The order of finite algebraic fundamental groups of surfaces with K^2<=3\chi-2
We study the structure of the algebraic fundamental group for minimal surfaces of general type S satisfying K_S^2<=3\chi-2$ and not having any irregular etale cover. We show that, if K_S^2<=3\chi-2, then then the algebraic fundamental group of S has order at most 5, and equality only occurs if S is a Godeaux surface. We also show that if K_S^2<= 3\chi-3 and the algebraic fundamental group of S is not trivial, then it is Z_2, or Z_2^2 or Z_3. Furthermore in this last case one has: 2<=\chi<=4, K^2=3\chi-3 and these possibilities do occur.
Hyperelliptic surfaces of general type with k2<4χ
If a minimal surface S of genral type with a hyperelliptic pencil satisfies K2<f(χ), where f(x) is a function asymptotic to 4x, then S has a pencil of hyperelliptic curves of bounded genus, with a bound depending only upon the ratio of K2 to χ. The Albanese map of S factors through the fibration corresponding to this pencil.
The Effectiveness of Two Approaches to the Teaching of High School American History
The purpose of this study was to test the effectiveness of studies in depth utilizing selected materials which had been developed for teaching high school American history.