Whis is this 'come' in this sentence for?

[dragontiger]

Hello, I'm James.

As a ESL student, I'm studying using some movie scripts.

Yesterday, I watched and studied the script of "You've got mail".
There was one sentence that I can't understand grammartically.

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INT. COMPUTER SCREEN - DAY

As we see the logo for America On Line come up and Kathleen's code name: Shopgirl. She logs on and the computer makes all its little modem noises as the computer dials the access number and connects and we hear the machine

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When I saw this sentence "we see the logo for America On Line come up", I thought the word 'come' had grammartical error.

I guess it's correct if this sentence is ' ~ On Line coming up '. Does it make sence? Or, It can just say correct because of an informal expression?

I looked up the dictionary that 'Come' is used as 'verb, linking-verb, prep'. I couldn't understand how the word is used.

Please give me advice :)
thanks beforehand

[Tdol]

If you see something coming up, you see it appear over time, but a logo on a screen is instantaneous, so the progressive form doesn't work as well.

[dragontiger]

Thanks, teacher Tdol :)
I apprecite your help.

[Tdol]

You're welcome- we often use the present simple for something that happens very fast- if you listen to sports sommentators, you'll here them use the present simple when the action is right now and very quick.