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We show that a complex planar curve homeomorphic to the projective line has at most four singular points. If it has exactly four then it has degree five and is unique up to a projective equivalence., 35 pages

We give examples of infinitely extendable (not as cones) arithmetically Cohen–Macaulay and arithmetically Gorenstein subvarieties of projective spaces and which are not complete intersections. The proof uses the computation of the dimension of the Hilbert scheme of codimension 2 subschemes of projective spaces due to G. Ellingsrud and of arithmetically Gorenstein codimension 3 subschemes due to J. O. Kleppe and R.-M. Miró-Roig.

For smooth families with maximal variation, whose general fibers have semi-ample canonical bundle, the generalized Viehweg hyperbolicity conjecture states that the base spaces of such families are of log general type. This deep conjecture was recently proved by Popa-Schnell using the theory of Hodge modules and a theorem by Campana-P\u{a}un. In this paper we prove that those base spaces are pseudo Kobayashi hyperbolic, as predicted by the Lang conjecture: any complex quasi-projective manifold is pseudo Kobayashi hyperbolic if it is of log general type. As a consequence, we prove the Brody hyperbolicity of moduli spaces of polarized manifolds with semi-ample canonical bundle. This answers a question by Viehweg-Zuo in 2003. We also prove the Kobayashi hyperbolicity of base spaces of effectively parametrized families of minimal projective manifolds of general type. This generalizes previous work by To-Yeung, in which they further assumed that these families are canonically polarized.

We prove that small deformations of a projective variety of general type are also projective varieties of general type, with the same plurigenera. Version 2: small changes in first half. Improved version of the second half is now a separate preprint (Seshadri's criterion and openness of projectivity). Version 3: Footnote corrects citation on p.4.

We show that a direct limit of projective contramodules (over a right linear topological ring) is projective if it has a projective cover. A similar result is obtained for $\infty$-strictly flat contramodules of projective dimension not exceeding $1$, using an argument based on the notion of the topological Jacobson radical. Covers and precovers of direct limits of more general classes of objects, both in abelian categories with exact and with nonexact direct limits, are also discussed, with an eye towards the Enochs conjecture about covers and direct limits, using locally split (mono)morphisms as the main technique. In particular, we offer a simple elementary proof of the Enochs conjecture for the left class of an $n$-tilting cotorsion pair in an abelian category with exact direct limits., Comment: LaTeX 2e with pb-diagram and xy-pic, 31 pages, 3 commutative diagrams; v.2: a new author joined, the title changed, paper greatly expanded, main result generalized from countable to uncountable direct limits; v.4: Introduction expanded, many references added, several misprints corrected, the numbering of sections shifted to agree with the journal version, former Remark 7.4 is now Corollary 8.4

We investigate the problem of classifying pencils of plane curves of degree d up to projective equivalence. We obtain explicit stability criteria in terms of the log canonical threshold by relating the stability of a pencil to the stability of the curves lying on it.

We show that the multipoint Seshadri constant determines the maximum possible radii of embeddings of Kähler balls and vice versa.

We prove that the profinite completion of a profinite projective group is projective.

We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of delta-admissible chains. Roughly speaking, a delta-admissible chain is a simplicial semi-algebraic chain meeting the "faces" properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via delta-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map., 2020. May. 20: Calculation in Section 4.2 is made clearer

Let $X_1$ and $X_2$ be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of $X_1$ is abelian if and only if the André motive of $X_2$ is abelian. Applying this to manifolds of $\mbox {K3}^{[n]}$, generalized Kummer and OG6 deformation types, we deduce that their André motives are abelian. As a consequence, we prove that all Hodge classes in arbitrary degree on such manifolds are absolute. We discuss applications to the Mumford–Tate conjecture, showing in particular that it holds for even degree cohomology of such manifolds.

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the following. (1) Over a quotient of a regular local ring by a regular sequence, every finitely generated module has finite quasi-projective dimension. (2) The Auslander--Buchsbaum formula and the depth formula for modules of finite projective dimension remain valid for modules of finite quasi-projective dimension. (3) Several results on vanishing of Tor and Ext hold for modules of finite quasi-projective dimension., Comment: Final version, to appear in Pacific Journal of Mathematics

Volumes of line bundles are known to exist as limits on generically reduced projective schemes. However, it is not known if they always exist as limits on more general projective schemes. We show that they do always exist as a limit on a codimension one subscheme of a nonsingular projective variety., 11 pages. Final version

This paper studies the class of rings in which every finitely projective ideal is projective (FPP-ring for short). We examine the transfer of this property to various context of commutative ring extensions such as direct product, homomorphic image, trivial ring extension and amalgamation ring. Our work is motivated by an attempt to generate new original classes of rings possessing this property.

Recently, Q. Han [7] proved a Second Main Theorem for algebraically nondegenerate meromorphic maps over p-Parabolic manifolds intersecting with hypersurfaces in general position in smooth projective algebraic variety, extending certain results of H. Cartan, L. Ahlfors, W. Stoll, M. Ru and Philip P. W. Wong. In this paper, we will prove a general form of Second Main Theorem for meromorphic maps from p-Parabolic manifold into smooth projective variety intersecting with hypersurfaces in subgeneral position. As an application of that result, we get a Second Main Theorem for meromorphic maps on p-Parabolic manifold intersecting with hypersurfaces in l-subgeneral position, which extends the result of Q. Han [7].

For the $p$-localized sphere $\mathbb {S}^{2m-1}_{(p)}$ with $p >3$ a prime, we prove that the homotopy nilpotency satisfies $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}, with respect to any homotopy associative $H$-structure on $\mathbb {S}^{2m-1}_{(p)}$. We also prove that $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}= 1$ for all but a finite number of primes $p >3$. Then, for the loop space of the associated $\mathbb {S}^{2m-1}_{(p)}$-projective space $\mathbb {S}^{2m-1}_{(p)}P(n-1)$, with $m,n\ge 2$ and $m\mid p-1$, we derive that $\mbox {nil}\ \Omega (\mathbb {S}^{2m-1}_{(p)}P (n-1))\le 3$.

A class of projective binary linear codes are constructed and their weight distributions are investigated using Weil sums. They have at most three nonzero weights, containing some optimal codes. Their dual codes are also studied and some of them are either optimal or almost optimal.

We show, among other things, that for each integer $$n \ge 3$$ , there is a smooth complex projective rational variety of dimension n, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.

We compute global sections of tensor products and symmetric products of several secant bundles of a nonsingular projective curve. We closely follow the approach of Gentiana Danila’s work on the similar problem for nonsingular projective surfaces.

This paper shall be concerned with the notion of von-Neumann regular acts. First, the notions of semiretraction homomorphisms and fully semiretractable (FSR) S-acts are introduced and some properties of them are investigated. Finally, using semiretraction notion, we investigate locally projective and von-Neumann regular S-acts.

The paper studies Rogers semilattices, i.e. upper semilattices induced by the reducibility between numberings. Under the assumption of Projective Determinacy, we prove that for every non-zero natural number $$n$$ , there are infinitely many pairwise elementarily non-equivalent Rogers semilattices for $$\Sigma^{1}_{n}$$ -computable families.

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated to a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal{T}$, there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal{T}$. Examples related to inflation categories and weighted projective lines are discussed., Comment: all comments are welcome!

We discuss a minimisation problem of the degree of the Chow–Mumford (CM) line bundle among all possible fillings of a polarised family with fixed general fibers, motivated by the study of the moduli space of K-stable Fano varieties. We show that such minimisation implies the slope semistability of the fiber if the central fiber is smooth.

In a recent paper, Dave Benson and Peter Symonds defined a new invariant γG(M) for a finite dimensional module M of a finite group G which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for permutation modules of the symmetric group corresponding to two-part partitions using tools from representation theory and combinatorics.

For $d\in\{5,6\}$, we classify arrangements of $d + 2$ points in $\mathbf{RP}^{d-1}$ for which the minimum distance is as large as possible. To do so, we leverage ideas from matrix and convex analysis to determine the best possible codes that contain equiangular lines, and we introduce a notion of approximate Positivstellensatz certificates that promotes numerical approximations of Stengle's Positivstellensatz certificates to honest certificates., Comment: code included in ancillary files

For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that its group of automorphisms faithfully acts on its cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and groups of automorphisms for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and Viehweg. In positive characteristic, a lifting criterion of automorphisms reduce the faithfulness problem to characteristic zero. To apply this criterion, we prove the degeneration of the Hodge-de Rham spectral sequences for a family of smooth cfinite yclic coverings, and the infinitesimal Torelli theorem for finite cyclic coverings defined over an arbitrary field.

We define the classes of strongly ω1-weak pω·2+n-projective, solidly ω1-weak pω·2+n-projective and nicely ω1-weak pω·2+n-projective abelian p-groups and study their crucial properties. This continues our recent investigations of this branch, published in Hacettepe J. Math. Stat. (2013) and Bull. Malaysian Math. Sci. Soc. (2014), respectively.

Motivated by a recent question of Peyre, we apply the Hardy-Littlewood circle method to count "sufficiently free" rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the rational numbers.

The fractional linear action of SO(3) on the projective space SO(3) is proven to be a (globally defined) projective action.

In this paper, we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well known flat torus theorem for ${\rm CAT}(0)$ spaces., Minor revisions, to appear in the Journal of the London Mathematical Society. 20 pages. Comments welcome

The notion of the Ricci curvature is defined for sprays on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. In this paper, we introduce the notion of projectively Ricci-flat sprays. We establish a global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature. Then we study and characterize projectively Ricci-flat Randers metrics.

We show that the boundary of any bounded strongly pseudoconvex complete circular domain in $\mathbb C^2$ must contain points that are exceptionally tangent to a projective image of the unit sphere.

Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S ⊗L X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X (possibly unbounded) with those of the S-complex S ⊗L X. It is shown that if R is a Noetherian ring of finite Krull dimension and ϕ: R → S is a faithfully flat ring homomorphism, then for any homologically degree-wise finite complex X, there is an equality GpdRX = GpdS(S ⊗L X). Similar result is obtained for Ding projective dimension of the S-complex S ⊗L X.

Let A be an artin algebra. An A-module M is semi-Gorenstein-projective provided that Exti(M,A) = 0 for all i ≥ 1. If M is Gorenstein-projective, then both M and its A-dual M∗ are semi-Gorenstein projective. As we have shown recently, the converse is not true, thus answering a question raised by Avramov and Martsinkovsky. The aim of the present note is to analyse in detail the modules M such that both M and M∗ are semi-Gorenstein-projective.

Projective modules play an important role in the study of the category of modules over rings and in the characterization of various classes of rings. Several characterizations of projective objects which are equivalent for modules over rings are not necessarily equivalent for semimodules over an arbitrary semiring. We study several of these notions, in particular the e-projectivity of semimodules introduced by the first author using his new notion of exact sequences of semimodules. As pushouts of semimodules play an important role in some of our proofs, we investigate them and give a constructive proof of their existence in a way that proved be very helpful.

We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed $\mathrm{PGL}_2(\mathbb{C})$ local systems on the associated marked bordered surface. We prove that the image of this map is contained in the union of the domains of the cluster charts. We discuss a number of open questions concerning this monodromy map., Comment: 54 pages. Version 2: Incorporated suggestions from referee

We generalize the results of Skorobogatov and Zarhin considering the commutativity of Brauer groups (and Brauer-Manin sets) with taking product of two varieties, by relaxing the condition that varieties are projective., 10 pages, 20190703/3e323ffe, to appear in the Bulletin of the London Mathematical Society

There is a question formulated by R. Ware in 1971. That is, if a projective right R R -module P P has a unique maximal submodule L L , then L L is the largest maximal submodule of P P . First we give a structure theorem for such a projective module. Next we define the notion of a minimal generator set and give an affirmative answer to this question by using this notion. In the last section, we give an example of a non-quasi-small module to discuss whether the answer to this question is affirmative or negative. The key theorem in Comm. Algebra 31 (2003), pp. 4195–4241 asserts that answer to this question is negative. But our example seems to be a counterexample of this key theorem.

The following problem was originally posed by B.H. Neumann and H. Neumann. Suppose that a group $G$ can be generated by $n$ elements and that $H$ is a homomorphic image of $G$. Does there exist, for every generating $n$-tuple $(h_1,\ldots, h_n)$ of $H$, a homomorphism $\vartheta \colon G \to H$ and a generating $n$-tuple $(g_1,\ldots,g_n)$ of $G$ such that $(g_1^\vartheta,\ldots,g_n^\vartheta) = (h_1,\ldots,h_n)$? M.J. Dunwoody gave a negative answer to this question, by means of a carefully engineered construction of an explicit pair of soluble groups. Via a new approach we produce, for $n = 2$, infinitely many pairs of groups $(G,H)$ that are negative examples to the Neumanns' problem. These new examples are easily described: $G$ is a free product of two suitable finite cyclic groups, such as $C_2 \ast C_3$, and $H$ is a suitable finite projective special linear group, such as $\mathrm{PSL}(2,p)$ for a prime $p \ge 5$. A small modification yields the first negative examples $(G,H)$ with $H$ infinite., Comment: 10 pages, minor corrections

It is well known that the decomposition of injective modules over Noether rings and the decomposition of projective modules over Artinian rings are some of the most beautiful and important results in commutative algebra. The goal of the paper is to prove similar results for graded rings; this is important to understand the structure of modules over graded rings. The obtained results include structural theorems for gr-injective modules over gr-Noetherian G-graded commutative rings and for gr-finitely generated gr-projective modules over gr-Artinian G-graded commutative rings. Bibliography: 13 titles.

Imagine playing tic-tac-toe to lose. Two players, Xavier and Olivia, alternate marking squares as usual. As soon as one player owns three squares in a row, they lose. The combinatorial game “misere...

We construct a relative projective compactification of Dolbeault moduli spaces of Higgs bundles for reductive algebraic groups on families of projective manifolds that is compatible with the Hitchin morphism.