A note on special cubic fourfolds of small discriminants

2021 ◽  
Vol 33 (5) ◽  
pp. 1137-1155
Author(s):  
Hoang Le Truong ◽  
Hoang Ngoc Yen
Keyword(s):  
Complete Intersection ◽  
Rational Surface ◽  
Explicit Construction ◽  
Smooth Surfaces ◽  
Liaison Class ◽  
Cubic Fourfold

Abstract In this paper, our purpose is to give a characterization of the generic special cubic fourfold which contains a smooth rational surface of degree 9 not homologous to a complete intersection. As corollaries, we will give an explicit construction of families of smooth surfaces in generic special cubic fourfolds X ∈ 𝒞 δ {X\in\mathcal{C}_{\delta}} for 6 < δ ≤ 30 {6<\delta\leq 30} and δ ≡ 0 ( mod 6 ) {\delta\equiv 0~{}(\bmod~{}6)} . This applies in particular to give an explicit construction of two different liaison class of smooth surfaces in all such special cubic fourfolds with the prescribed invariants.

Double cover K3 surfaces of Hirzebruch surfaces

2021 ◽  
Vol 21 (2) ◽  
pp. 221-225
Author(s):  
Taro Hayashi
Keyword(s):  
Rational Surface ◽  
K3 Surfaces ◽  
Double Cover ◽  
Double Covering ◽  
Smooth Surfaces ◽  
Branch Locus ◽  
Hirzebruch Surfaces ◽  
Double Covers

Abstract General K3 surfaces obtained as double covers of the n-th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces.

scholarly journals Kodaira dimension of moduli of special cubic fourfolds

2019 ◽  
Vol 2019 (752) ◽  
pp. 265-300 ◽  
Author(s):  
Sho Tanimoto ◽  
Anthony Várilly-Alvarado
Keyword(s):  
Complete Intersection ◽  
Cusp Form ◽  
Moduli Space ◽  
General Type ◽  
Kodaira Dimension ◽  
Countable Family ◽  
Prior Work ◽  
Low Weight ◽  
Modular Varieties ◽  
Cubic Fourfold

Abstract A special cubic fourfold is a smooth hypersurface of degree 3 and dimension 4 that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether–Lefschetz divisors {{\mathcal{C}}_{d}} in the moduli space {{\mathcal{C}}} of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the “low-weight cusp form trick” of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of {{\mathcal{C}}_{d}} . For example, if {d=6n+2} , then we show that {{\mathcal{C}}_{d}} is of general type for {n>18} , {n\notin\{20,21,25\}} ; it has nonnegative Kodaira dimension if {n>13} and {n\neq 15} . In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of {{\mathcal{C}}_{d}} is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.

UHF FLOWS AND COCYCLES

2012 ◽  
Vol 23 (01) ◽  
pp. 1250018
Author(s):  
A. KISHIMOTO
Keyword(s):  
Explicit Construction ◽  
Inductive Limits ◽  
Full Matrix ◽  
Matrix Algebras

UHF flows are the flows obtained as inductive limits of flows on full matrix algebras. We will revisit universal UHF flows and give an explicit construction of such flows on a UHF algebra Mk∞ for any k and also present a characterization of such flows. Those flows are UHF flows whose cocycle perturbations are almost conjugate to themselves.

Novel Noncontact Thickness Metrology for Partially Transparent and Nontransparent Wafers for Backend Semiconductor Manufacturing

2004 ◽  
Vol 829 ◽  
Author(s):  
Wojciech J. Walecki ◽  
Vitali Souchkov ◽  
Kevin Lai ◽  
Phuc Van ◽  
Manuel Santos ◽  
...  
Keyword(s):  
Semiconductor Manufacturing ◽  
Rough Surfaces ◽  
Optical Interferometry ◽  
Smooth Surfaces ◽  
Low Coherence ◽  
Low Coherence Interferometry ◽  
Single Probe ◽  
Thin Semiconductor ◽  
Compound Materials

ABSTRACTSingle probe infrared low coherence optical interferometry has been proven to be an effective tool for characterization of thin and ultra-thin semiconductor Si and compound materials wafers. Its application was however limited to wafers transparent at probing wavelength, and having relatively smooth surfaces. Purpose of this paper is to present an extension of low coherence interferometry to characterization of non-transparent wafers, and wafers with rough surfaces.

Castelnuovo-Mumford regularity bound for smooth threefolds in ℙ5 and extremal examples

1999 ◽  
Vol 1999 (509) ◽  
pp. 21-34
Author(s):  
Si-Jong Kwak
Keyword(s):  
Complete Intersection ◽  
Number Field ◽  
Complex Number ◽  
Smooth Surfaces ◽  
Veronese Surface ◽  
Optimal Bound ◽  
Integral Curves ◽  
Complex Number Field

Abstract Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙN defined over the complex number field ℂ. X is said to be k-regular if Hi(ℙN, ℐX (k – i)) = 0 for all i ≧ 1, where ℐX is the sheaf of ideals of ℐℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ5 and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ5 is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ4 other than for the Veronese surface.

scholarly journals Laser Optical Characterization of Smooth Surfaces with Respect to Roughness and Defects in the Micrometer Range

2011 ◽  
Vol 19 ◽  
pp. 235-240 ◽  
Author(s):  
D. Kruse ◽  
S. Patzelt ◽  
C. Dollinger ◽  
A. Tausendfreund ◽  
G. Goch
Keyword(s):  
Optical Characterization ◽  
Smooth Surfaces
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