scholarly journals Simple cyclic covers of the plane and the Seshadri constants of some general hypersurfaces in weighted projective space

2021 ◽  
Author(s):  
Alex Küronya ◽  
Sönke Rollenske
Keyword(s):  
Projective Space ◽  
Irrational Number ◽  
Weighted Projective Space ◽  
Main Step ◽  
General Point ◽  
Cyclic Cover ◽  
Seshadri Constants ◽  
Cyclic Covers ◽  
Seshadri Constant

AbstractLet $$X \subset {\mathbb P}(1,1,1,m)$$ X ⊂ P ( 1 , 1 , 1 , m ) be a general hypersurface of degree md for some for $$d\ge 2$$ d ≥ 2 and $$m\ge 3$$ m ≥ 3 . We prove that the Seshadri constant $$\varepsilon ( {\mathcal O}_X(1), x)$$ ε ( O X ( 1 ) , x ) at a general point $$x\in X$$ x ∈ X lies in the interval $$\left[ \sqrt{d}- \frac{d}{m}, \sqrt{d}\right] $$ d - d m , d and thus approaches the possibly irrational number $$\sqrt{d}$$ d as m grows. The main step is a detailed study of the case where X is a simple cyclic cover of the plane.

scholarly journals Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes

2014 ◽  
Vol 66 (3) ◽  
pp. 505-524 ◽  
Author(s):  
Donu Arapura
Keyword(s):  
Projective Space ◽  
Prime Number ◽  
Hyperplane Arrangement ◽  
Hodge Theory ◽  
Cyclic Cover ◽  
Hodge Conjecture ◽  
Cyclic Covers ◽  
Generalized Hodge Conjecture ◽  
Ramification Locus

AbstractSuppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D and that U is the complement of the ramification locus in Y. The first theorem in this paper implies that the Beilinson-Hodge conjecture holds for U if certain multiplicities of D are coprime to the degree of the cover. For instance, this applies when D is reduced with normal crossings. The second theorem shows that when D has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of Y. The last section contains some partial extensions to more general nonabelian covers.

scholarly journals Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jacob L. Bourjaily ◽  
Andrew J. McLeod ◽  
Cristian Vergu ◽  
Matthias Volk ◽  
Matt von Hippel ◽  
...  
Keyword(s):  
Projective Space ◽  
Feynman Integral ◽  
Weighted Projective Space

scholarly journals THE COHOMOLOGICAL CREPANT RESOLUTION CONJECTURE FOR ℙ(1,3,4,4)

2009 ◽  
Vol 20 (06) ◽  
pp. 791-801 ◽  
Author(s):  
S. BOISSIÈRE ◽  
E. MANN ◽  
F. PERRONI
Keyword(s):  
Projective Space ◽  
Cohomology Ring ◽  
Weighted Projective Space ◽  
Crepant Resolution ◽  
Crepant Resolution Conjecture

We prove the cohomological crepant resolution conjecture of Ruan for the weighted projective space ℙ(1,3,4,4). To compute the quantum corrected cohomology ring, we combine the results of Coates–Corti–Iritani–Tseng on ℙ(1,1,1,3) and our previous results.

scholarly journals On the arithmetic of a family of degree - two K3 surfaces

2018 ◽  
Vol 166 (3) ◽  
pp. 523-542 ◽  
Author(s):  
FLORIAN BOUYER ◽  
EDGAR COSTA ◽  
DINO FESTI ◽  
CHRISTOPHER NICHOLLS ◽  
MCKENZIE WEST
Keyword(s):  
Projective Space ◽  
Function Field ◽  
Weighted Projective Space ◽  
Module Structure ◽  
K3 Surface ◽  
Galois Module ◽  
Generic Element ◽  
The Family ◽  
Formula Group ◽  
Image Position

AbstractLet ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field ℚ(t). In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family X.

scholarly journals Circle of Sarkisov Links on a Fano 3-Fold

2016 ◽  
Vol 60 (1) ◽  
pp. 1-16
Author(s):  
Hamid Ahmadinezhad ◽  
Francesco Zucconi
Keyword(s):  
Projective Space ◽  
Weighted Projective Space ◽  
Birational Maps

AbstractFor a general Fano 3-fold of index 1 in the weighted projective space ℙ(1, 1, 1, 1, 2, 2, 3) we construct two new birational models that are Mori fibre spaces in the framework of the so-called Sarkisov program. We highlight a relation between the corresponding birational maps, as a circle of Sarkisov links, visualizing the notion of relations in the Sarkisov program.

scholarly journals Toric complete intersections and weighted projective space

2003 ◽  
Vol 46 (2) ◽  
pp. 159-173 ◽  
Author(s):  
Maximilian Kreuzer ◽  
Erwin Riegler ◽  
David A. Sahakyan
Keyword(s):  
Projective Space ◽  
Weighted Projective Space ◽  
Complete Intersections

Is a linear space contained in a submanifold? – On the number of derivatives needed to tell

1999 ◽  
Vol 1999 (508) ◽  
pp. 53-60
Author(s):  
J. M Landsberg
Keyword(s):  
Projective Space ◽  
Linear Space ◽  
General Problem ◽  
General Point ◽  
Classical Theorem ◽  
Linear Spaces ◽  
Dual Varieties

Abstract Let Xn ⊂ or Xn ⊂ ℙn + a be a patch of a C∞ submanifold of an affine or projective space such that through each point x ∈ X there exists a line osculating to order n + 1 at x. We show that X is uniruled by lines, generalizing a classical theorem for surfaces. We describe two circumstances that imply linear spaces of dimension k osculating to order two must be contained in X, shedding light on some of Ein's results on dual varieties. We present some partial results on the general problem of finding the integer m0 = m0(k, n, a) such that there exist examples of patches Xn ⊂ ℙn + a, having a linear space L of dimension k osculating to order m0 — 1 at each point such that L is not locally contained in X, but if there are k-planes osculating to order m0 at each point, they are locally contained in X. The same conclusions hold in the analytic category and complex analytic category if there is a linear space osculating to order m at one general point x ∈ X.

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