On the irreducibility of Hilbert scheme of surfaces of minimal degree

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Fedor Bogomolov ◽  
Viktor Kulikov
Keyword(s):  
Projective Plane ◽  
Hilbert Scheme ◽  
Special Class ◽  
Main Idea ◽  
Rational Curves ◽  
Minimal Degree ◽  
Irreducible Components

AbstractThe article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.

scholarly journals Schemes of modules over gentle algebras and laminations of surfaces

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Christof Geiß ◽  
Daniel Labardini-Fragoso ◽  
Jan Schröer
Keyword(s):  
Special Class ◽  
Cluster Algebra ◽  
Irreducible Components ◽  
Gentle Algebras

AbstractWe study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces. For these we obtain a bijection between the set of generically $$\tau $$ τ -reduced decorated irreducible components and the set of laminations of the surface. As an application, we get that the set of bangle functions (defined by Musiker–Schiffler–Williams) in the upper cluster algebra associated with the surface coincides with the set of generic Caldero-Chapoton functions (defined by Geiß–Leclerc–Schröer).

scholarly journals Characteristic Varieties for a Class of Line Arrangements

2011 ◽  
Vol 54 (1) ◽  
pp. 56-67 ◽  
Author(s):  
Thi Anh Thu Dinh
Keyword(s):  
Projective Plane ◽  
Complex Projective Plane ◽  
Line Arrangement ◽  
Line Arrangements ◽  
Irreducible Components ◽  
Characteristic Varieties ◽  
Non Local ◽  
Complex Projective

AbstractLet be a line arrangement in the complex projective plane ℙ2, having the points of multiplicity ≥ 3 situated on two lines in , say H0 and H∞. Then we show that the non-local irreducible components of the first resonance variety are 2-dimensional and correspond to parallelograms ℙ in ℂ2 = ℙ2 \ H∞ whose sides are in and for which H0 is a diagonal.

An algorithm for a lifted Massey triple product of a smooth projective plane curve

2020 ◽  
Vol 30 (08) ◽  
pp. 1651-1669
Author(s):  
Younggi Lee ◽  
Jeehoon Park ◽  
Junyeong Park ◽  
Jaehyun Yim
Keyword(s):  
Projective Plane ◽  
Homogeneous Polynomial ◽  
Plane Curve ◽  
Main Idea ◽  
Triple Product ◽  
Cup Product ◽  
Smooth Projective Plane ◽  
Defining System ◽  
Polynomial Formula ◽  
Massey Triple Product

We provide an explicit algorithm to compute a lifted Massey triple product relative to a defining system for a smooth projective plane curve [Formula: see text] defined by a homogeneous polynomial [Formula: see text] over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for [Formula: see text] in terms of the multiplications inside the Jacobian ring of [Formula: see text] and the Cech–deRham complex of [Formula: see text]. Our algorithm gives a criterion whether a lifted Massey triple product vanishes or not in [Formula: see text] under a particular nontrivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in [Formula: see text] of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program.

Coinvariant Algebras of Finite Subgroups of SL(3;C)

2004 ◽  
Vol 56 (3) ◽  
pp. 495-528 ◽  
Author(s):  
Yasushi Gomi ◽  
Iku Nakamura ◽  
Ken-ichi Shinoda
Keyword(s):  
Hilbert Scheme ◽  
Dual Graph ◽  
Orbit Space ◽  
Rational Curves ◽  
Finite Subgroup ◽  
Crepant Resolution ◽  
Mckay Correspondence ◽  
Finite Subgroups ◽  
Explicit Formulae ◽  
Molien Series

AbstractFor most of the finite subgroups of SL(3; C) we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae [McKay99] for subgroups of SU(2). We also study the G-orbit Hilbert scheme HilbG(C3) for any finite subgroup G of SO(3), which is known to be a minimal (crepant) resolution of the orbit space C3/G. In this case the fiber over the origin of the Hilbert-Chow morphism from HilbG(C3) to C3/G consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of G. This is an SO(3) version of the McKay correspondence in the SU(2) case.

Harbourne–Hirschowitz surfaces whose anticanonical divisors consist only of three irreducible components

2018 ◽  
Vol 29 (12) ◽  
pp. 1850072 ◽  
Author(s):  
J. B. Frías-Medina ◽  
M. Lahyane
Keyword(s):  
Rational Curves ◽  
Finitely Generated ◽  
Generating Sets ◽  
Cox Rings ◽  
Irreducible Components

In this paper, we provide new families of Harbourne–Hirschowitz surfaces whose effective monoids are finitely generated, and consequently, their Cox rings are finitely generated. Indeed, these properties are achieved by imposing some reasonable numerical conditions. Our method gives an efficient way of computing the minimal generating sets whenever the effective monoids are finitely generated. These surfaces are anticanonical ones having triangle anticanonical divisors consisting of smooth projective rational curves. Moreover, we present some families that do not satisfy the imposed numerical conditions but their effective monoids are still finitely generated by giving explicitly the minimal generating sets.

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 RATIONAL FAMILIES OF VECTOR BUNDLES ON CURVES

2004 ◽  
Vol 15 (01) ◽  
pp. 13-45 ◽  
Author(s):  
ANA-MARIA CASTRAVET
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Rational Curves ◽  
Complex Curve ◽  
Stable Vector Bundles ◽  
Irreducible Components ◽  
Rationally Connected ◽  
Rank 2 ◽  
Rank 2 Vector Bundles

Let C be a smooth projective complex curve of genus g≥2 and let M be the moduli space of rank 2, stable vector bundles on C, with fixed determinant of degree 1. For any k≥1, we find all the irreducible components of the space of rational curves on M, of degree k. In particular, we find the maximal rationally connected fibrations of these components. We prove that there is a one-to-one correspondence between moduli spaces of rational curves on M and moduli spaces of rank 2 vector bundles on ℙ1×C.

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