The Proportion of Weierstrass Semigroups

We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitz' necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. We also show that the family of semigroups known to be Weierstrass semigroups using a result of Eisenbud and Harris, has zero density in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup.Comment: 15 pages. Corrected typos, some minors mathematical changes, added some discussion. To appear in J. Algebr

Re-evaluation of pachycormid fishes from the Late Jurassic of Southwestern Germany

Pachycormidae is an extinct group of Mesozoic fishes that exhibits extensive body size and shape disparity. The Late Jurassic record of the group is dominated by fossils from the lithographic limestone of Bavaria, Germany that, although complete and articulated, are not well characterized anatomically. In addition, stratigraphic and geographical provenance are often only approximately known, making these taxa difficult to place in a global biogeographical context. In contrast, the late Kimmeridgian Nusplingen Plattenkalk of Baden-Württemberg is a well-constrained locality yielding hundreds of exceptionally preserved and prepared vertebrate fossils. Pachycormid fishes are rare, but these finds have the potential to broaden our understanding of anatomical variation within this group, as well as provide new information regarding the trophic complexity of the Nusplingen lagoonal ecosystem. Here, we review the fossil record of Pachycormidae from Nusplingen, including one fragmentary and two relatively complete skulls, a largely complete fish, and a fragment of a caudal fin. These finds can be referred to three taxa: Orthocormus sp., Hypsocormus posterodorsalis sp. nov., and Simocormus macrolepidotus gen. et sp. nov. The latter taxon was erected to replace “Hypsocormus” macrodon, here considered to be a nomen dubium. Hypsocormus posterodorsalis is known only from Nusplingen, and is characterized by teeth lacking apicobasal ridging at the bases, a dorsal fin positioned opposite the anterior edge of the anal fin, and a hypural plate consisting of a fused parhypural and hypurals. The holotype specimen contributes additional palaeobiological information, with small teleosteans preserved as gastric contents and ribs showing signs of callus formation. These new findings extend our knowledge of the anatomy and diversity of Pachycormidae, and represent an important first step in understanding factors controlling their distribution and morphological variation in the Late Jurassic of Europe