groups above hyperplane sections.

Relations of Normal Bundles and Flips of Small Contractions on Projective Manifolds

Let X be a smooth projective variety over the complex number field. Let f : X → Y be a small contraction, and suppose that each irreducible component Ei of the exceptional locus E of f is a smooth subvariety. Assume that dim E ≤ ½ ( dim X + 1), and the normal bundle [Formula: see text]. Then each Ei ≅ P dim Ei or Q dim Ei. Moreover, the flip f+ : X+ → Y of f exists.

Singularities on the 2-Dimensional Moduli Spaces of Stable Sheaves on K3 Surfaces

We consider the singuralities of 2-dimensional moduli spaces of semi-stable sheaves on k3 surfaces. We show that the moduli space is normal, in particular the siguralities are rational double points. We also describe the exceptional locus on the resolution in terms of exceptional sheaves.

A STRENGTHENING OF RESOLUTION OF SINGULARITIES IN CHARACTERISTIC ZERO

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.2000 Mathematical Subject Classification: 14E15.

A MUTATIONAL ANALYSIS OF THE TRIPLO-LETHAL REGION OF DROSOPHILA MELANOGASTER

ABSTRACT The extensive analysis of the impact of segmental aneuploidy by LINDSLEY et al. (1972) showed that there are relatively few haplo-lethal loci in the genome and that, with one exception, all loci are triplo-viable. The exceptional locus, which lies in salivary gland chromosome region 83D-E, is associated with lethality when present in either one or three doses in an otherwise diploid individual (DENEU 1976). The genetic nature of the phenomenon has been studied by examining the rates of induction, by ionizing radiation and chemical mutagens, of mutations affecting the dose-sensitive behavior. For both types of mutagens, the frequency of inactivation of the locus is relatively low, and a high proportion of such mutations is associated with chromosomal deficiencies. These data indicate that the locus is infrequently and perhaps never inactivated by a DNA base-pair substitution and thus that the triplo-lethal phenomenon is not associated with a "typical" structural gene. It is possible that the triplo-lethal locus is very small, is reiterated or otherwise complex or is functionally insensitive to base-pair substitutions. The result that all mutations that complement a duplication of the triplo-lethal locus are lethal in heterozygous combination with a normal third chromosome argues that triplo- and haplo-lethality are concomitants of the same phenomenon. Salivary gland chromosome analysis of newly induced deficiencies and duplications localizes the locus to 83D4,5-83E1,2, and further cytogenetic manipulation shows that the dose-sensitive behavior is independent of the position of the locus in the genome.