6. Projective Varieties I: Topology
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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. -
Arxiv:2006.16553V2 [Math.AG] 26 Jul 2020
ON ULRICH BUNDLES ON PROJECTIVE BUNDLES ANDREAS HOCHENEGGER Abstract. In this article, the existence of Ulrich bundles on projective bundles P(E) → X is discussed. In the case, that the base variety X is a curve or surface, a close relationship between Ulrich bundles on X and those on P(E) is established for specific polarisations. This yields the existence of Ulrich bundles on a wide range of projective bundles over curves and some surfaces. 1. Introduction Given a smooth projective variety X, polarised by a very ample divisor A, let i: X ֒→ PN be the associated closed embedding. A locally free sheaf F on X is called Ulrich bundle (with respect to A) if and only if it satisfies one of the following conditions: • There is a linear resolution of F: ⊕bc ⊕bc−1 ⊕b0 0 → OPN (−c) → OPN (−c + 1) →···→OPN → i∗F → 0, where c is the codimension of X in PN . • The cohomology H•(X, F(−pA)) vanishes for 1 ≤ p ≤ dim(X). • For any finite linear projection π : X → Pdim(X), the locally free sheaf π∗F splits into a direct sum of OPdim(X) . Actually, by [18], these three conditions are equivalent. One guiding question about Ulrich bundles is whether a given variety admits an Ulrich bundle of low rank. The existence of such a locally free sheaf has surprisingly strong implications about the geometry of the variety, see the excellent surveys [6, 14]. Given a projective bundle π : P(E) → X, this article deals with the ques- tion, what is the relation between Ulrich bundles on the base X and those on P(E)? Note that answers to such a question depend much on the choice arXiv:2006.16553v3 [math.AG] 15 Aug 2021 of a very ample divisor. -
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. -
A Zariski Topology for Modules
A Zariski Topology for Modules∗ Jawad Abuhlail† Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals [email protected] October 18, 2018 Abstract Given a duo module M over an associative (not necessarily commutative) ring R, a Zariski topology is defined on the spectrum Specfp(M) of fully prime R-submodules of M. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration. 1 Introduction The interplay between the properties of a given ring and the topological properties of the Zariski topology defined on its prime spectrum has been studied intensively in the literature (e.g. [LY2006, ST2010, ZTW2006]). On the other hand, many papers considered the so called top modules, i.e. modules whose spectrum of prime submodules attains a Zariski topology (e.g. [Lu1984, Lu1999, MMS1997, MMS1998, Zah2006-a, Zha2006-b]). arXiv:1007.3149v1 [math.RA] 19 Jul 2010 On the other hand, different notions of primeness for modules were introduced and in- vestigated in the literature (e.g. [Dau1978, Wis1996, RRRF-AS2002, RRW2005, Wij2006, Abu2006, WW2009]). In this paper, we consider a notions that was not dealt with, from the topological point of view, so far. Given a duo module M over an associative ring R, we introduce and topologize the spectrum of fully prime R-submodules of M. Motivated by results on the Zariski topology on the spectrum of prime ideals of a commutative ring (e.g. [Bou1998, AM1969]), we investigate the interplay between the topological properties of the obtained space and the module under consideration. -
Lecture 1 Course Introduction, Zariski Topology
18.725 Algebraic Geometry I Lecture 1 Lecture 1: Course Introduction, Zariski topology Some teasers So what is algebraic geometry? In short, geometry of sets given by algebraic equations. Some examples of questions along this line: 1. In 1874, H. Schubert in his book Calculus of enumerative geometry proposed the question that given 4 generic lines in the 3-space, how many lines can intersect all 4 of them. The answer is 2. The proof is as follows. Move the lines to a configuration of the form of two pairs, each consists of two intersecting lines. Then there are two lines, one of them passing the two intersection points, the other being the intersection of the two planes defined by each pair. Now we need to show somehow that the answer stays the same if we are truly in a generic position. This is answered by intersection theory, a big topic in AG. 2. We can generalize this statement. Consider 4 generic polynomials over C in 3 variables of degrees d1; d2; d3; d4, how many lines intersect the zero sets of each polynomial? The answer is 2d1d2d3d4. This is given in general by \Schubert calculus." 4 3. Take C , and 2 generic quadratic polynomials of degree two, how many lines are on the common zero set? The answer is 16. 4. For a generic cubic polynomial in 3 variables, how many lines are on the zero set? There are exactly 27 of them. (This is related to the exceptional Lie group E6.) Another major development of AG in the 20th century was on counting the numbers of solutions for n 2 3 polynomial equations over Fq where q = p . -
Package 'Rgbif'
Package ‘rgbif’ January 21, 2017 Title Interface to the Global 'Biodiversity' Information Facility 'API' Description A programmatic interface to the Web Service methods provided by the Global Biodiversity Information Facility ('GBIF'; <http://www.gbif.org/developer/summary>). 'GBIF' is a database of species occurrence records from sources all over the globe. 'rgbif' includes functions for searching for taxonomic names, retrieving information on data providers, getting species occurrence records, and getting counts of occurrence records. Version 0.9.7 License MIT + file LICENSE URL https://github.com/ropensci/rgbif BugReports https://github.com/ropensci/rgbif/issues LazyData true LazyLoad true VignetteBuilder knitr Imports utils, stats, xml2, ggplot2, httr (>= 1.0.0), data.table, whisker, magrittr, jsonlite (>= 0.9.16), V8, oai (>= 0.2.2), geoaxe, tibble Suggests testthat, knitr, reshape2, maps, sp, rgeos RoxygenNote 5.0.1 NeedsCompilation no Author Scott Chamberlain [aut, cre], Vijay Barve [ctb], Dan Mcglinn [ctb] Maintainer Scott Chamberlain <[email protected]> Repository CRAN Date/Publication 2017-01-21 01:27:16 1 2 R topics documented: R topics documented: rgbif-package . .3 check_wkt . .4 count_facet . .4 datasets . .6 dataset_metrics . .7 dataset_search . .8 dataset_suggest . 10 downloads . 12 elevation . 13 enumeration . 14 gbifmap . 15 gbif_bbox2wkt . 16 gbif_citation . 17 gbif_issues . 19 gbif_names . 19 gbif_oai . 20 gbif_photos . 22 installations . 23 isocodes . 24 name_backbone . 25 name_lookup . 26 name_suggest . 30 name_usage . 31 networks . 33 nodes . 35 occ_count . 37 occ_data . 40 occ_download . 53 occ_download_cancel . 56 occ_download_get . 57 occ_download_import . 58 occ_download_list . 59 occ_download_meta . 60 occ_facet . 60 occ_fields . 61 occ_get . 62 occ_issues . 63 occ_issues_lookup . 64 occ_metadata . 65 occ_search . 66 occ_spellcheck . 82 organizations . 83 parsenames . 84 read_wkt . -
Projective Varieties and Their Sheaves of Regular Functions
Math 6130 Notes. Fall 2002. 4. Projective Varieties and their Sheaves of Regular Functions. These are the geometric objects associated to the graded domains: C[x0;x1; :::; xn]=P (for homogeneous primes P ) defining“global” complex algebraic geometry. Definition: (a) A subset V ⊆ CPn is algebraic if there is a homogeneous ideal I ⊂ C[x ;x ; :::; x ] for which V = V (I). 0 1 n p (b) A homogeneous ideal I ⊂ C[x0;x1; :::; xn]isradical if I = I. Proposition 4.1: (a) Every algebraic set V ⊆ CPn is the zero locus: V = fF1(x1; :::; xn)=F2(x1; :::; xn)=::: = Fm(x1; :::; xn)=0g of a finite set of homogeneous polynomials F1; :::; Fm 2 C[x0;x1; :::; xn]. (b) The maps V 7! I(V ) and I 7! V (I) ⊂ CPn give a bijection: n fnonempty alg sets V ⊆ CP g$fhomog radical ideals I ⊂hx0; :::; xnig (c) A topology on CPn, called the Zariski topology, results when: U ⊆ CPn is open , Z := CPn − U is an algebraic set (or empty) Proof: Note the similarity with Proposition 3.1. The proof is the same, except that Exercise 2.5 should be used in place of Corollary 1.4. Remark: As in x3, the bijection of (b) is \inclusion reversing," i.e. V1 ⊆ V2 , I(V1) ⊇ I(V2) and irreducibility and the features of the Zariski topology are the same. Example: A projective hypersurface is the zero locus: n n V (F )=f(a0 : a1 : ::: : an) 2 CP j F (a0 : a1 : ::: : an)=0}⊂CP whose irreducible components, as in x3, are obtained by factoring F , and the basic open sets of CPn are, as in x3, the complements of hypersurfaces. -
The Riemann-Roch Theorem
The Riemann-Roch Theorem by Carmen Anthony Bruni A project presented to the University of Waterloo in fulfillment of the project requirement for the degree of Master of Mathematics in Pure Mathematics Waterloo, Ontario, Canada, 2010 c Carmen Anthony Bruni 2010 Declaration I hereby declare that I am the sole author of this project. This is a true copy of the project, including any required final revisions, as accepted by my examiners. I understand that my project may be made electronically available to the public. ii Abstract In this paper, I present varied topics in algebraic geometry with a motivation towards the Riemann-Roch theorem. I start by introducing basic notions in algebraic geometry. Then I proceed to the topic of divisors, specifically Weil divisors, Cartier divisors and examples of both. Linear systems which are also associated with divisors are introduced in the next chapter. These systems are the primary motivation for the Riemann-Roch theorem. Next, I introduce sheaves, a mathematical object that encompasses a lot of the useful features of the ring of regular functions and generalizes it. Cohomology plays a crucial role in the final steps before the Riemann-Roch theorem which encompasses all the previously developed tools. I then finish by describing some of the applications of the Riemann-Roch theorem to other problems in algebraic geometry. iii Acknowledgements I would like to thank all the people who made this project possible. I would like to thank Professor David McKinnon for his support and help to make this project a reality. I would also like to thank all my friends who offered a hand with the creation of this project. -
8. Grassmannians
66 Andreas Gathmann 8. Grassmannians After having introduced (projective) varieties — the main objects of study in algebraic geometry — let us now take a break in our discussion of the general theory to construct an interesting and useful class of examples of projective varieties. The idea behind this construction is simple: since the definition of projective spaces as the sets of 1-dimensional linear subspaces of Kn turned out to be a very useful concept, let us now generalize this and consider instead the sets of k-dimensional linear subspaces of Kn for an arbitrary k = 0;:::;n. Definition 8.1 (Grassmannians). Let n 2 N>0, and let k 2 N with 0 ≤ k ≤ n. We denote by G(k;n) the set of all k-dimensional linear subspaces of Kn. It is called the Grassmannian of k-planes in Kn. Remark 8.2. By Example 6.12 (b) and Exercise 6.32 (a), the correspondence of Remark 6.17 shows that k-dimensional linear subspaces of Kn are in natural one-to-one correspondence with (k − 1)- n− dimensional linear subspaces of P 1. We can therefore consider G(k;n) alternatively as the set of such projective linear subspaces. As the dimensions k and n are reduced by 1 in this way, our Grassmannian G(k;n) of Definition 8.1 is sometimes written in the literature as G(k − 1;n − 1) instead. Of course, as in the case of projective spaces our goal must again be to make the Grassmannian G(k;n) into a variety — in fact, we will see that it is even a projective variety in a natural way. -
An Introduction to Volume Functions for Algebraic Cycles
AN INTRODUCTION TO VOLUME FUNCTIONS FOR ALGEBRAIC CYCLES Abstract. DRAFT VERSION DRAFT VERSION 1. Introduction One of the oldest problems in algebraic geometry is the Riemann-Roch problem: given a divisor L on a smooth projective variety X, what is the dimension of H0(X; L)? Even better, one would like to calculate the entire 0 section ring ⊕m2ZH (X; mL). It is often quite difficult to compute this ring precisely; for example, section rings need not be finitely generated. Even if we cannot completely understand the section ring, we can still extract information by studying its asymptotics { how does H0(X; mL) behave as m ! 1? This approach is surprisingly fruitful. On the one hand, asymptotic invariants are easier to compute and have nice variational properties. On the other, they retain a surprising amount of geometric information. Such invariants have played a central role in the development of birational geometry over the last forty years. Perhaps the most important asymptotic invariant for a divisor L is the volume, defined as dim H0(X; mL) vol(L) = lim sup n : m!1 m =n! In other words, the volume is the section ring analogue of the Hilbert-Samuel multiplicity for graded rings. As we will see shortly, the volume lies at the intersection of many fields of mathematics and has a variety of interesting applications. This expository paper outlines recent progress in the construction of \volume-type" functions for cycles of higher codimension. The main theme is the relationship between: • the positivity of cycle classes, • an asymptotic approach to enumerative geometry, and • the convex analysis of positivity functions.