scholarly journals Fundamental group schemes of Hilbert scheme of n points on a smooth projective surface

2020 ◽  
Vol 164 ◽  
pp. 102898
Author(s):  
Arjun Paul ◽  
Ronnie Sebastian
Keyword(s):  
Fundamental Group ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Group Schemes ◽  
Smooth Projective Surface

scholarly journals Stability of tautological bundles on the Hilbert scheme of two points on a surface

2014 ◽  
Vol 214 ◽  
pp. 79-94 ◽  
Author(s):  
Malte Wandel
Keyword(s):  
Vector Bundle ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Stable Vector Bundle ◽  
Free Sheaf ◽  
Smooth Projective Surface ◽  
Rank 2 ◽  
Torsion Free

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

scholarly journals FUNDAMENTAL GROUP OF SOME GENUS-2 FIBRATIONS AND APPLICATIONS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250080 ◽  
Author(s):  
R. V. GURJAR ◽  
SAGAR KOLTE
Keyword(s):  
Fundamental Group ◽  
Elliptic Curve ◽  
Universal Cover ◽  
Projective Surface ◽  
Smooth Projective Surface ◽  
Shafarevich Conjecture ◽  
Genus 2 ◽  
Holomorphic Convexity

We will prove that given a genus-2 fibration f : X → C on a smooth projective surface X such that b1(X) = b1(C) + 2, the fundamental group of X is almost isomorphic to π1(C) × π1(E), where E is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces X with genus-2 fibration X → C such that b1(X) > b1(C).

scholarly journals Stability of tautological bundles on the Hilbert scheme of two points on a surface

2014 ◽  
Vol 214 ◽  
pp. 79-94
Author(s):  
Malte Wandel
Keyword(s):  
Vector Bundle ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Stable Vector Bundle ◽  
Free Sheaf ◽  
Smooth Projective Surface ◽  
Rank 2 ◽  
Torsion Free

AbstractLet (X, H) be a polarized smooth projective surface satisfyingH1(Χ, OΧ) = 0, and letƑbe either a rank 1 torsion-free sheaf or a rank 2μH-stable vector bundle onΧ. Assume thatc1(Ƒ) ≠ 0. This article shows that the rank 2—respectively, rank 4—tautological sheafƑ[2]associated withƑon the Hilbert squareΧ[2]isμ-stable with respect to a certain polarization.

scholarly journals The degree of the tangent and secant variety to a projective surface

2020 ◽  
Vol 20 (2) ◽  
pp. 233-248
Author(s):  
Andrea Cattaneo
Keyword(s):  
Projective Space ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Secant Variety ◽  
Smooth Projective Surface

AbstractWe present a way of computing the degree of the secant (resp. tangent) variety of a smooth projective surface, under the assumption that the divisor giving the embedding in the projective space is 3-very ample. This method exploits the link between these varieties and the Hilbert scheme 0-dimensional subschemes of length 2 of the surface.

scholarly journals The integral cohomology of the Hilbert scheme of points on a surface

2020 ◽  
Vol 8 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Natural Number ◽  
Hilbert Scheme ◽  
Obstruction Theory ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Complex Projective Surface ◽  
Smooth Complex ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Torsion Free

Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

scholarly journals Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

2019 ◽  
Vol 163 (3-4) ◽  
pp. 361-373
Author(s):  
Roberto Laface ◽  
Piotr Pokora
Keyword(s):  
Line Bundle ◽  
Characteristic Zero ◽  
Algebraic Surface ◽  
Intersection Number ◽  
Algebraic Surfaces ◽  
The Self ◽  
Projective Surface ◽  
Intersection Numbers ◽  
Weighted Version ◽  
Smooth Projective Surface

AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.

The sheaf of relative differentials of a fibred surface

1993 ◽  
Vol 114 (3) ◽  
pp. 461-470
Author(s):  
Fernando Serrano
Keyword(s):  
Linear Systems ◽  
Smooth Curve ◽  
Base Point ◽  
Projective Surface ◽  
Positive Part ◽  
Rational Singularities ◽  
Smooth Projective Surface ◽  
Genus 2

AbstractLet Φ: S → C denote a fibration from a smooth projective surface onto a smooth curve, with fibres of genus ≥2. The double dual of the sheaf of relative differentials has been studied by F. Serrano [14]. There, it was proved that dim grows asymptotically as the square of n in case Φ is not isotrivial (i.e. fibres vary in modulus), and the converse holds true in most cases, in a way that can be made precise. In the non-isotrivial case, the present paper provides further information about by analysing the linear systems for large n. If P denotes the positive part of in its Zariski decomposition, then it is shown that |rP| is eventually base-point free for some r > 0. Furthermore, Proj is a normal projective surface, fibred over C, birational to S, and with only rational singularities.

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