scholarly journals The geometry of the moduli space of one-dimensional sheaves

2014 ◽  
Vol 58 (3) ◽  
pp. 487-500 ◽  
Author(s):  
Jinwon Choi ◽  
Kiryong Chung
Keyword(s):  
Moduli Space ◽  
One Dimensional

scholarly journals Numerical criteria for divisors on to be ample

2009 ◽  
Vol 145 (5) ◽  
pp. 1227-1248 ◽  
Author(s):  
Angela Gibney
Keyword(s):  
Moduli Space ◽  
General Type ◽  
Topological Type ◽  
Rational Curves ◽  
Canonical Divisor ◽  
Content Type ◽  
One Dimensional ◽  
Stable Curves ◽  
The One ◽  
Ample Divisors

AbstractThe moduli space $\M _{g,n}$ of n-pointed stable curves of genus g is stratified by the topological type of the curves being parameterized: the closure of the locus of curves with k nodes has codimension k. The one-dimensional components of this stratification are smooth rational curves called F-curves. These are believed to determine all ample divisors. F-conjecture A divisor on $\M _{g,n}$ is ample if and only if it positively intersects theF-curves. In this paper, proving the F-conjecture on $\M _{g,n}$ is reduced to showing that certain divisors on $\M _{0,N}$ for N⩽g+n are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary. Numerical criteria and an algorithm are given to check whether a divisor is ample. By using a computer program called the Nef Wizard, written by Daniel Krashen, one can verify the conjecture for low genus. This is done on $\M _g$ for g⩽24, more than doubling the number of cases for which the conjecture is known to hold and showing that it is true for the first genera such that $\M _g$ is known to be of general type.

scholarly journals Strange Duality on Rational Surfaces II: Higher-Rank Cases

2018 ◽  
Vol 2020 (10) ◽  
pp. 3153-3200 ◽  
Author(s):  
Yao Yuan
Keyword(s):  
Exact Sequence ◽  
Moduli Space ◽  
Chern Class ◽  
Rational Surface ◽  
Rational Surfaces ◽  
One Dimensional ◽  
Hirzebruch Surfaces ◽  
Duality Map ◽  
Higher Rank ◽  
Strange Duality

Abstract We study Le Potier’s strange duality conjecture on a rational surface. We focus on the strange duality map $SD_{c_n^r,L}$ that involves the moduli space of rank $r$ sheaves with trivial 1st Chern class and 2nd Chern class $n$, and the moduli space of one-dimensional sheaves with determinant $L$ and Euler characteristic 0. We show there is an exact sequence relating the map $SD_{c_r^r,L}$ to $SD_{c^{r-1}_{r},L}$ and $SD_{c_r^r,L\otimes K_X}$ for all $r\geq 1$ under some conditions on $X$ and $L$ that applies to a large number of cases on $\mathbb{P}^2$ or Hirzebruch surfaces. Also on $\mathbb{P}^2$ we show that for any $r>0$, $SD_{c^r_r,dH}$ is an isomorphism for $d=1,2$, injective for $d=3,$ and moreover $SD_{c_3^3,rH}$ and $SD_{c_3^2,rH}$ are injective. At the end we prove that the map $SD_{c_n^2,L}$ ($n\geq 2$) is an isomorphism for $X=\mathbb{P}^2$ or Fano rational-ruled surfaces and $g_L=3$, and hence so is $SD_{c_3^3,L}$ as a corollary of our main result.

scholarly journals Cohomology bounds for sheaves of dimension one

2014 ◽  
Vol 25 (11) ◽  
pp. 1450103 ◽  
Author(s):  
Jinwon Choi ◽  
Kiryong Chung
Keyword(s):  
Projective Plane ◽  
Moduli Space ◽  
Hilbert Scheme ◽  
Closed Subset ◽  
Projective Variety ◽  
Global Section ◽  
Sharp Bounds ◽  
One Dimensional ◽  
Dimension One

We find sharp bounds on h0(F) for one-dimensional semistable sheaves F on a projective variety X. When X is the projective plane ℙ2, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a closed subset of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.

scholarly journals Lamé polynomials, hyperelliptic reductions and Lamé band structure

2007 ◽  
Vol 366 (1867) ◽  
pp. 1115-1153 ◽  
Author(s):  
Robert S Maier
Keyword(s):  
Differential Equation ◽  
Dispersion Relation ◽  
Band Structure ◽  
Elliptic Curve ◽  
General Formula ◽  
Moduli Space ◽  
Periodic Potential ◽  
Hyperelliptic Curve ◽  
One Dimensional ◽  
Lamé Equation

The band structure of the Lamé equation, viewed as a one-dimensional Schrödinger equation with a periodic potential, is studied. At integer values of the degree parameter ℓ , the dispersion relation is reduced to the ℓ =1 dispersion relation, and a previously published ℓ =2 dispersion relation is shown to be partly incorrect. The Hermite–Krichever Ansatz, which expresses Lamé equation solutions in terms of ℓ =1 solutions, is the chief tool. It is based on a projection from a genus- ℓ hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lamé equation parameters take complex values, are investigated. If the Lamé equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lamé polynomials, i.e. band-edge solutions, are given. A table in the earlier literature is corrected.

scholarly journals Abelian Surfaces with an Automorphism and Quaternionic Multiplication

2016 ◽  
Vol 68 (1) ◽  
pp. 24-43 ◽  
Author(s):  
Matteo Alfonso Bonfanti ◽  
Bert van Geemen
Keyword(s):  
Moduli Space ◽  
Abelian Surfaces ◽  
Shimura Curve ◽  
One Dimensional ◽  
Real Multiplication ◽  
Genus Two

AbstractWe construct one-dimensional families of Abelian surfaces with quaternionic multiplication, which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)- polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space that parametrizes Abelian surfaces with real multiplication by .

scholarly journals One-dimensional Schubert Problems with Respect to Osculating Flags

2017 ◽  
Vol 69 (1) ◽  
pp. 143-185 ◽  
Author(s):  
Jake Levinson
Keyword(s):  
Moduli Space ◽  
Special Interest ◽  
Combinatorial Proof ◽  
Young Tableaux ◽  
Normal Curve ◽  
Jeu De Taquin ◽  
One Dimensional ◽  
First Order ◽  
Real Geometry ◽  
Rational Normal Curve

AbstractWe consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real. In this case, for zerodimensional Schubert problems, the solutions are “ as real as possible”. Recent work by Speyer has extended the theory to the moduli space allowing the points to collide. This gives rise to smooth covers (ℝ), with structure and monodromy described by Young tableaux and jeu de taquin.In this paper, we give analogous results on one-dimensional Schubert problems over .Their(real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over M 0,r, our results show that the real points of the solution curves are smooth.We also find a new identity involving “first-order” K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.

scholarly journals The One-Dimensional Stratum in the Boundary of the Moduli Stack of Stable Curves

2009 ◽  
Vol 196 ◽  
pp. 27-66
Author(s):  
Jörg Zintl
Keyword(s):  
Moduli Space ◽  
Fixed Number ◽  
One Dimensional ◽  
Stable Curves ◽  
Moduli Stack ◽  
The One

It is well-known that the moduli space of Deligne-Mumford stable curves of genus g admits a stratification by the loci of stable curves with a fixed number i of nodes, where 0 ≤ i ≤ 3g - 3. There is an analogous stratification of the associated moduli stack .

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Electron-Mirror Microscopic Aspects of Ferroelectric Domains of BaTiO3 And Ca2Sr (C2H5CO2)6

1970 ◽  
Vol 28 ◽  
pp. 478-479
Author(s):  
Teruo Someya ◽  
Jinzo Kobayashi
Keyword(s):  
Surface Layer ◽  
Electric Fields ◽  
Quantitative Study ◽  
Recent Progress ◽  
Ferroelectric Domain ◽  
Resolving Power ◽  
Surface Pattern ◽  
Crystal Surfaces ◽  
One Dimensional ◽  
Ferroelectric Domains

Recent progress in the electron-mirror microscopy (EMM), e.g., an improvement of its resolving power together with an increase of the magnification makes it useful for investigating the ferroelectric domain physics. English has recently observed the domain texture in the surface layer of BaTiO3. The present authors ) have developed a theory by which one can evaluate small one-dimensional electric fields and/or topographic step heights in the crystal surfaces from their EMM pictures. This theory was applied to a quantitative study of the surface pattern of BaTiO3).

Sign in / Sign up

Export Citation Format

Share Document