For grammar, it is correct. But, I would translate it on the basis of your English sentence, 机の上にいろいろな本があります。 It's because your English text begins "There is" and no context with that. I am sorry I could not explain it in English well. I hope some one will help your understanding.

Like "various" is a adj. for "books", 「いろいろ」 is also to be used before 「本」. A) If to say "There are various books on the desk.", then it would be 「机の上にいろいろな本があります。」 B) If to say "Various books are on the desk", it would be 「いろいろな本は机の上にあります。」 However, in my opinion, both of the sentences (in English and in Japanese) in B are weird. Will it sound more normal if was said like this? "Your books are on the desk" 「君の本は机の上にあります。」