scholarly journals On Stability of Logarithmic Tangent Sheaves: Symmetric and Generic Determinants

2021 ◽  
Author(s):  
Daniele Faenzi ◽  
Simone Marchesi
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Characteristic Zero ◽  
Main Application

Abstract We prove stability of logarithmic tangent sheaves of singular hypersurfaces $D$ of the projective space with constraints on the dimension and degree of the singularities of $D$. As the main application, we prove that determinants and symmetric determinants have simple (in characteristic zero, stable) logarithmic tangent sheaves and we describe an open dense piece of the associated moduli space.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

scholarly journals Codimension One Holomorphic Distributions on the Projective Three-space

2018 ◽  
Vol 2020 (23) ◽  
pp. 9011-9074 ◽  
Author(s):  
Omegar Calvo-Andrade ◽  
Maurício Corrêa ◽  
Marcos Jardim
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Tangent Bundle ◽  
Projective Variety ◽  
Arbitrary Degree ◽  
Codimension One ◽  
Quot Scheme ◽  
Space Of Distributions ◽  
Three Space

Abstract We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves.

scholarly journals On the geometry of bifurcation currents for quadratic rational maps

2014 ◽  
Vol 35 (5) ◽  
pp. 1369-1379 ◽  
Author(s):  
FRANÇOIS BERTELOOT ◽  
THOMAS GAUTHIER
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Complex Projective Space ◽  
The Self ◽  
Rational Maps ◽  
Two Dimensional ◽  
Lelong Numbers ◽  
Complex Projective

We describe the behaviour at infinity of the bifurcation current in the moduli space of quadratic rational maps. To this purpose, we extend it to some closed, positive $(1,1)$-current on a two-dimensional complex projective space and then compute the Lelong numbers and the self-intersection of the extended current.

scholarly journals COMPLETE INTERSECTIONS AND MOVABLE CURVES ON THE MODULI SPACE OF SIX-POINTED RATIONAL CURVES

2012 ◽  
Vol 22 (06) ◽  
pp. 1250049
Author(s):  
PAUL L. LARSEN
Keyword(s):  
Projective Space ◽  
Complete Intersection ◽  
Moduli Space ◽  
Complex Projective Space ◽  
Projective Variety ◽  
Rational Curves ◽  
Complete Intersections ◽  
Family Of Curves ◽  
Complex Projective

A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves and the permutohedral or Losev–Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of this family, but differ for the non-toric member, the moduli space of six-pointed rational curves. The proof is via an algorithm that applies in greater generality. We also give an example of a projective toric threefold for which these two cones differ.

scholarly journals MODULI SPACE OF STABLE MAPS TO PROJECTIVE SPACE VIA GIT

2010 ◽  
Vol 21 (05) ◽  
pp. 639-664 ◽  
Author(s):  
YOUNG-HOON KIEM ◽  
HAN-BOM MOON
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Projective Variety ◽  
Blow Up ◽  
Stable Maps ◽  
Homogeneous Polynomials ◽  
Birational Map

We compare the Kontsevich moduli space [Formula: see text] of stable maps to projective space with the quasi-map space ℙ( Sym d(ℂ2) ⊗ ℂn)//SL(2). Consider the birational map [Formula: see text] which assigns to an n tuple of degree d homogeneous polynomials f1, …, fn in two variables, the map f = (f1 : ⋯ : fn) : ℙ1 → ℙn-1. In this paper, for d = 3, we prove that [Formula: see text] is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces [Formula: see text] with d = 1, 2. In particular, [Formula: see text] is the SL(2)-quotient of a smooth rational projective variety. The degree two case [Formula: see text], which is the blow-up of ℙ( Sym 2ℂ2 ⊗ ℂn)//SL(2) along ℙn-1, is worked out as a preliminary example.

scholarly journals Counting the number of trigonal curves of genus 5 over finite fields

2020 ◽  
Vol 208 (1) ◽  
pp. 31-48
Author(s):  
Thomas Wennink
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Euler Characteristic ◽  
Finite Fields ◽  
Moduli Space ◽  
Plane Curves ◽  
Sieve Method ◽  
Main Application ◽  
Trigonal Curves

AbstractThe trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

Basepoint Free Cycles on $\overline{\operatorname{\textbf{M}}}_{\boldsymbol{0,n}}$ from Gromov–Witten Theory

2019 ◽  
Author(s):  
P Belkale ◽  
A Gibney
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Homogeneous Spaces ◽  
Type A ◽  
Rational Curves ◽  
Conformal Blocks ◽  
Witten Theory ◽  
Level 1

Abstract Basepoint free cycles on the moduli space $\overline{\operatorname{M}}_{0,n}$ of stable $n$-pointed rational curves, defined using Gromov–Witten invariants of smooth projective homogeneous spaces are introduced and studied. Intersection formulas to find classes are given. Gromov–Witten divisors for projective space are shown equivalent to conformal blocks divisors for type A at level 1.

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