scholarly journals I-surfaces with one T-singularity

2021 ◽  
Author(s):  
Marco Franciosi ◽  
Rita Pardini ◽  
Julie Rana ◽  
Sönke Rollenske
Keyword(s):  
Singular Point ◽  
Irreducible Component ◽  
Moduli Space ◽  
Main Component ◽  
Stable Surfaces

AbstractWe classify normal stable surfaces with $$K_X^2 = 1$$ K X 2 = 1 , $$p_g = 2$$ p g = 2 and $$q=0$$ q = 0 with a unique singular point which is a non-canonical T-singularity, thus exhibiting two divisors in the main component and a new irreducible component of the moduli space of stable surfaces $${{\overline{{{\mathfrak {M}}}}}}_{1,3}$$ M ¯ 1 , 3 .

scholarly journals Instanton bundles on two Fano threefolds of index 1

2020 ◽  
Vol 32 (5) ◽  
pp. 1315-1336
Author(s):  
Gianfranco Casnati ◽  
Ozhan Genc
Keyword(s):  
Irreducible Component ◽  
Moduli Space ◽  
Blow Up ◽  
Explicit Construction ◽  
Fano Threefolds ◽  
Instanton Bundles

AbstractWe deal with instanton bundles on the product {\mathbb{P}^{1}\times\mathbb{P}^{2}} and the blow up of {\mathbb{P}^{3}} along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space.

ℤ3-actions on Horikawa surfaces

2021 ◽  
pp. 2150097
Author(s):  
Vicente Lorenzo
Keyword(s):  
Moduli Space ◽  
General Type ◽  
Admissible Pair ◽  
Algebraic Surfaces ◽  
The Other ◽  
Connected Components ◽  
Surfaces Of General Type ◽  
Canonical Models ◽  
Normal Surfaces ◽  
Stable Surfaces

Minimal algebraic surfaces of general type [Formula: see text] such that [Formula: see text] are called Horikawa surfaces. In this note, [Formula: see text]-actions on Horikawa surfaces are studied. The main result states that given an admissible pair [Formula: see text] such that [Formula: see text], all the connected components of Gieseker’s moduli space [Formula: see text] contain surfaces admitting a [Formula: see text]-action. On the other hand, the examples considered allow us to produce normal stable surfaces that do not admit a [Formula: see text]-Gorenstein smoothing. This is illustrated by constructing non-smoothable normal surfaces in the KSBA-compactification [Formula: see text] of Gieseker’s moduli space [Formula: see text] for every admissible pair [Formula: see text] such that [Formula: see text]. Furthermore, the surfaces constructed belong to connected components of [Formula: see text] without canonical models.

scholarly journals HALF NIKULIN SURFACES AND MODULI OF PRYM CURVES

2019 ◽  
pp. 1-38
Author(s):  
Andreas Leopold Knutsen ◽  
Margherita Lelli-Chiesa ◽  
Alessandro Verra
Keyword(s):  
Irreducible Component ◽  
Moduli Space ◽  
Smooth Curve ◽  
Complete Picture ◽  
Boundary Points

Let ${\mathcal{F}}_{g}^{\mathbf{N}}$ be the moduli space of polarized Nikulin surfaces $(Y,H)$ of genus $g$ and let ${\mathcal{P}}_{g}^{\mathbf{N}}$ be the moduli of triples $(Y,H,C)$ , with $C\in |H|$ a smooth curve. We study the natural map $\unicode[STIX]{x1D712}_{g}:{\mathcal{P}}_{g}^{\mathbf{N}}\rightarrow {\mathcal{R}}_{g}$ , where ${\mathcal{R}}_{g}$ is the moduli space of Prym curves of genus $g$ . We prove that it is generically injective on every irreducible component, with a few exceptions in low genus. This gives a complete picture of the map $\unicode[STIX]{x1D712}_{g}$ and confirms some striking analogies between it and the Mukai map $m_{g}:{\mathcal{P}}_{g}\rightarrow {\mathcal{M}}_{g}$ for moduli of triples $(Y,H,C)$ , where $(Y,H)$ is any genus $g$ polarized $K3$ surface. The proof is by degeneration to boundary points of a partial compactification of ${\mathcal{F}}_{g}^{\mathbf{N}}$ . These represent the union of two surfaces with four even nodes and effective anticanonical class, which we call half Nikulin surfaces. The use of this degeneration is new with respect to previous techniques.

A boundary divisor in the moduli spaces of stable quintic surfaces

2017 ◽  
Vol 28 (04) ◽  
pp. 1750021 ◽  
Author(s):  
Julie Rana
Keyword(s):  
Algebraic Geometry ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Deformation Theory ◽  
General Type ◽  
Topological Invariants ◽  
Surfaces Of General Type ◽  
Wide Range ◽  
Stable Surfaces ◽  
Standing Question

We give a bound on which singularities may appear on Kollár–Shepherd-Barron–Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with [Formula: see text]) whose unique non-Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space [Formula: see text] corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry.

Quantization of Bending Deformations of Polygons In , Hypergeometric Integrals and the Gassner Representation

2001 ◽  
Vol 44 (1) ◽  
pp. 36-60 ◽  
Author(s):  
Michael Kapovich ◽  
John J. Millson
Keyword(s):  
Singular Point ◽  
Lie Algebra ◽  
Moduli Space ◽  
Braid Group ◽  
Vector Fields ◽  
Flat Connection ◽  
Hamiltonian Action ◽  
Pure Braid Group ◽  
Hypergeometric Integrals

AbstractThe Hamiltonian potentials of the bending deformations of n-gons instudied in [KM] and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra𝓟nof the pure braid groupPnon the moduli spaceMrofn-gon linkages with the side-lengthsr = (r1, … , rn)in. Ife∈Mris a singular point wemay linearize the vector fields in𝓟nate. This linearization yields a flat connection ∇ on the spaceof n distinct points on. We show that the monodromy of ∇ is the dual of a quotient of a specialized reduced Gassner representation.

scholarly journals On vector bundles over reducible curves with a node

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Filippo F. Favale ◽  
Sonia Brivio
Keyword(s):  
Irreducible Component ◽  
Euler Characteristic ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Closed Subset ◽  
Vector Bundles ◽  
Projective Bundle ◽  
Single Node ◽  
Semistable Vector Bundles

AbstractLet C be a curve with two smooth components and a single node, and let 𝓤C(w, r, χ) be the moduli space of w-semistable classes of depth one sheaves on C having rank r on both components and Euler characteristic χ. In this paper, under suitable assumptions, we produce a projective bundle over the product of the moduli spaces of semistable vector bundles of rank r on each component and we show that it is birational to an irreducible component of 𝓤C(w, r, χ). Then we prove the rationality of the closed subset containing vector bundles with given fixed determinant.

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