The Picard group of a coarse moduli space of vector bundles in positive characteristic

Abstract Let X be a smooth projective complex curve of genus g ≥ 2 and let M X (2,Λ) be the moduli space of semi-stable rank-2 vector bundles over X with fixed determinant Λ. We show that the wobbly locus, i.e. the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field, is a union of divisors 𝓦 k ⊂ M X (2,Λ). We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors 𝓦 k in the Picard group of M X (2, Λ).

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THE ℚ-PICARD GROUP OF THE MODULI SPACE OF CURVES IN POSITIVE CHARACTERISTIC

International Journal of Mathematics ◽

10.1142/s0129167x01000964 ◽

2001 ◽

Vol 12 (05) ◽

pp. 519-534 ◽

Cited By ~ 3

Author(s):

ATSUSHI MORIWAKI

Keyword(s):

Moduli Space ◽

Cohomology Group ◽

Positive Characteristic ◽

Picard Group ◽

Closed Field ◽

Moduli Space Of Curves ◽

Algebraically Closed Field ◽

Stable Curves ◽

Étale Cohomology ◽

Cycle Map

In this note, we prove that the ℚ-Picard group of the moduli space of n-pointed stable curves of genus g over an algebraically closed field is generated by the tautological classes. We also prove that the cycle map to the second étale cohomology group is bijective.

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THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

Nagoya Mathematical Journal ◽

10.1017/nmj.2020.37 ◽

2020 ◽

pp. 1-23

Author(s):

MICHELE BOLOGNESI ◽

NÉSTOR FERNÁNDEZ VARGAS

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Vector Bundles ◽

Hyperelliptic Curve ◽

Geometric Description ◽

Birational Model ◽

Rank 2 ◽

Semistable Vector Bundles ◽

Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

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EXTENSIONS AND RANK-2 VECTOR BUNDLES ON IRREDUCIBLE NODAL CURVES

International Journal of Mathematics ◽

10.1142/s0129167x05003284 ◽

2005 ◽

Vol 16 (10) ◽

pp. 1081-1118

Author(s):

D. ARCARA

Keyword(s):

Moduli Space ◽

Vector Bundles ◽

Rational Map ◽

Nodal Curves ◽

Nodal Curve ◽

Stable Vector Bundles ◽

Rank 2 ◽

Rank 2 Vector Bundles

We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use the natural rational map [Formula: see text] defined by [Formula: see text] to study a compactification [Formula: see text] of the moduli space [Formula: see text] of semi-stable vector bundles of rank 2 and determinant L on C. In particular, we resolve the indeterminancy of ϕL in the case deg L = 3,4 via a sequence of three blow-ups with smooth centers.

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Moduli spaces of orthogonal and symplectic bundles over an algebraic curve

Compositio Mathematica ◽

10.1112/s0010437x07003247 ◽

2008 ◽

Vol 144 (3) ◽

pp. 721-733 ◽

Cited By ~ 7

Author(s):

Olivier Serman

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Algebraic Curve ◽

Vector Bundles ◽

Explicit Description ◽

Representation Space ◽

Classical Groups ◽

Projective Curve ◽

Smooth Projective Curve ◽

Polynomial Invariants

AbstractWe prove that, given a smooth projective curve C of genus g≥2, the forgetful morphism $\mathcal {M}_{\mathbf {O}_r} \longrightarrow \mathcal {M}_{\mathbf {GL}_r}$ (respectively $\mathcal M_{\mathbf {Sp}_{2r}}\longrightarrow \mathcal M_{\mathbf {GL}_{2r}}$) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.

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