Learn Russian: Вчера мы купили десять слив и пять груш. - Yesterday we bought ten plums and five pears.

They are in the genitive plural case! Both слива (plum) and груша (pear) are feminine nouns ending in -а. To form the genitive plural for these nouns, you drop the -а completely, leaving a zero ending: слив and груш.

Why use the genitive plural here if the plums and pears are the direct objects we bought?

When you use quantities in Russian, the number controls the case of the noun. Numbers 5 and above (like десять and пять) require the following noun to be in the genitive plural. This overrides the accusative case you would normally expect to use for a direct object.

Why isn't there a soft sign (ь) at the end of груш?

Because груша is a feminine noun ending in -а, you simply drop the -а to form the genitive plural. You never add a soft sign to a genitive plural noun ending in a husher consonant (ш, ж, ч, or щ). You only see soft signs after hushers on feminine nouns that already have them in their basic dictionary form (like вещь or мышь).

Could I use покупали instead of купили in this sentence?