Surfaces with canonical map of maximum degree

We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the Z / 3 \mathbb {Z}/3 -invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth.

TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.

Some infinite sequences of canonical covers of degree 2

In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points.

Canonical maps of general hypersurfaces in Abelian varieties

<p style='text-indent:20px;'>The main theorem of this paper is that, for a general pair <inline-formula><tex-math id="M1">\begin{document}$ (A,X) $\end{document}</tex-math></inline-formula> of an (ample) hypersurface <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula> in an Abelian Variety <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula>, the canonical map <inline-formula><tex-math id="M4">\begin{document}$ \Phi_X $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M5">\begin{document}$ X $\end{document}</tex-math></inline-formula> is birational onto its image if the polarization given by <inline-formula><tex-math id="M6">\begin{document}$ X $\end{document}</tex-math></inline-formula> is not principal (i.e., its Pfaffian <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> is not equal to <inline-formula><tex-math id="M8">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>).</p><p style='text-indent:20px;'>We also easily show that, setting <inline-formula><tex-math id="M9">\begin{document}$ g = dim (A) $\end{document}</tex-math></inline-formula>, and letting <inline-formula><tex-math id="M10">\begin{document}$ d $\end{document}</tex-math></inline-formula> be the Pfaffian of the polarization given by <inline-formula><tex-math id="M11">\begin{document}$ X $\end{document}</tex-math></inline-formula>, then if <inline-formula><tex-math id="M12">\begin{document}$ X $\end{document}</tex-math></inline-formula> is smooth and</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Phi_X : X {\rightarrow } {\mathbb{P}}^{N: = g+d-2} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is an embedding, then necessarily we have the inequality <inline-formula><tex-math id="M13">\begin{document}$ d \geq g + 1 $\end{document}</tex-math></inline-formula>, equivalent to <inline-formula><tex-math id="M14">\begin{document}$ N : = g+d-2 \geq 2 \ dim(X) + 1. $\end{document}</tex-math></inline-formula></p><p style='text-indent:20px;'>Hence we formulate the following interesting conjecture, motivated by work of the second author: if <inline-formula><tex-math id="M15">\begin{document}$ d \geq g + 1, $\end{document}</tex-math></inline-formula> then, for a general pair <inline-formula><tex-math id="M16">\begin{document}$ (A,X) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M17">\begin{document}$ \Phi_X $\end{document}</tex-math></inline-formula> is an embedding.</p>