Gorenstein weighted projective spaces and Egyptian fractions
During our seminar on Wednesday about weighted projective spaces we wondered how singular weighted projective spaces are precisely. First of all, they are always Cohen–Macaulay. So the next question could be: when are they Gorenstein? For this question there is a purely numerical criterion in terms of the weights:
Fact The weighted projective space
So I quickly wrote a silly Python script which iterates over all possible weights up to a certain limit, only considered the normalised ones (i.e. to consider them up to isomorphism) and see how many of these satisfy the condition. It turned out that there are 3 Gorenstein weighted
Actually, looking for weights
Then as explained at the end of section 2 of Samuel Bossière, Étienne Mann and Fabio Perroni, On the cohomological crepant resolution conjecture for weighted projective spaces we have that
The solutions to this equation are known as Egyptian fractions, and there is an OEIS sequence telling us precisely how many there are.
Followup question When is
For something completely unrelated using Egyptian fractions, today 0xDE published Egyptian fractions with practical denominators.