Complete Linear System Research Articles

Syzygies of secant varieties of curves of genus 2

Ein, Niu and Park showed in [6] that if the degree of the line bundle L on a curve of genus g is at least 2g+2k+1, the k-th secant variety of the curve via the embedding defined by the complete linear system of L is normal, projectively normal and arithmetically Cohen-Macaulay, and they also proved some vanishing of the Betti diagrams. However, the length of the linear strand of weight k+1 of the resolution of the secant variety Σk of a curve of g≥2 is still mysterious. In this paper we calculate the complete Betti diagrams of the secant varieties of curves of genus 2 using Boij-Söderberg theory. The main idea is to find the pure diagrams that contribute to the Betti diagram of the secant variety via calculating some special positions of the Betti diagram.

Classification of subpencils for hyperplane sections on certain K3 surfaces

AbstractWe classify the subpencils of complete linear systems for the hyperplane sections on K3 surfaces obtained as the complete intersection of a hyperquadric and a hypercubic. The classification is done from three points of view, namely, the type of a general fibre, the base locus and the Horikawa index of the essential member. This classification shows the distinct phenomenons depending on the rank of the hyperquadrics containing the surface.

Integral Points on Varieties With Infinite Étale Fundamental Group

Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi \colon Y \to X$ of degree at least $2\dim (X)^{2}$, there exists an ample line bundle $\mathscr{L}$ on $Y$ such that for a general member $D$ of the complete linear system $|\mathscr{L}|$, $D$ is geometrically irreducible and any set of $\varphi (D)$-integral points on $X$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.

Geometry of points satisfying Cayley–Bacharach conditions and applications

In this paper, we study the geometry of points in complex projective space that satisfy the Cayley–Bacharach condition with respect to the complete linear system of hypersurfaces of given degree. In particular, we improve a result by Lopez and Pirola and we show that, if k≥1 and Γ={P1,…,Pd}⊂Pn is a set of distinct points satisfying the Cayley–Bacharach condition with respect to |OPn(k)|, with d≤h(k−h+3)−1 and 3≤h≤5, then Γ lies on a curve of degree h−1. Then we apply this result to the study of linear series on curves on smooth surfaces in P3. Moreover, we discuss correspondences with null trace on smooth hypersurfaces of Pn and on codimension 2 complete intersections.

Galois quotients of tropical curves and invariant linear systems

For a map $\varphi : \Gamma \rightarrow \Gamma'$ between tropical curves and an isometric action on $\Gamma$ of a finite group $K$, $\varphi$ is a $K$-Galois covering on $\Gamma'$ if $\varphi$ is a harmonic morphism, the degree of $\varphi$ coincides with the order of $K$ and the action of $K$ induces a transitive action on every fibre. We prove that for a tropical curve $\Gamma$ with an isometric action of a finite group $K$, there exists a rational map, from $\Gamma$ to a tropical projective space, which induces a $K$-Galois covering on the image with proper edge-multiplicities. As an application, we also prove that for a hyperelliptic tropical curve without one valent points and of genus at least two, the invariant linear system of the hyperelliptic involution $\iota$ of the canonical linear system, the complete linear system associated with the canonical divisor, induces an $\langle \iota \rangle$-Galois covering on a tree. This is an analogy of the fact that a compact Riemann surface is hyperelliptic if and only if the canonical map, the rational map induced by the canonical linear system, is a double covering on a projective line $\boldsymbol{P}^1$.

On linear stability and syzygy stability

In previous works, the authors investigated the relationships between linear stability of a generated linear series |V| on a curve C, and slope stability of the syzygy vector bundle M_{V,L} := ker (V otimes mathcal {O}_C rightarrow L). In particular, the second named author and L. Stoppino conjecture that, for a complete linear system |L|, linear (semi)stability is equivalent to slope (semi)stability of M_{L}. The first and third named authors proved that this conjecture holds in the two opposite cases: hyperelliptic and generic curves. In this work we provide a counterexample to this conjecture on any smooth plane curve of degree 7.

The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Let f:S'longrightarrow S be a cyclic branched covering of smooth projective surfaces over {mathbb {C}} whose branch locus Delta subset S is a smooth ample divisor. Pick a very ample complete linear system |{mathcal {H}}| on S, such that the polarized surface (S, |{mathcal {H}}|) is not a scroll nor has rational hyperplane sections. For the general member [C]in |{mathcal {H}}| consider the mu _{n}-equivariant isogeny decomposition of the Prym variety {{,mathrm{Prym},}}(C'/C) of the induced covering f:C'{:}{=}f^{-1}(C)longrightarrow C: Prym(C′/C)∼∏d|n,d≠1Pd(C′/C).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {{\\,\\mathrm{Prym}\\,}}(C'/C)\\sim \\prod _{d|n,\\ d\\ne 1}{\\mathcal {P}}_{d}(C'/C). \\end{aligned}$$\\end{document}We show that for the very general member [C]in |{mathcal {H}}| the isogeny component {mathcal {P}}_{d}(C'/C) is mu _{d}-simple with {{,mathrm{End},}}_{mu _{d}}({mathcal {P}}_{d}(C'/C))cong {mathbb {Z}}[zeta _{d}]. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map {mathcal {P}}_{d}(C'/C)subset {{,mathrm{Jac},}}(C')longrightarrow {{,mathrm{Alb},}}(S').

Effective Bertini theorem and formulas for multiplicity and the local Łojasiewicz exponent

The classical Bertini theorem on generic intersection of an algebraic set with hyperplanes states the following: Let X be a nonsingular closed subvariety ofPkn, where k is an algebraically closed field. Then there exists a hyperplaneH⊂Pknnot containing X and such that the schemeH∩Xis regular at every point. Furthermore, the set of hyperplanes with this property forms an open dense subset of the complete linear system|H|considered as a projective space. We show that one can effectively indicate a finite family of hyperplanes H such that at least one of them satisfies the assertion of the Bertini theorem, provided the characteristic of the field k is equal to zero. As an application of the method used in the proof we give effective formulas for the multiplicity and the Łojasiewicz exponent of polynomial mappings.

Generators of invariant linear system on tropical curves for finite isometry group

For a tropical curve $\Gamma$ and a finite subgroup $K$ of the isometry group of $\Gamma$, we prove, extending the work by Haase, Musiker and Yu ([6]), that the $K$-invariant part of the complete linear system associated to a $K$-invariant effective divisor on $\Gamma$ is finitely generated

Galois subspaces for smooth projective curves

Given an embedding of a smooth projective curve X of genus g≥1 into PN, we study the locus of linear subspaces of PN of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism X→P1. For genus g≥2 we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If g=1 and the embedding is given by a complete linear system, we prove that this locus is also a smooth projective variety whose positive-dimensional components are isomorphic to projective bundles over étale quotients of the elliptic curve, and we describe these components explicitly.

Enumerating linear systems on graphs

The divisor theory of graphs views a finite connected graph G as a discrete version of a Riemann surface. Divisors on G are formal integral combinations of the vertices of G, and linear equivalence of divisors is determined by the discrete Laplacian operator for G. As in the case of Riemann surfaces, we are interested in the complete linear system |D| of a divisor D—the collection of nonnegative divisors linearly equivalent to D. Unlike the case of Riemann surfaces, the complete linear system of a divisor on a graph is always finite. We compute generating functions encoding the sizes of all complete linear systems on G and interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the (dual of the) Jacobian group of G. If G is a cycle graph, our results lead to a bijection between complete linear systems and binary necklaces. Our results also apply to a model in which the Laplacian is replaced by an invertible, integral M-matrix.

Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

Abstract In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree $d$ rational curves in $\mathbb{P}^n$ when $d-n\leq 3$ and $d<2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d<2n$, the arithmetic genus of any non-degenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d-n$.

Geometry of ℙ2 blown up at seven points

Abstract In this paper, we prove that blown up at seven general points admits a conic bundle structure over ℙ1 and it can be embedded as (2, 2) divisor in ℙ1 × ℙ2. Conversely, any smooth surface in the complete linear system |(2, 2)| of ℙ1 × ℙ2 can be obtained as an embedding of blowing up ℙ2 at seven points. We also show that smooth surface linearly equivalent to (2, 2) in ℙ1 × ℙ2 has at most four (−2) curves.

Amoebas of curves and the Lyashko–Looijenga map

For any curve $\mathcal{V}$ in a toric surface $X$, we study the critical locus $S(\mathcal{V})$ of the moment map $\mu$ from $\mathcal{V}$ to its compactified amoeba $\mu(\mathcal{V})$. We show that for curves $\mathcal{V}$ in a fixed complete linear system, the critical locus $S(\mathcal{V})$ is smooth apart from some real codimension $1$ walls. We then investigate the topological classification of pairs $(\mathcal{V},S(\mathcal{V}))$ when $\mathcal{V}$ and $S(\mathcal{V})$ are smooth. As a main tool, we use the Lyashko-Looijenga mapping ($\mathcal{LL}$) relative to the logarithmic Gauss map $\gamma : \mathcal{V} \rightarrow \mathbb{C}P^1$. We prove two statements concerning $\mathcal{LL}$ that are crucial for our study: the map $\mathcal{LL}$ is algebraic; the map $\mathcal{LL}$ extends to nodal curves. It allows us to construct many examples of pairs $(\mathcal{V},S(\mathcal{V}))$ by perturbing nodal curves.

Monodromy and vanishing cycles in toric surfaces

Given an ample line bundle on a toric surface, a question of Donaldson asks which simple closed curves can be vanishing cycles for nodal degenerations of smooth curves in the complete linear system. This paper provides a complete answer. This is accomplished by reformulating the problem in terms of the mapping class group-valued monodromy of the linear system, and giving a precise determination of this monodromy group.

Line arrangements and configurations of points with an unexpected geometric property

We propose here a generalization of the problem addressed by the SHGH conjecture. The SHGH conjecture posits a solution to the question of how many conditions a general union$X$of fat points imposes on the complete linear system of curves in$\mathbb{P}^{2}$of fixed degree$d$, in terms of the occurrence of certain rational curves in the base locus of the linear subsystem defined by$X$. As a first step towards a new theory, we show that rational curves play a similar role in a special case of a generalized problem, which asks how many conditions are imposed by a general union of fat points on linear subsystems defined by imposed base points. Moreover, motivated by work of Di Gennaro, Ilardi and Vallès and of Faenzi and Vallès, we relate our results to the failure of a strong Lefschetz property, and we give a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements.

The vanishing cycles of curves in toric surfaces I

This article is the first in a series of two in which we study the vanishing cycles of curves in toric surfaces. We give a list of possible obstructions to contract vanishing cycles within a given complete linear system. Using tropical means, we show that any non-separating simple closed curve is a vanishing cycle whenever none of the listed obstructions appears.

Base-point-freeness of double-point divisors of smooth birational-divisors on conical rational scrolls

We work over an algebraically closed field of characteristic zero. The purpose of this paper is to prove that the complete linear system of the double point divisors of smooth birational-divisors on conical rational scrolls are base-point-free. A smooth birational-divisor on a conical rational scroll has a nonbirational inner center, that is a point on it from which the linear projection gives nonbirational map to its image. In the previous paper by the author, it was shown that for a projective variety without nonbirational inner centers, its double point divisor is base-point-free.

The canonical volume of minimal 3-folds of general type

Let [Formula: see text] be a nonsingular projective [Formula: see text]-fold of general type. Denote by [Formula: see text] the [Formula: see text]-canonical map of [Formula: see text] which is the rational map naturally associated to the complete linear system [Formula: see text]. Suppose that [Formula: see text] be a minimal [Formula: see text]-fold of [Formula: see text] and [Formula: see text] the pluricanonical section index. In this paper, we obtain the lower bounds of the canonical volume [Formula: see text] in term of [Formula: see text] for [Formula: see text]. In addition, we also classify the weighted baskets [Formula: see text] of [Formula: see text] satisfying [Formula: see text].

Lazarsfeld-Mukai reflexive sheaves and their stability

ABSTRACTConsider an ample and globally generated line bundle L on a smooth projective variety X of dimension N≥2 over ℂ. Let D be a smooth divisor in the complete linear system of L. We construct reflexive sheaves on X by an elementary transformation of a trivial bundle on X along certain globally generated torsion-free sheaves on D. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the μL-(semi)stability of such reflexive sheaves under certain conditions.