The current paper is devoted to the study of integral curves of constant type in parabolic homogeneous spaces. We construct a canonical moving frame bundle for such curves and give the criterium when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to general parabolic geometries are discussed.

We analyze Weierstrass cycles and tautological rings in moduli spaces of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus ≤ 6. In particular, we show that our general formula gives a good estimate for the dimension of Weierstrass cycles for low genera.

We construct families of hyperelliptic curves over Q of arbitrary genus g with (at least) g integral elements in K2. We also verify the Beilinson conjectures about K2 numerically for several curves with g = 2, 3, 4 and 5. The first few sections of the paper also provide an elementary introduction to the Beilinson conjectures for K2 of curves.

The problem of several finite moving cracks in a functionally graded material is solved by dislocation technique under the condition of anti-plane deformation. By using the Fourier transform the stress fields are obtained for a functionally graded strip containing a screw dislocation. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is em...

These lectures concern the arithmetic of modular curves, and in particular the geometry of integral models of modular curves in the neighborhood of their singular points. These singularities only appear modulo p. If X is a modular curve and x is a singular point, then the nature of the singularity is measured by the completed local ring ÔX,x. Let us first review the basics of integral models of...

Bezout’s theorem gives us the degree of intersection of two properly intersecting projective varieties. As two curves in P never intersect properly, Bezout’s theorem cannot be directly used to bound the number of intersection points of such curves. In this work, we bound the maximum number of intersection points of two integral ACM curves in P. The bound that we give is in many cases optimal as...