Subspecies, Semispecies, Superspecies
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BASTERIA, 50: 93-150, 1986 Alphabetical revision of the (sub)species in recent Conidae. 9. ebraeus to extraordinarius with the description of Conus elegans ramalhoi, nov. subspecies H.E. Coomans R.G. Moolenbeek& E. Wils Institute of Taxonomic Zoology (Zoological Museum) University of Amsterdam INTRODUCTION In this ninth part of the revision all names of recent Conus taxa beginning with the letter e are discussed. Amongst these are several nominal species of tent-cones with a C.of close-set lines, the shell a darker pattern consisting very giving appearance (e.g. C. C. The elisae, euetrios, eumitus). phenomenon was also mentioned for C. castaneo- fasciatus, C. cholmondeleyi and C. dactylosus in former issues. This occurs in populations where with normal also that consider them specimens a tent-pattern are found, so we as colour formae. The effect is known shells in which of white opposite too, areas are present, leaving 'islands' with the tent-pattern (e.g. C. bitleri, C. castrensis, C. concatenatus and C. episco- These colour formae. patus). are also art. Because of a change in the rules of the ICZN (3rd edition, 1985: 73-74), there has risen a disagreement about the concept of the "type series". In cases where a museum type-lot consists of more than one specimen, although the original author(s) did not indicate that more than one shell was used for the description, we will designate the single originally mentioned and/or figured specimen as the "lectotype". Never- theless a number of taxonomists will consider that "lectotype" as the holotype, and disregard the remaining shells in the lot as type material. -
INTRODUCTION to ALGEBRAIC GEOMETRY, CLASS 25 Contents 1
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 25 RAVI VAKIL Contents 1. The genus of a nonsingular projective curve 1 2. The Riemann-Roch Theorem with Applications but No Proof 2 2.1. A criterion for closed immersions 3 3. Recap of course 6 PS10 back today; PS11 due today. PS12 due Monday December 13. 1. The genus of a nonsingular projective curve The definition I’m going to give you isn’t the one people would typically start with. I prefer to introduce this one here, because it is more easily computable. Definition. The tentative genus of a nonsingular projective curve C is given by 1 − deg ΩC =2g 2. Fact (from Riemann-Roch, later). g is always a nonnegative integer, i.e. deg K = −2, 0, 2,.... Complex picture: Riemann-surface with g “holes”. Examples. Hence P1 has genus 0, smooth plane cubics have genus 1, etc. Exercise: Hyperelliptic curves. Suppose f(x0,x1) is a polynomial of homo- geneous degree n where n is even. Let C0 be the affine plane curve given by 2 y = f(1,x1), with the generically 2-to-1 cover C0 → U0.LetC1be the affine 2 plane curve given by z = f(x0, 1), with the generically 2-to-1 cover C1 → U1. Check that C0 and C1 are nonsingular. Show that you can glue together C0 and C1 (and the double covers) so as to give a double cover C → P1. (For computational convenience, you may assume that neither [0; 1] nor [1; 0] are zeros of f.) What goes wrong if n is odd? Show that the tentative genus of C is n/2 − 1.(Thisisa special case of the Riemann-Hurwitz formula.) This provides examples of curves of any genus. -
X(V, M) = Dim Lm + 1 LEMMA 1. a Specialization of The
34 MA THEMA TICS: J. IGUSA PROC. N. A. S. 4R.Bellman, Dynamic Programming and ContinuousProcesses (RAND Monograph R-271, 1954). 5R. Bellman, "Dynamic Programming and a New Formalism in the Calculus of Variations," these PROCEEDINGS, 40, 231-235, 1954. 6 R. Bellman, "Monotone Convergence in Dynamic Programming and the Calculus of Vari- ations," ibid., (these PROCEEDINGS, 40, 1073-1075, 1954). 7 E. Hille, Functional Analysis and Semi-groups ("American Mathematical Society Colloquium Publications," Vol. XXXI [1948]). 8 Cf. ibid., p. 71. 9 Ibid., p. 388. 10 V. Volterra, Leqons sur les fonctions des lignes (Paris: Gauthier-Villars, 1913). ARITHMETIC GENERA OF NORMAL VARIETIES IN AN ALGEBRAIC FAMILY* BY JUN-ICHI IGUSA DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, AND KYOTO UNIVERSITY, JAPAN Communicated by Oscar Zariski, October 28, 1954 It is well known that linear equivalence of divisors on a fixed ambient variety is preserved by specialization.' In this paper we shall show that the above assertion remains valid even when the ambient variety varies under specialization. This fact will be used as a lemma in our later paper. Here we shall derive the following theorem as an immediate consequence. If a normal variety V' is a specialization of a positive cycle V, then V is also a normal variety, and they have the same arith- metic genus. In the case of curves this assertion was proved by Chow and Nakai,2 and in our proof we shall use some of their ideas. We note also that a slightly less general result was proved in the classical case by Spencer and Kodaira quite recently.3 Let yr be a variety of dimension r in a projective space L'. -
Applications of the Hyperbolic Ax-Schanuel Conjecture
Applications of the hyperbolic Ax-Schanuel conjecture Christopher Daw Jinbo Ren Abstract In 2014, Pila and Tsimerman gave a proof of the Ax-Schanuel conjecture for the j- function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax-Schanuel conjecture. In this article, we show that the hyperbolic Ax-Schanuel conjecture can be used to reduce the Zilber-Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila- Wilkie counting theorem and several arithmetic conjectures. Our methods are inspired by previous applications of the Pila-Zannier method and, in particular, the recent proof by Habegger and Pila of the Zilber-Pink conjecture for curves in abelian varieties. 2010 Mathematics Subject Classification: 11G18, 14G35 Keywords: hyperbolic Ax-Schanuel conjecture, Zilber-Pink conjecture, Shimura varieties Contents 1 Introduction 2 2 Special and weakly special subvarieties 5 3 The Zilber-Pink conjecture 7 4 The defect condition 8 5 The hyperbolic Ax-Schanuel conjecture 10 6 A finiteness result for weakly optimal subvarieties 13 7 Anomalous subvarieties 20 arXiv:1703.08967v5 [math.NT] 11 Oct 2018 8 Main results (part 1): Reductions to point counting 23 9 The counting theorem 25 10 Complexity 26 11 Galois orbits 27 12 Further arithmetic hypotheses 29 13 Products of modular curves 31 14 Main results (part 2): Conditional solutions to the counting problems 35 1 15 A brief note on special anomalous subvarieties 40 References 44 1 Introduction The Ax-Schanuel theorem [2] is a result regarding the transcendence degrees of fields generated over the complex numbers by power series and their exponentials. -
Receptor-Like Kinases from Arabidopsis Form a Monophyletic Gene Family Related to Animal Receptor Kinases
Receptor-like kinases from Arabidopsis form a monophyletic gene family related to animal receptor kinases Shin-Han Shiu and Anthony B. Bleecker* Department of Botany and Laboratory of Genetics, University of Wisconsin, Madison, WI 53706 Edited by Elliot M. Meyerowitz, California Institute of Technology, Pasadena, CA, and approved July 6, 2001 (received for review March 22, 2001) Plant receptor-like kinases (RLKs) are proteins with a predicted tionary relationship between the RTKs and RLKs within the signal sequence, single transmembrane region, and cytoplasmic recognized superfamily of related eukaryotic serine͞threonine͞ kinase domain. Receptor-like kinases belong to a large gene family tyrosine protein kinases (ePKs). An earlier phylogenetic analysis with at least 610 members that represent nearly 2.5% of Arabi- (22), using the six RLK sequences available at the time, indicated dopsis protein coding genes. We have categorized members of this a close relationship between plant sequences and animal RTKs, family into subfamilies based on both the identity of the extracel- although RLKs were placed in the ‘‘other kinase’’ category. A more lular domains and the phylogenetic relationships between the recent analysis using only plant sequences led to the conclusion that kinase domains of subfamily members. Surprisingly, this structur- the 18 RLKs sampled seemed to form a separate family among the ally defined group of genes is monophyletic with respect to kinase various eukaryotic kinases (23). The recent completion of the domains when compared with the other eukaryotic kinase families. Arabidopsis genome sequence (5) provides an opportunity for a In an extended analysis, animal receptor kinases, Raf kinases, plant more comprehensive analysis of the relationships between these RLKs, and animal receptor tyrosine kinases form a well supported classes of receptor kinases. -