arXiv subjects

Aldi Nestor de Souza

Publications and source records attributed to Aldi Nestor de Souza.

2 recordsLinked to original sources

arXiv2018

On the degree-1 Abel map for nodal curves

Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $α_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right] $ in the Jacobian of $C$. In this work we extend $α_{L}$ to a morphism $\overlineα_{L}:C\rightarrow\overline{J}_{E}^{P}$ taking values on Esteves' compactified Jacobian for any given polarization $E$. The maps $\overlineα_{L}$ are limits of Abel maps of smooth curves of the type $α_{L}$.

math.AG↗

arXiv2018

Where do odd perfect numbers live?

The existence of a perfect odd number is an old open problem of number theory. An Euler's theorem states that if an odd integer $ n $ is perfect, then $ n $ is written as $ n = p ^ rm ^ 2 $, where $ r, m $ are odd numbers, $ p $ is a prime number of the form $ 4 k + 1 $ and $ (p, m) = 1 $, where $ (x, y) $ denotes the greatest common divisor of $ x $ and $ y $. In this article we show that the exponent $ r $, of $ p $, in this equation, is necessarily equal to 1. That is, if $ n $ is an odd perfect number, then $ n $ is written as $ n = pm ^ 2. $

math.NT↗