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Register a new userProjective spaces over $mathbb{F}_{1^{ell}}$
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Authors
Koen Thas
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In this essay we study various notions of projective space (and other schemes) over $mathbb{F}_{1^ell}$, with $mathbb{F}_1$ denoting the field with one element. Our leading motivation is the Hiden Points Principle, which shows a huge deviation between the set of rational points as closed points defined over $mathbb{F}_{1^ell}$, and the set of rational points defined as morphisms $texttt{Spec}(mathbb{F}_{1^ell}) mapsto mathcal{X}$. We also introduce, in the same vein as Kurokawa [13], schemes of $mathbb{F}_{1^ell}$-type, and consider their zeta functions.
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This paper considers the moduli spaces (stacks) of parabolic bundles (parabolic logarithmic flat bundles with given spectrum, parabolic regular Higgs bundles) with rank 2 and degree 1 over $mathbb{P}^1$ with five marked points. The stratification structures on these moduli spaces (stacks) are investigated. We confirm Simpsons foliation conjecture of moduli space of parabolic logarithmic flat bundles for our case.
We give a general structure theorem for affine A 1-fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth A 1-fibered affine surface non-isomorphic to the total space of a line bundle over a smooth affine curve fails the Zariski Cancellation Problem. The present note is an expanded version of a talk given at the Kinosaki Algebraic Geometry Symposium in October 2019.
Let $mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $mathcal{R}=mathbb{F}_p+umathbb{F}_p+u^2mathbb{F}_p+u^{3}mathbb{F}_{p}+cdots+u^{k}mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.
We assume that $mathcal{E}$ is a rank $r$ Ulrich bundle for $(P^n, mathcal{O}(d))$. The main result of this paper is that $mathcal{E}(i)otimes Omega^{j}(j)$ has natural cohomology for any integers $i in mathbb{Z}$ and $0 leq j leq n$, and every Ulrich bundle $mathcal{E}$ has a resolution in terms of $n$ of the trivial bundle over $P^n$. As a corollary, we can give a necessary and sufficient condition for Ulrich bundles if $n leq 3$, which can be used to find some new examples, i.e., rank $2$ bundles for $(P^3, mathcal{O}(2))$ and rank $3$ bundles for $(P^2, mathcal{O}(3))$.
We study matrix factorizations of locally free coherent sheaves on a scheme. For a scheme that is projective over an affine scheme, we show that homomorphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we give another proof of Orlovs theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. We also give a complete description of the image of this functor.
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