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Skoda's 1972 result on ideal generation is a crucial ingredient in the analytic approach to the finite generation of the canonical ring and the abundance conjecture. Special analytic techniques developed by Skoda, other than applications of the usual vanishing theorems and L2 estimates for the d-bar equation, are required for its proof. This note (which is part of a lecture given in the 60th birthday conference for Lawrence Ein) gives a simpler, more straightforward proof of Skoda's result, which makes it a natural consequence of the standard techniques in vanishing theorems and solving d-bar equation with L2 estimates. The proof involves the following three ingredients: (i) one particular Cauchy-Schwarz inequality for tensors with a special factor which accounts for the exponent of the denominator in the formulation of the integral condition for Skoda's ideal generation, (ii) the nonnegativity of Nakano curvature of the induced metric of a special co-rank-1 subbundle of a trivial vector bundle twisted by a special scalar weight function, and (iii) the vanishing theorem and solvability of d-bar equation with L2 estimates for vector bundles of nonnegative Nakano curvature on a strictly pseudoconvex domain. Our proof gives readily other similar results on ideal generation.

Let k be an algebraically closed field of characteristic zero and K a finitely generated field over k. Let Σ be a central simple K-algebra, X a normal projective model of K and Λ a sheaf of maximal O X -orders in Σ. There is a ramification Q -divisor Δ on X, which is related to the canonical bimodule ω Λ by an adjunction formula. It only depends on the class of Σ in the Brauer group of K. When the numerical abundance conjecture holds true, or when Σ is a central simple algebra, we show that the Gelfand-Kirillov dimension (or GK dimension) of the canonical ring of Λ is one more than the Iitaka dimension (or D-dimension) of the log pair ( X , Δ ) . In the case that Σ is a division algebra, we further show that this GK dimension is also one more than the transcendence degree of the division algebra of degree zero fractions of the canonical ring of Λ. We prove that these dimensions are birationally invariant when the b-log pair determined by the ramification divisor has b-canonical singularities. In that case we refer to the Iitaka (or D-dimension) of ( X , Δ ) as the Kodaira dimension of the order Λ. For this, we establish birational invariance of the Kodaira dimension of b-log pairs with b-canonical singularities. We also show that the Kodaira dimension can not decrease for an embedding of central simple algebras, finite dimensional over their centres, which induces a Galois extension of their centres, and satisfies a condition on the ramification which we call an effective embedding. For example, this condition holds if the target central simple algebra has the property that its period equals its index. In proving our main result, we establish existence of equivariant b-terminal resolutions of G-b-log pairs and we also find two variants of the Riemann-Hurwitz formula. The first variant applies to effective embeddings of central simple algebras with fixed centres while the second applies to the pullback of a central simple algebra by a Galois extension of its centre. We also give two different local characterizations of effective embeddings. The first is in terms of complete local invariants, while the second uses Galois cohomology.

The famous structure theorem of Buchsbaum and Eisenbud gives a complete characterization of Gorenstein ideals of codimension 3 and their minimal free resolutions. We generalize the ideas of Buchsbaum and Eisenbud from Gorenstein ideals to Gorenstein algebras and present a description of Gorenstein algebras of any odd codimension. As an application we study the canonical ring of a numerical Godeaux surface., 14 pages

The automorphism group of a curve is studied from the viewpoint of the canonical embedding and Petri's theorem. A criterion for identifying the automorphism group as an algebraic subgroup the general linear group is given. Furthermore the action of the automorphism group is extended to an action of the minimal free resolution of the canonical ring of the curve $X$., 16 pages

We prove that the graded ring of modular forms of weight divisible by 3 is naturally isomorphic to a certain log canonical ring of the associated elliptic modular surface. This extends the Shioda correspondence between weight 3 cusp forms and holomorphic 2-forms.

We will prove the following results for $3$-fold pairs $(X,B)$ over an algebraically closed field $k$ of characteristic $p>5$: log flips exist for $\Q$-factorial dlt pairs $(X,B)$; log minimal models exist for projective klt pairs $(X,B)$ with pseudo-effective $K_X+B$; the log canonical ring $R(K_X+B)$ is finitely generated for projective klt pairs $(X,B)$ when $K_X+B$ is a big $\Q$-divisor; semi-ampleness holds for a nef and big $\Q$-divisor $D$ if $D-(K_X+B)$ is nef and big and $(X,B)$ is projective klt; $\Q$-factorial dlt models exist for lc pairs $(X,B)$; terminal models exist for klt pairs $(X,B)$; ACC holds for lc thresholds; etc., 48 pages, final version to appear in Annales Scientifiques de l'ENS

Consider modular forms arising from a finite-area quotient of the upper-half plane by a Fuchsian group. By the classical results of Kodaira-Spencer, this ring of modular forms may be viewed as the log spin canonical ring of a stacky curve. In this paper, we tightly bound the degrees of minimal generators and relations of log spin canonical rings. As a consequence, we obtain a tight bound on the degrees of minimal generators and relations for rings of modular forms of arbitrary integral weight., Comment: 36 pages. To appear in the Annales de l'Insitut Fourier

The log canonical ring of a projective plt pair with the Kodaira dimension two is finitely generated., Comment: 11 pages, v2: revision following the referee's comments

The canonical ring \({S_{D} = \oplus_{d\geq0}H^{0}(X, \lfloor dD\rfloor)}\) of a divisor D on a curve X is a natural object of study; when D is a \({\mathbb{Q}}\)-divisor, it has connections to projective embeddings of stacky curves and rings of modular forms. We study the generators and relations of \({S_D}\) for the simplest curve \({X = \mathbb{P}^1}\). When D contains at most two points, we give a complete description of \({S_D}\); for general D, we give bounds on the generators and relations. We also show that the generators (for at most five points) and a Grobner basis of relations between them (for at most four points) depend only on the coefficients in the divisor D, not its points or the characteristic of the ground field; we conjecture that the minimal system of relations varies in a similar way. Although stated in terms of algebraic geometry, our results are proved by translating to the combinatorics of lattice points in simplices and cones.

Topos theory is a category-theoretical axiomatization of set theory. Model categories are a category-theoretical framework for abstract homotopy theory. They are complete and cocomplete categories endowed with three classes of morphisms (called brations, co brations and weak equivalences) satisfying certain axioms. We would like to present an abstract framework for classical di erential geometry as an extension of topos theory, hopefully comparable with model categories for homotopy theory. Functors from the category W of Weil algebras to the category Sets of sets are called Weil spaces by Wolfgang Bertram and form the Weil topos after Eduardo J. Dubuc. The Weil topos is endowed intrinsically with the Dubuc functor, a functor from a larger category eW of cahiers algebras to the Weil topos standing for the incarnation of each algebraic entity of eW in the Weil topos. The Weil functor and the canonical ring object are to be de ned in terms of the Dubuc functor. The principal objective of this paper is to present a category-theoretical axiomatization of theWeil topos with the Dubuc functor intended to be an adequate framework for axiomatic classical di erential geometry. We will give an appropriate formulation and a rather complete proof of a generalization of the familiar and desired fact that the tangent space of a microlinear Weil space is a module over the canonical ring object.

We prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are klt if and only if R(-K_X) is finitely generated. We introduce a notion of nefness for non-Q-Gorenstein varieties and study some of its properties. We then focus on these properties for non-Q-Gorenstein toric varieties., Comment: Updated version

Let $C$ be a smooth curve which is complete intersection of a quadric and a degree $k>2$ surface in $\mathbb{P}^3$ and let $C^{(2)}$ be its second symmetric power. In this paper we study the finite generation of the extended canonical ring $R(\Delta,K) := \bigoplus_{(a,b)\in\mathbb{Z}^2}H^0(C^{(2)},a\Delta+bK)$, where $\Delta$ is the image of the diagonal and $K$ is the canonical divisor. We first show that $R(\Delta,K)$ is finitely generated if and only if the difference of the two $g_k^1$ on $C$ is torsion non-trivial and then show that this holds on an analytically dense locus of the moduli space of such curves., Comment: 16 pages

RING-between-RING (RBR) E3s contain RING1 domains that are structurally similar yet mechanistically distinct from canonical RING domains. Both types of E3 bind E2∼ubiquitin (E2∼Ub) via their RINGs but canonical RING E3s promote closed E2∼Ub conformations required for direct Ub transfer from the E2 to substrate, while RBR RING1s promote open E2∼Ub to favor Ub transfer to the E3 active site. This different RING/E2∼Ub conformation determines its direct target, which for canonical RING E3s is typically a substrate or substrate-linked Ub, but is the E3 active-site cysteine in the case of RBR-type E3s. Here we show that a short extension of HHARI RING1, namely Zn2+-loop II, not present in any RING E3s, acts as a steric wedge to disrupt closed E2∼Ub, providing a structural explanation for the distinctive RING1-dependent conformational restriction mechanism utilized by RBR E3s.

RING E3 ligases catalyze the transfer of ubiquitin (Ub) from E2 ubiquitin-conjugating enzyme thioesterified with Ub (E2~Ub) to substrate. For RING E3 dimers, the RING domain of one subunit and tail of the second cooperate to prime Ub, but how this is accomplished by monomeric RING E3s in the absence of a tail-like component is currently unknown. Here, we present a crystal structure of a monomeric RING E3, Tyr363-phosphorylated human CBL-B, bound to a stabilized Ub-linked E2, revealing a similar mechanism in activating E2~Ub. Both pTyr363 and the pTyr363-induced element interact directly with Ub's Ile36 surface, improving the catalytic efficiency of Ub transfer by ~200-fold. Hence, interactions outside the canonical RING domain are crucial for optimizing Ub transfer in both monomeric and dimeric RING E3s. We propose that an additional non-RING Ub-priming element may be a common RING E3 feature.

We consider a family of surfaces of general type $S$ with $K_S$ ample, having $K^2_S = 24, p_g (S) = 6, q(S)=0$. We prove that for these surfaces the canonical system is base point free and yields an embedding $\Phi_1 : S \rightarrow \mathbb{P}^5$. This result answers a question posed by G. and M. Kapustka. We discuss some related open problems, concerning also the case $p_g(S) = 5$, where one requires the canonical map to be birational onto its image., Comment: 10 pages, dedicated to the 60th birthday of Philippe Ellia

This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in [EGA,III.2.6.2] and raised again by Viehweg., Comment: Version 2: Several details corrected

Let R be a commutative Noetherian ring and a a proper ideal of R. We show that if n := gradeRa, then . We also prove that, for a nonnegative integer n such that = 0 for every i ≠ n, if for all i > 0 and z ∈ a, then is a homomorphic image of R, where Rz is the ring of fractions of R with respect to a multiplicatively closed subset ﹛z j | j ⩾ 0﹜ of R. Moreover, if HomR(Rz , R) = 0 for all z ∈ a, then is an isomorphism, where is the canonical ring homomorphism R → .

We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance theorem for irregular fourfolds. We obtain (a) as a direct consequence of the existence of four-dimensional log minimal models by using Fukuda's theorem on the four-dimensional log abundance conjecture. We can prove (b) only by using traditional arguments. More precisely, we prove the abundance conjecture for irregular $(n+1)$-folds on the assumption that the minimal model conjecture and the abundance conjecture hold in dimension $\leq n$., 14 pages; v2: completely revised and expanded version, v3: Section 5 in v2 was removed because it contained a conceptual mistake

We produce a family of numerical Campedelli surfaces with Z / 6 \mathbb {Z}/6 torsion by constructing the canonical ring of the étale 6 to 1 cover using serial unprojection. In Section 2 we develop the necessary algebraic machinery. Section 3 contains the numerical Campedelli surface construction, while Section 4 contains remarks and open questions.

The purpose of this note is to review an algebraic proof of the finite generation theorem due to Birkar-Cascini-Hacon-McKernan whose method is based on the Minimal Model Program. A survey article for Current Development in Mathematics 2007., 45 pages

We prove that the log canonical ring of a klt pair of dimension $3$ with $\mathbb{Q}$-boundary over an algebraically closed field of characteristic $p>5$ is finitely generated. In the process we prove log abundance for such pairs in the case $\kappa=2$.

This article discusses the geometric application of the method of multiplier ideal sheaves. It first briefly describes its application to effective problems in algebraic geometry and then presents and explains its application to the deformational invariance of plurigenera for general compact algebraic manifolds. Finally its application to the conjecture of the finite generation of the canonical ring is explored and the use of complex algebraic geometry in complex Neumann estimates is discussed., This article is an expanded version of a talk given on August 23, 2004 in the International Conference on Several Complex Variables in Capital Normal University, Beijing, China and will appear in "Science in China," Series A Mathematics 2005 Volume 48, as part of the proceedings of the conference

We shall work over the complex number field C. Let X be a non-singular projective curve of genus g. We always assume that it is non-hyperelliptic and sometimes identify it with its canonical image in PH(X,KX). By Max Noether’s theorem, the canonical ring of X is generated in degree one and the canonical map is a projectively normal embedding. Furthermore, a well-known theorem of Enriques-Petri states that X is cut out by hyperquadrics if Cliff(X), the Clifford index of X, is bigger than one. Then Mark Green [4] conjectured that the non-vanishing of a certain higher syzygy can be characterized by the Clifford index, which has been verified in many cases. From these, we learn that Cliff(X) reflects the algebraic structure on X better than the gonality gon(X), while two invariants are almost equivalent [2]. For a non-negative integer k, we denote by X the k-th symmetric product of X whose points are considered as effective divisors of degree k on X. When k = 0, we understand that X is one point corresponding to the zero divisor. Let Dk be the open subset of X consisting of effective divisors D which spans scheme theoretically a (k − 1)-plane 〈D〉 in PH(X,KX). For D ∈ Dk, we put KX,D = KX − [D], where [D] denotes the line bundle associated to D. We let ΦKX,D : X → PH(X,KX,D) be the rational map associated to the complete linear system |KX,D|. In this article, we consider two kinds of uniformity questions with respect to Dk

We construct from a general del Pezzo surface of degree 1 a Gorenstein stable surfaces $X$ with $K_X^2=1$ and $p_g(X)=q(X)=0$. These surfaces are not smoothable but give an open subset of an irreducible component of the moduli space of stable Godeaux surfaces. In a particular example we also compute the canonical ring explicitly and discuss the behaviour of pluricanonical maps., Comment: 12 pages, 4 figures

Let C be a Gorenstein curve which is either reduced or contained in a smooth algebraic surface. We show that the canonical ring R(C, ωC) = L k≥0 H 0 (C, ωC ⊗k ) is generated in degree 1 if C is 3-connected and not (honestly) hyperelliptic; we show moreover that R(C, L) = L k≥0 H 0 (C, L ⊗k ) is generated in degree 1 if C is reduced and L is an invertible sheaf such that deg L|B ≥ 2pa(B)+ 1 for every B ⊆ C. keyword: algebraic curve, Noether’s theorem, canonical ring Mathematics Subject Classification (2010) 14H20, 14C20, 14H51

The $K$-theory of the stable Higson corona of a coarse space carries a canonical ring structure. This ring is the domain of an unreduced version of the coarse co-assembly map of Emerson and Meyer. We show that the target also carries a ring structure and co-assembly is a ring homomorphism, provided that the given coarse space is contractible in a coarse sense.

In this paper we construct families of algebraic nonsingular 3- folds X of general type having minimal $K_X^3$ for their $p_g$, in order that the canonical morphism is finite of degree 3. For such 3-folds it is $K_X^3=3(p_g-3)$ and $p_g$ is odd.

Upper bounds for the degree of the generators of the canonical rings of surfaces of general type were found by Ciliberto [7]. In particular it was established that the canonical ring of a minimal surface of general type with p g = 0 is generated by its elements of degree lesser or equal to 6, ([7];, Th. (3.6)). This was the best bound possible to obtain at the time, since Reider's results, [11], were not yet available. In this note, this bound is improved in some cases (Theorems (3.1), (3.2)). ¶ In particular it is shown that if K 2≥ 5, or if K 2≥ 2 and |2 K S | is base point free this bound can be lowered to 4. This result is proved by showing first that, under the same hypothesis, the degree of the bicanonical map is lesser or equal to 4 if K 2≥ 3, (Theorem (2.1)), implying that the hyperplane sections of the bicanonical image have not arithmetic genus 0. The result on the generation of the canonical ring then follows by the techniques utilized in [7].

This survey is an invitation to recent developments in higher dimensional birational geometry.

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated., to appear in Duke Math. J

The human ubiquitin-conjugating enzyme Rad6 (E2), with ubiquitin ligase enzyme Rad18 (RING E3), monoubiquitinates proliferating cell nuclear antigen at stalled replication forks in DNA translesion synthesis. Here, we determine the structure of the homodimeric Rad18 RING domains by X-ray crystallography and classify it to RING-RING dimers that dimerize through helices adjacent to the RING domains and through the canonical RING domains. Using NMR spectroscopy and site-directed mutagenesis, we demonstrate that the Rad6b binding site, for the Rad18 RING domain, strongly resembles that of other E2/E3 RING/U-box complexes. We show that the homodimeric Rad18 RING domain can recruit two Rad6b E2 enzymes, whereas the full-length Rad18 homodimer binds only to a single Rad6b molecule. Such asymmetry is a common feature of RING-RING heterodimers and has been observed for the CHIP U-box homodimer. We propose that asymmetry may be a common feature of dimeric RING E3 ligases.

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring \({ R(C, \omega_C)=\bigoplus_{k\geq 0} H^0(C, {\omega_C}^{\otimes k})}\) is generated in degree 1 if C is numerically four-connected, not hyperelliptic and even (i.e. with ωC of even degree on every component). As a corollary we show that on a smooth algebraic surface of general type with pg(S) ≥ 1 and q(S) = 0 the canonical ring R(S, KS) is generated in degree ≤ 3 if there exists a curve \({C \in |K_S|}\) numerically three-connected and not hyperelliptic.

We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an explicit construction of the transgression map $T_1: H^*(\Gamma;A) \to H^{*-1}((N\rtimes \Gamma;A)$ for any crossed module $N\to \Gamma$ and prove that any element in the image is $\infty$-multiplicative. As a consequence, we prove that, under some mild conditions, for a crossed module $N \to \gm$ and any $e \in \check{Z}^3(\Gamma;S^1)$, that the equivariant twisted K-theory group $K^*_{e,\Gamma}(N)$ admits a ring structure. As an application, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted K-theory group $K_{[c], G}^* (G)$ is endowed with a canonical ring structure $K^{i+d}_{[c],G}(G)\otimes K^{j+d}_{[c],G}(G)\to K^{i+j+d}_{[c], G}(G)$, where $d=dim G$ and $[c]\in H^2(G\rtimes G;S^1)$., Comment: 47 pages. To appear in Crelle

Let p denote an odd prime. We show that the spectrum [ E(n), the In-adic completion of Johnson and Wilson’s E(n), admits a unique topological A∞ structure compatible with its canonical ring spectrum structure. Furthermore, the canonical morphism of ring spectra [ E(n) −→ K(n) admits an A∞ structure whichever of the uncountably many A∞ structures of A. Robinson is imposed upon K(n), the n th Morava K-theory at the prime p. We construct an inverse system of A∞ module spectra over [ E(n) · · · −→ E(n)/I n −→ E(n)/I n −→ · · · −→ E(n)/In = K(n) for which holim ←− k E(n)/I n ' [ E(n). §0 Introduction. Recently, A. Robinson has described a theory of A∞ ring spectra, their module spectra and the associated derived categories (see [9], [10], [11], [12]). As a special case, in [12] he showed that at an odd prime p the n th Morava K-theory spectrum K(n) admits uncountably many distinct A∞ structures compatible with its canonical multiplication. The principal result of the present work is to show that E(n), the (Noetherian) In-adic completion of the spectrum E(n) defined by D. C. Johnson and W. S. Wilson, admits a unique topological A∞ structure compatible with its canonical ring spectrum structure; moreover, the canonical morphism of ring spectra E(n) −→ K(n) can be given the structure of an A∞ morphism whichever of Robinson’s A∞ structures we take. As an application, we construct an inverse system of A∞ module spectra over E(n) · · · −→ E(n)/I n −→ E(n)/I n −→ · · · −→ E(n)/In = K(n)

In this article we prove a non-vanishing statement, as well as several properties of metrics with minimal singularities of adjoint bundles. Our arguments involve many ideas from Y.-T. Siu's analytic proof of the finite generation of the canonical ring. An important technical tool is the notion of relative critical exponent of two closed positive currents with respect to a measure., No figures

In the 80th birthday conference for Professor LU Qikeng in June 2006 I gave a talk on the analytic approach to the finite generation of the canonical ring for a compact complex algebraic manifold of general type. This article is my contribution to the proceedings of that conference from my talk. In this article I give an overview of the analytic proof and focus on explaining how the analytic method handles the problem of infinite number of interminable blow-ups in the intuitive approach to prove the finite generation of the canonical ring. The proceedings of the LU Qikeng conference will appear as Issue No. 4 of Volume 51 of Science in China Series A: Mathematics (www.springer.com/math/applications/journal/11425).

This article is written for the Proceedings of the Conference on Current Developments in Mathematics in Harvard University, November 16-17, 2007. It is an exposition of the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type. It lists and discusses the main techniques and explains how they are put together in the proof. Of the various main techniques some special attention is given to (i) the technique of discrepancy subspaces and (ii) the technique of subspaces of minimum additional vanishing.

We introduce the concept of polynomial operation from the Burnside ring functor A to other representation functors R, which includes operations such as symmetric powers, φ-operations and Adams operations. The set of polynomial operations from A to R, Pol(A,R), has a canonical ring structure. We give a complete description of the additive structure of this ring as well as a family of generating operations.

In this article we present a new technique to handle the study of homogeneous rings of a projective variety endowed with a finite or a generically finite morphism to another variety Y whose geometry is easier to handle. Under these circumstances it is possible to use the information given by the algebra structure of \( {{\mathcal{O}}_{X}} \) over \( {{\mathcal{O}}_{Y}} \) to describe the homogeneous ring associated to line bundles which are pullback of line bundles on Y. In this article we illustrate our technique to study the canonical ring of curves (a well-known ring that we revisit with this new technique) equipped with a suitable finite morphism and homogeneous rings of certain class of Calabi-Yau threefolds.

In one of the main results of this paper, we find the degrees of the generators of the canonical ring of a regular algebraic surface X X of general type defined over a field of characteristic 0 0 , under the hypothesis that the canonical divisor of X X determines a morphism φ \varphi from X X to a surface of minimal degree Y Y . As a corollary of our results and results of Ciliberto and Green, we obtain a necessary and sufficient condition for the canonical ring of X X to be generated in degree less than or equal to 2 2 . We construct new examples of surfaces satisfying the hypothesis of our theorem and prove results which show that many a priori plausible examples cannot exist. Our methods are to exploit the O Y \mathcal {O}_{Y} -algebra structure on φ ∗ O X \varphi _{*}\mathcal {O}_{X} . These methods have other applications, including those on Calabi-Yau threefolds. We prove new results on homogeneous rings associated to a polarized Calabi-Yau threefold and also prove some existence theorems for Calabi-Yau covers of threefolds of minimal degree. These have consequences towards constructing new examples of Calabi-Yau threefolds.

A higher dimensional analogue of Kodaira's canonical bundle formula is obtained. As applications, we prove that the log-canonical ring of a klt pair with κ ≤ 3 is finitely generated, and that there exists an effectively computable natural number M such that |MKX| induces the Iitaka fibering for every algebraic threefold X with Kodaira dimension κ = 1.