CLASSIFICATION of COMPLEX ALGEBRAIC SURFACES from The
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Ribbons and Their Canonical Embeddings
transactions of the american mathematical society Volume 347, Number 3, March 1995 RIBBONS AND THEIR CANONICAL EMBEDDINGS DAVE BAYER AND DAVID EISENBUD Abstract. We study double structures on the projective line and on certain other varieties, with a view to having a nice family of degenerations of curves and K3 surfaces of given genus and Clifford index. Our main interest is in the canonical embeddings of these objects, with a view toward Green's Conjecture on the free resolutions of canonical curves. We give the canonical embeddings explicitly, and exhibit an approach to determining a minimal free resolution. Introduction What is the limit of the canonical model of a smooth curve as the curve degenerates to a hyperelliptic curve? "A ribbon" — more precisely "a ribbon on P1 " — may be defined as the answer to this riddle. A ribbon on P1 is a double structure on the projective line. Such ribbons represent a little-studied degeneration of smooth curves that shows promise especially for dealing with questions about the Clifford indices of curves. The theory of ribbons is in some respects remarkably close to that of smooth curves, but ribbons are much easier to construct and work with. In this paper we discuss the classification of ribbons and their maps. In particular, we construct the "holomorphic differentials" — sections of the canonical bundle — of a ribbon, and study properties of the canonical embedding. Aside from the genus, the main invariant of a ribbon is a number we call the "Clifford index", although the definition for it that we give is completely different from the definition for smooth curves. -
Curve Counting on Abelian Surfaces and Threefolds
Algebraic Geometry 5 (4) (2018) 398{463 doi:10.14231/AG-2018-012 Curve counting on abelian surfaces and threefolds Jim Bryan, Georg Oberdieck, Rahul Pandharipande and Qizheng Yin Abstract We study the enumerative geometry of algebraic curves on abelian surfaces and three- folds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov{Witten theory of K3 surfaces. We prove complete results in all genera for primitive classes. The generating series are quasi-modular forms of pure weight. Conjectures for imprimitive classes are presented. In genus 2, the counts in all classes are proven. Special counts match the Euler characteristic calculations of the moduli spaces of stable pairs on abelian surfaces by G¨ottsche{Shende. A formula for hyperelliptic curve counting in terms of Jacobi forms is proven (modulo a transversality statement). For abelian threefolds, complete conjectures in terms of Jacobi forms for the gen- erating series of curve counts in primitive classes are presented. The base cases make connections to classical lattice counts of Debarre, G¨ottsche, and Lange{Sernesi. Further evidence is provided by Donaldson{Thomas partition function computations for abelian threefolds. A multiple cover structure is presented. The abelian threefold conjectures open a new direction in the subject. 1. Introduction 1.1 Vanishings Let A be a complex abelian variety of dimension d. The Gromov{Witten invariants of A in genus g and class β 2 H2(A; Z) are defined by integration against the virtual class of the moduli space of stable maps M g;n(A; β), Z A ∗ a1 ∗ an τa1 (γ1) ··· τan (γn) = ev1(γ1) 1 ··· evn(γn) n ; g,β vir [M g;n(A,β)] see [PT14] for an introduction. -
Canonical Bundle Formulae for a Family of Lc-Trivial Fibrations
CANONICAL BUNDLE FORMULAE FOR A FAMILY OF LC-TRIVIAL FIBRATIONS Abstract. We give a sufficient condition for divisorial and fiber space adjunc- tion to commute. We generalize the log canonical bundle formula of Fujino and Mori to the relative case. Contents 1. Introduction 1 2. Preliminaries 2 3. Lc-trivial fibrations 3 4. Proofs 6 References 6 1. Introduction A lc-trivial fibration consists of a (sub-)pair (X; B), and a contraction f : X −! Z, such that KZ + B ∼Q;Z 0, and (X; B) is (sub) lc over the generic point of Z. Lc-trivial fibrations appear naturally in higher dimensional algebraic geometry: the Minimal Model Program and the Abundance Conjecture predict that any log canonical pair (X; B), such that KX + B is pseudo-effective, has a birational model with a naturally defined lc-trivial fibration. Intuitively, one can think of (X; B) as being constructed from the base Z and a general fiber (Xz;Bz). This relation is made more precise by the canonical bundle formula ∗ KX + B ∼Q f (KZ + MZ + BZ ) a result first proven by Kodaira for minimal elliptic surfaces [11, 12], and then gen- eralized by the work of Ambro, Fujino-Mori and Kawamata [1, 2, 6, 10]. Here, BZ measures the singularities of the fibers, while MZ measures, at least conjecturally, the variation in moduli of the general fiber. It is then possible to relate the singu- larities of the total space with those of the base: for example, [5, Proposition 4.16] shows that (X; B) and (Z; MZ + BZ ) are in the same class of singularities. -
Foundations of Algebraic Geometry Class 46
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 46 RAVI VAKIL CONTENTS 1. Curves of genus 4 and 5 1 2. Curves of genus 1: the beginning 3 1. CURVES OF GENUS 4 AND 5 We begin with two exercises in general genus, and then go back to genus 4. 1.A. EXERCISE. Suppose C is a genus g curve. Show that if C is not hyperelliptic, then the canonical bundle gives a closed immersion C ,! Pg-1. (In the hyperelliptic case, we have already seen that the canonical bundle gives us a double cover of a rational normal curve.) Hint: follow the genus 3 case. Such a curve is called a canonical curve, and this closed immersion is called the canonical embedding of C. 1.B. EXERCISE. Suppose C is a curve of genus g > 1, over a field k that is not algebraically closed. Show that C has a closed point of degree at most 2g - 2 over the base field. (For comparison: if g = 1, it turns out that there is no such bound independent of k!) We next consider nonhyperelliptic curves C of genus 4. Note that deg K = 6 and h0(C; K) = 4, so the canonical map expresses C as a sextic curve in P3. We shall see that all such C are complete intersections of quadric surfaces and cubic surfaces, and con- versely all nonsingular complete intersections of quadrics and cubics are genus 4 non- hyperelliptic curves, canonically embedded. By Riemann-Roch, h0(C; K⊗2) = deg K⊗2 - g + 1 = 12 - 4 + 1 = 9: 0 P3 O ! 0 K⊗2 2 K 4+1 We have the restriction map H ( ; (2)) H (C; ), and dim Sym Γ(C; ) = 2 = 10. -
1 Real-Time Algebraic Surface Visualization
1 Real-Time Algebraic Surface Visualization Johan Simon Seland1 and Tor Dokken2 1 Center of Mathematics for Applications, University of Oslo [email protected] 2 SINTEF ICT [email protected] Summary. We demonstrate a ray tracing type technique for rendering algebraic surfaces us- ing programmable graphics hardware (GPUs). Our approach allows for real-time exploration and manipulation of arbitrary real algebraic surfaces, with no pre-processing step, except that of a possible change of polynomial basis. The algorithm is based on the blossoming principle of trivariate Bernstein-Bezier´ func- tions over a tetrahedron. By computing the blossom of the function describing the surface with respect to each ray, we obtain the coefficients of a univariate Bernstein polynomial, de- scribing the surface’s value along each ray. We then use Bezier´ subdivision to find the first root of the curve along each ray to display the surface. These computations are performed in parallel for all rays and executed on a GPU. Key words: GPU, algebraic surface, ray tracing, root finding, blossoming 1.1 Introduction Visualization of 3D shapes is a computationally intensive task and modern GPUs have been designed with lots of computational horsepower to improve performance and quality of 3D shape visualization. However, GPUs were designed to only pro- cess shapes represented as collections of discrete polygons. Consequently all other representations have to be converted to polygons, a process known as tessellation, in order to be processed by GPUs. Conversion of shapes represented in formats other than polygons will often give satisfactory visualization quality. However, the tessel- lation process can easily miss details and consequently provide false information. -
On the Euler Number of an Orbifold 257 Where X ~ Is Embedded in .W(X, G) As the Set of Constant Paths
Math. Ann. 286, 255-260 (1990) Imam Springer-Verlag 1990 On the Euler number of an orbifoid Friedrich Hirzebruch and Thomas HSfer Max-Planck-Institut ffir Mathematik, Gottfried-Claren-Strasse 26, D-5300 Bonn 3, Federal Republic of Germany Dedicated to Hans Grauert on his sixtieth birthday This short note illustrates connections between Lothar G6ttsche's results from the preceding paper and invariants for finite group actions on manifolds that have been introduced in string theory. A lecture on this was given at the MPI workshop on "Links between Geometry and Physics" at Schlol3 Ringberg, April 1989. Invariants of quotient spaces. Let G be a finite group acting on a compact differentiable manifold X. Topological invariants like Betti numbers of the quotient space X/G are well-known: i a 1 b,~X/G) = dlmH (X, R) = ~ g~)~ tr(g* I H'(X, R)). The topological Euler characteristic is determined by the Euler characteristic of the fixed point sets Xa: 1 Physicists" formula. Viewed as an orbifold, X/G still carries someinformation on the group action. In I-DHVW1, 2; V] one finds the following string-theoretic definition of the "orbifold Euler characteristic": e X, 1 e(X<~.h>) Here summation runs over all pairs of commuting elements in G x G, and X <g'h> denotes the common fixed point set ofg and h. The physicists are mainly interested in the case where X is a complex threefold with trivial canonical bundle and G is a finite subgroup of $U(3). They point out that in some situations where X/G has a resolution of singularities X-~Z,X/G with trivial canonical bundle e(X, G) is just the Euler characteristic of this resolution [DHVW2; St-W]. -
Threefolds of Kodaira Dimension One Such That a Small Pluricanonical System Do Not Define the Iitaka fibration
THREEFOLDS OF KODAIRA DIMENSION ONE HSIN-KU CHEN Abstract. We prove that for any smooth complex projective threefold of Kodaira di- mension one, the m-th pluricanonical map is birational to the Iitaka fibration for every m ≥ 5868 and divisible by 12. 1. Introduction By the result of Hacon-McKernan [14], Takayama [25] and Tsuji [26], it is known that for any positive integer n there exists an integer rn such that if X is an n-dimensional smooth complex projective variety of general type, then |rKX| defines a birational morphism for all r ≥ rn. It is conjectured in [14] that a similar phenomenon occurs for any projective variety of non-negative Kodaira dimension. That is, for any positive integer n there exists a constant sn such that, if X is an n-dimensional smooth projective variety of non-negative Kodaira dimension and s ≥ sn is sufficiently divisible, then the s-th pluricanonical map of X is birational to the Iitaka fibration. We list some known results related to this problem. In 1986, Kawamata [17] proved that there is an integer m0 such that for any terminal threefold X with Kodaira dimension zero, the m0-th plurigenera of X is non-zero. Later on, Morrison proved that one can 5 3 2 take m0 =2 × 3 × 5 × 7 × 11 × 13 × 17 × 19. See [21] for details. In 2000, Fujino and Mori [12] proved that if X is a smooth projective variety with Kodaira dimension one and F is a general fiber of the Iitaka fibration of X, then there exists a integer M, which depends on the dimension of X, the middle Betti number of some finite covering of F and the smallest integer so that the pluricanonical system of F is non-empty, such that the M-th pluricanonical map of X is birational to the Iitaka fibration. -
Algebraic Curves and Surfaces
Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces . -
General Introduction to K3 Surfaces
General introduction to K3 surfaces Svetlana Makarova MIT Mathematics Contents 1 Algebraic K3 surfaces 1 1.1 Definition of K3 surfaces . .1 1.2 Classical invariants . .3 2 Complex K3 surfaces 9 2.1 Complex K3 surfaces . .9 2.2 Hodge structures . 11 2.3 Period map . 12 References 16 1 Algebraic K3 surfaces 1.1 Definition of K3 surfaces Let K be an arbitrary field. Here, a variety over K will mean a separated, geometrically integral scheme of finite type over K.A surface is a variety of dimension two. If X is a variety over of dimension n, then ! will denote its canonical class, that is ! =∼ Ωn . K X X X=K For a sheaf F on a scheme X, I will write H• (F) for H• (X; F), unless that leads to ambiguity. Definition 1.1.1. A K3 surface over K is a complete non-singular surface X such that ∼ 1 !X = OX and H (X; OX ) = 0. Corollary 1.1.1. One can observe several simple facts for a K3 surface: ∼ 1.Ω X = TX ; 2 ∼ 0 2.H (OX ) = H (OX ); 3. χ(OX ) = 2 dim Γ(OX ) = 2. 1 Fact 1.1.2. Any smooth complete surface over an algebraically closed field is projective. This fact is an immediate corollary of the Zariski{Goodman theorem which states that for any open affine U in a smooth complete surface X (over an algebraically closed field), the closed subset X nU is connected and of pure codimension one in X, and moreover supports an ample effective divisor. -
MINIMAL MODEL THEORY of NUMERICAL KODAIRA DIMENSION ZERO Contents 1. Introduction 1 2. Preliminaries 3 3. Log Minimal Model Prog
MINIMAL MODEL THEORY OF NUMERICAL KODAIRA DIMENSION ZERO YOSHINORI GONGYO Abstract. We prove the existence of good minimal models of numerical Kodaira dimension 0. Contents 1. Introduction 1 2. Preliminaries 3 3. Log minimal model program with scaling 7 4. Zariski decomposition in the sense of Nakayama 9 5. Existence of minimal model in the case where º = 0 10 6. Abundance theorem in the case where º = 0 12 References 14 1. Introduction Throughout this paper, we work over C, the complex number ¯eld. We will make use of the standard notation and de¯nitions as in [KM] and [KMM]. The minimal model conjecture for smooth varieties is the following: Conjecture 1.1 (Minimal model conjecture). Let X be a smooth pro- jective variety. Then there exists a minimal model or a Mori ¯ber space of X. This conjecture is true in dimension 3 and 4 by Kawamata, Koll¶ar, Mori, Shokurov and Reid (cf. [KMM], [KM] and [Sho2]). In the case where KX is numerically equivalent to some e®ective divisor in dimen- sion 5, this conjecture is proved by Birkar (cf. [Bi1]). When X is of general type or KX is not pseudo-e®ective, Birkar, Cascini, Hacon Date: 2010/9/21, version 4.03. 2000 Mathematics Subject Classi¯cation. 14E30. Key words and phrases. minimal model, the abundance conjecture, numerical Kodaira dimension. 1 2 YOSHINORI GONGYO and McKernan prove Conjecture 1.1 for arbitrary dimension ([BCHM]). Moreover if X has maximal Albanese dimension, Conjecture 1.1 is true by [F2]. In this paper, among other things, we show Conjecture 1.1 in the case where º(KX ) = 0 (for the de¯nition of º, see De¯nition 2.6): Theorem 1.2. -
Arxiv:2008.08852V2 [Math.AG] 15 Oct 2020 Assuming Theorem 2, We Conclude That M16 Cannot Be of General Type, Thus Es- Tablishing Theorem 1
ON THE KODAIRA DIMENSION OF M16 GAVRIL FARKAS AND ALESSANDRO VERRA ABSTRACT. We prove that the moduli space of curves of genus 16 is not of general type. The problem of determining the nature of the moduli space Mg of stable curves of genus g has long been one of the key questions in the field, motivating important developments in moduli theory. Severi [Sev] observed that Mg is unirational for g ≤ 10, see [AC] for a modern presentation. Much later, in the celebrated series of papers [HM], [H], [EH], Harris, Mumford and Eisenbud showed that Mg is of general type for g ≥ 24. Very recently, it has been showed in [FJP] that both M22 and M23 are of general type. On the other hand, due to work of Sernesi [Ser], Chang-Ran [CR1], [CR2] and Verra [Ve] it is known that Mg is unirational also for 11 ≤ g ≤ 14. Finally, Bruno and Verra [BV] proved that M15 is rationally connected. Our result is the following: Theorem 1. The moduli space M16 of stable curves of genus 16 is not of general type. A few comments are in order. The main result of [CR3] claims that M16 is unir- uled. It has been however recently pointed out by Tseng [Ts] that the key calculation in [CR3] contains a fatal error, which genuinely reopens this problem (after 28 years)! Before explaining our strategy of proving Theorem 1, recall the standard notation ∆0;:::; ∆ g for the irreducible boundary divisors on Mg, see [HM]. Here ∆0 denotes b 2 c the closure in Mg of the locus of irreducible 1-nodal curves of arithmetic genus g. -
Algebraic Surfaces with Minimal Betti Numbers
P. I. C. M. – 2018 Rio de Janeiro, Vol. 2 (717–736) ALGEBRAIC SURFACES WITH MINIMAL BETTI NUMBERS JH K (금종해) Abstract These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called Q-homology projective planes. Fake projective planes and the complex projective plane are smooth examples. We describe recent progress in the study of such surfaces, singular ones and fake projective planes. We also discuss open questions. 1 Q-homology Projective Planes and Montgomery-Yang problem A normal projective surface with the Betti numbers of the complex projective plane CP 2 is called a rational homology projective plane or a Q-homology CP 2. When a normal projective surface S has only rational singularities, S is a Q-homology CP 2 if its second Betti number b2(S) = 1. This can be seen easily by considering the Albanese fibration on a resolution of S. It is known that a Q-homology CP 2 with quotient singularities (and no worse singu- larities) has at most 5 singular points (cf. Hwang and Keum [2011b, Corollary 3.4]). The Q-homology projective planes with 5 quotient singularities were classified in Hwang and Keum [ibid.]. In this section we summarize progress on the Algebraic Montgomery-Yang problem, which was formulated by J. Kollár. Conjecture 1.1 (Algebraic Montgomery–Yang Problem Kollár [2008]). Let S be a Q- homology projective plane with quotient singularities. Assume that S 0 := S Sing(S) is n simply connected. Then S has at most 3 singular points. This is the algebraic version of Montgomery–Yang Problem Fintushel and Stern [1987], which was originated from pseudofree circle group actions on higher dimensional sphere.