A falcon chased a pigeon into my garden on Sunday and killed it.

I did not know that there were any in our area.

Saw a kestrel hovering over the long grass on Blythe Hill a couple of summers ago, and more recently saw a Sparrowhawk in Burgess Park in SE5. I guess that may be what you saw: they like to eat pigeons.

I spotted a Kestrel hovering above the railway embankment behind Devonshire Road in April and another in Peckham last year along the old Surrey Canal footpath.