Zero character shown in billet-doux (4,6)
I believe the answer is:
love letter
'billet-doux' is the definition.
(thesaurus)
'zero character' is the wordplay.
'zero' becomes 'love'.
'character' becomes 'letter' (letter is a kind of character).
'love'+'letter'='LOVE-LETTER'
'shown in' is the link.
(Other definitions for love letter that I've seen before include "Billet-doux" , "caring note!" , "romantic reading material" .)