Figure 1: Red color indicates higher elevation. Given a domain X, its topology provides a rather coarse description of it. However, functions or maps defined on X give us powerful ways to encode various both geometric and other type of information of X. For example, the Gaussian curvature function κ : X → IR describes the local geometric shape of X. Or imagine that X is a terrain on earth, and ...

We describe in terms of the j-invariant all elliptic surfaces π : X → C with a section, such that h(X) = rankNS(X) and the Mordell-Weil group of π is finite. We use this to give a complete solution to infinitesimal Torelli for elliptic surfaces over P with a section.

Let 2 be a manifold and X, Yr , Ya equidimensional submanifolds, all intersecting at 0 E X. Y, and Yz are said to be contact eqil&.~lent (r&A respect to X) at 0 if there exists a germ of diffeomorphism f: (2, 0) + (Z, 0) mapping X into X and Yr into Yz . The notion of contact equivalence is due to John Mather and plays an important role in his theory of singularities of differentiable mappings....

FIGURE 4 Section reveals three mitochondrial complexes (arrows) . In one mitochondrion a divalent cation granule (G) can also be seen . X 15,000 . FIGURE 3 Longitudinal section of mitochondrion . The complex appears to be arranged in a helical manner . X 35,000 . FIGURE 4 Cross section of mitochondrion showing complex . X 35,000 . FIGURES 5-8 Sections of mitochondria from cells treated for 8 h ...

In this article we discuss torsion sections of semistable elliptic surfaces defined over the complex field C. Recall that a semistable elliptic surface is a fibration π : X → C, where X is a smooth compact surface, C is a smooth curve, the general fiber of π is a smooth curve of genus one, and all the singular fibers of π are semistable, that is, all are of type Im in Kodaira’s notation (see [K...