Bridge trisections in rational surfaces

We study smooth isotopy classes of complex curves in complex surfaces from the perspective of the theory of bridge trisections, with a special focus on curves in $\mathbb{CP}^2$ and $\mathbb{CP}^1\times\mathbb{CP}^1$. We are especially interested in bridge trisections and trisections that are as simple as possible, which we call "efficient". We show that any curve in $\mathbb{CP}^2$ or $\mathbb{CP}^1\times\mathbb{CP}^1$ admits an efficient bridge trisection. Because bridge trisections and trisections are nicely related via branched covering operations, we are able to give many examples of complex surfaces that admit efficient trisections. Among these are hypersurfaces in $\mathbb{CP}^3$, the elliptic surfaces $E(n)$, the Horikawa surfaces $H(n)$, and complete intersections of hypersurfaces in $\mathbb{CP}^N$. As a corollary, we observe that, in many cases, manifolds that are homeomorphic but not diffeomorphic have the same trisection genus, which is consistent with the conjecture that trisection genus is additive under connected sum. We give many trisection diagrams to illustrate our examples.Comment: 46 pages, 28 color figure

Competition on MARS? A study of broker-dealer competition in the U.S. municipal auction rate securities market

The relationship between competitive bidding in auctions and its impact on price and interest rates has long been of importance for a wide range of market practitioners. Although research has shown that increased competition among broker-dealers and bidders results in lower municipal interest rates, the amount of literature addressing auction rate securities is almost non-existent. The U.S. municipal auction rate securities market (MARS) offers an opportunity to expand the growing but limited empirical analysis of auctions. In particular, researchers can study the impact of market power and competitive search on interest rates using this uniform pricing, multi-unit, frequently repeated dutch auction process. Furthermore, in general, previous cross-sectional models measuring relationships in standard municipal markets are quite static in that they mostly assume some form of dynamic parametric homogeneity. Using a novel empirical approach, i.e., one that doesn't assume the time-constancy of cross-sectional parameters, our research shows that greater underwriter competition and search for potential investors in the form of multiple broker-dealers does indeed lead to lower municipal auction rates. This outcome does not hold for the entire sample life of the security--a result that was captured clearly using our methodology.MARS, municipal auctions, securities, parametric heterogeneity