scholarly journals Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces

2017 ◽  
Vol 2017 (730) ◽  
Author(s):  
Marian Aprodu ◽  
Gavril Farkas ◽  
Angela Ortega
Keyword(s):  
Projective Variety ◽  
K3 Surfaces ◽  
Minimal Resolution ◽  
Chow Forms ◽  
Minimal Resolutions

AbstractThe Minimal Resolution Conjecture (MRC) for points on a projective variety

scholarly journals Elliptic Fibrations and Kodaira Dimensions of Schreieder’s Varieties

2021 ◽  
Author(s):  
Alice Garbagnati
Keyword(s):  
K3 Surfaces ◽  
Base Change ◽  
Kodaira Dimension ◽  
Elliptic Surface ◽  
Minimal Resolution ◽  
Elliptic Fibrations ◽  
Smooth Model ◽  
Birational Model ◽  
Hodge Numbers ◽  
Rational Elliptic Surface

Abstract We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_{(n)}^{(l)}$, which is the quotient of the product of a certain curve $C_{(n)}$ by itself $l$ times by a group $G\simeq \left ({\mathbb{Z}}/n{\mathbb{Z}}\right )^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved that the Kodaira dimension of this smooth model is 1, if $c>1$; if $l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on $n$ and $l$ we prove that $Y_{(n)}^{(l)}$ admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if $l=2$, its minimal resolution is a modular elliptic surface, obtained by a base change of order $n$ on a specific extremal rational elliptic surface; if $l\geq 3$ it has a birational model that admits a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of Kodaira dimension at most 0.

scholarly journals COX RINGS OF MINIMAL RESOLUTIONS OF SURFACE QUOTIENT SINGULARITIES

2015 ◽  
Vol 58 (2) ◽  
pp. 325-355 ◽  
Author(s):  
MARIA DONTEN-BURY
Keyword(s):  
Toric Variety ◽  
Affine Space ◽  
Explicit Description ◽  
Quotient Singularity ◽  
Coordinate Ring ◽  
Minimal Resolution ◽  
Cox Rings ◽  
Cox Ring ◽  
Quotient Singularities ◽  
Minimal Resolutions

AbstractWe investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.

scholarly journals A One-Dimensional Family of K3 Surfaces with a ℤ4 Action

2009 ◽  
Vol 52 (4) ◽  
pp. 493-510 ◽  
Author(s):  
Michela Artebani
Keyword(s):  
K3 Surfaces ◽  
Minimal Resolution ◽  
Cyclic Cover ◽  
One Dimensional ◽  
Plane Quartics ◽  
Symplectic Action ◽  
The One

AbstractThe minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a K3 surface with a non symplectic action of ℤ4. In this paper we study the geometry of the one-dimensional family of K3 surfaces associated to the locus of plane quartics with five nodes.

scholarly journals A smörgåsbord of scalar-flat Kähler ALE surfaces

2019 ◽  
Vol 2019 (746) ◽  
pp. 171-208 ◽  
Author(s):  
Michael T. Lock ◽  
Jeff A. Viaclovsky
Keyword(s):  
Complex Structure ◽  
Finite Subgroup ◽  
Circle Action ◽  
Minimal Resolution ◽  
Kähler Metrics ◽  
Circle Actions ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Minimal Resolutions ◽  
Small Deformations

Abstract There are many known examples of scalar-flat Kähler ALE surfaces, all of which have group at infinity either cyclic or contained in {{\rm{SU}}(2)} . The main result in this paper shows that for any non-cyclic finite subgroup Γ \subset U(2) containing no complex reflections, there exist scalar-flat Kähler ALE metrics on the minimal resolution of \mathbb{C}^{2} /Γ, for which Γ occurs as the group at infinity. Furthermore, we show that these metrics admit a holomorphic isometric circle action. It is also shown that there exist scalar-flat Kähler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal Kähler metrics admitting holomorphic isometric circle actions in certain Kähler classes on the complex analytic compactifications of the minimal resolutions.

scholarly journals Free minimal resolutions and the Betti numbers of the suspension of ann-gon

2000 ◽  
Vol 23 (3) ◽  
pp. 211-216 ◽  
Author(s):  
Tilak de Alwis
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Minimal Resolution ◽  
Minimal Resolutions

Consider the generaln-gon with vertices at the points1,2,…,n. Then its suspension involves two more vertices, say atn+1andn+2. LetRbe the polynomial ringk[x1,x2,…,xn], wherekis any field. Then we can associate an idealIto our suspension in the Stanley-Reisner sense. In this paper, we find a free minimal resolution and the Betti numbers of theR-moduleR/I.

scholarly journals KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four

2021 ◽  
Vol 64 (1) ◽  
pp. 99-127
Author(s):  
Han-Bom Moon ◽  
Luca Schaffler
Keyword(s):  
Finite Group ◽  
Moduli Space ◽  
K3 Surfaces ◽  
Double Cover ◽  
Minimal Resolution ◽  
Finite Group Action ◽  
Git Quotient ◽  
Stable Pairs ◽  
A Finite Group ◽  
Order Four

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus 2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb {P}^{1}\times \mathbb {P}^{1}$ branched along a specific $(4,\,4)$ curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$ with the symmetric linearization.

scholarly journals Motivic integral of K3 surfaces over a non-archimedean field

2011 ◽  
Vol 228 (5) ◽  
pp. 2688-2730 ◽  
Author(s):  
Allen J. Stewart ◽  
Vadim Vologodsky
Keyword(s):  
K3 Surfaces

scholarly journals Verlinde formulae on complex surfaces: K-theoretic invariants

2021 ◽  
Vol 9 ◽  
Author(s):  
L. Göttsche ◽  
M. Kool ◽  
R. A. Williams
Keyword(s):  
Moduli Space ◽  
K3 Surfaces ◽  
Complex Surfaces ◽  
Type Formula ◽  
Verlinde Formula ◽  
Donaldson Invariants

Abstract We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).

scholarly journals Curves on K3 surfaces in divisibility 2

2021 ◽  
Vol 9 ◽  
Author(s):  
Younghan Bae ◽  
Tim-Henrik Buelles
Keyword(s):  
Modular Forms ◽  
K3 Surfaces ◽  
Initial Condition ◽  
Smooth Curves ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation ◽  
Degeneration Formula ◽  
The Relationship ◽  
Level 2

Abstract We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.

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