The title of this anthology comes, according to its editor, from the poetry of Catullus, who he claims “was the first poet in the ancient world to write about a personal love affair in an extended way”. Two of Catullus’ poems concern the death of the sparrow beloved by Lesbia, the object of the poet’s obsession and Eugenides adds, “Despite the multiplicity of subjects and situations treated here, one Catullan requirement remains in force throughout. In each of these 26 love stories, either there is a sparrow or the sparrow is dead”
He speaks figuratively, of course, but his point is that a love story “can never be about full possession”, that it depends on “disappointment, on unequal births and feuding families, on matrimonial boredom and at least one cold heart”. Disappointment and incompleteness are central to the editor’s conception of the love story.
Here's part of the poem where Catullus mourns for Lesbia's sparrow:
Lugete, o Veneres Cupidinesque...
passer mortuus est meae puellae,
passer, deliciae meae puellae,
quem plus illa oculis suis amabat.
nam mellitus erat suamque norat
ipsam tam bene quam puella matrem,
nec sese a gremio illius movebat,
sed circumsiliens modo huc modo illuc
ad solam dominam usque pipiabat.
the sparrow of my girlfriend has died,
the sparrow, the darling of my girl,
whom she loved more than her eyes.
He was sweet as honey, and he knew his mistress
as well as a girl her own mother,
and he would not budge from her lap,
but jumping around, now here, now there,
he was always chirping for his mistress alone.
You can read the two sparrow poems (in English) here and here. As far as I can tell, apart from the title the book doesn't have much to do with Catullus' poetry, or Latin in general.