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Thanks ChrisE, all good

Just 1d and 10a to go but been̈ staring at them now for ages. Is the definition of 1d the first word?

rocky, yes, first word. See my hint above.
Sorry, 10a was my FOI, though it did rely on me knowing an obscure word for a digging implement

Yes, rocky, 1st word. Not the most obvious definition, perhaps.

Yes re 10a - not only did I not know it, but I didn’t believe it could be until there seemed to be no alternative.

Thanks for that both. I even had the answer to 1d but dismissed it because I couldn't parse it. But in fact it's simpler than it seems from the clue.

COD I liked 4d. FOI 2d. LOI 10a.

I learned an Italian poet, the name for an operatic piece and a work by Kant in the process. Thanks Imogen.

Yes, Rocky: on 1d I spent too long thinking of the river and the University as separate elements, and juggling with abbreviations before I tumbled to the simplicity of it. So I nice bit of misdirection, I’d say!

27 across: there seems to be only one possible answer, but I can't parse it - please someone put me out of my misery!

Hello judygs
Defn is 1st word. Now think about what Heyerdahl was famous for and remove the usual abbreviation for river

Thank you very much, BuzzyBee - I was looking at the wrong end of the solution!