scholarly journals A finite group acting on the moduli space of K3 surfaces

2008 ◽  
Vol 360 (12) ◽  
pp. 6631-6642 ◽  
Author(s):  
Paolo Stellari
Keyword(s):  
Finite Group ◽  
Moduli Space ◽  
K3 Surfaces ◽  
A Finite Group

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 The KSBA compactification of the moduli space of $$D_{1,6}$$-polarized Enriques surfaces

2021 ◽  
Author(s):  
Luca Schaffler
Keyword(s):  
Finite Group ◽  
Moduli Space ◽  
Blow Up ◽  
Unit Cube ◽  
Finite Group Action ◽  
Stable Pair ◽  
Enriques Surfaces ◽  
Secondary Polytope ◽  
Stable Pairs ◽  
A Finite Group

AbstractWe describe a compactification by stable pairs (also known as KSBA compactification) of the 4-dimensional family of Enriques surfaces which arise as the $${\mathbb {Z}}_2^2$$ Z 2 2 -covers of the blow up of $${\mathbb {P}}^2$$ P 2 at three general points branched along a configuration of three pairs of lines. Up to a finite group action, we show that this compactification is isomorphic to the toric variety associated to the secondary polytope of the unit cube. We relate the KSBA compactification considered to the Baily–Borel compactification of the same family of Enriques surfaces. Part of the KSBA boundary has a toroidal behavior, another part is isomorphic to the Baily–Borel compactification, and what remains is a mixture of these two. We relate the stable pair compactification studied here with Looijenga’s semitoric compactifications.

scholarly journals Non-rational varieties with the Hilbert Property

2019 ◽  
Vol 16 (04) ◽  
pp. 803-822
Author(s):  
Julian Lawrence Demeio
Keyword(s):  
Finite Group ◽  
K3 Surfaces ◽  
Sufficient Condition ◽  
Elliptic Fibrations ◽  
Rational Varieties ◽  
A Finite Group

A variety [Formula: see text] over a field [Formula: see text] is said to have the Hilbert Property if [Formula: see text] is not thin. We shall exhibit some examples of varieties, for which the Hilbert Property is a new result. We give a sufficient condition for descending the Hilbert Property to the quotient of a variety by the action of a finite group. Applying this result to linear actions of groups, we exhibit some examples of non-rational unirational varieties with the Hilbert Property, providing positive instances of a conjecture posed by Colliot–Thélèene and Sansuc. We also give a sufficient condition for a surface with two elliptic fibrations to have the Hilbert Property, and use it to prove that a certain class of K3 surfaces have the Hilbert Property, generalizing a result of Corvaja and Zannier.

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).

ON HIGHER FROBENIUS–SCHUR INDICATORS

2021 ◽  
pp. 1-11
Author(s):  
YANJUN LIU ◽  
WOLFGANG WILLEMS
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Irreducible Characters ◽  
A Finite Group

Abstract Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators $\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$ for primes p and $n \in \mathbb {N}$ , where G is a finite group and $\chi $ is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

scholarly journals Random additions in urns of integers

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper
Keyword(s):  
Finite Group ◽  
Random Process ◽  
Uniform Distribution ◽  
Discrete Time ◽  
Random Variable ◽  
Urn Model ◽  
Infinite Type ◽  
The Mean ◽  
A Finite Group ◽  
Proof Of Convergence

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

𝒫-characters and the structure of finite solvable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiakuan Lu ◽  
Kaisun Wu ◽  
Wei Meng
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Group ◽  
Irreducible Character ◽  
Solvable Groups ◽  
Irreducible Constituent ◽  
A Finite Group

AbstractLet 𝐺 be a finite group. An irreducible character of 𝐺 is called a 𝒫-character if it is an irreducible constituent of (1_{H})^{G} for some maximal subgroup 𝐻 of 𝐺. In this paper, we obtain some conditions for a solvable group 𝐺 to be 𝑝-nilpotent or 𝑝-closed in terms of 𝒫-characters.

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