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It's hard to say what's going on with your Arrowhead Vine http://davesgarden.com/guides/pf/go/93772/ without a photo. Can you possibly post a picture of your plant? Curling leaves could mean an insect problem but I'm not sure what you mean by dark breaks in the leaf stems.
That plant looks very healthy to me. I really can't see anything wrong. I have one that looks very similar to that one and the curling leaves are newer leaves that will eventually open fully. I think what you are referring to as breaks in the stems are actually nodes where aerial roots will come out - this plant is like a vine and will climb if given a support! I have the solid green planted in the ground outside my screened pool enclosure and it attaches itself to the screen and climbs up the side of the screen.
Here's a photo of the solid green Arrowhead Vine. You can see the holes in the screen where I've pulled it off in places! It's growing back up again. Sometime in the next couple of months I'm going to dig it all up and hubby is going to replace the screen.
I don't have a clue as to what the problem could be. I've never had a problem with stems of any plant breaking off like that. The leaves on mine are staying curled ... it takes awhile for them to open. I have heard of plant leaves curling when there is an infestation of spider mites or caterpillar type worms. Looking at your photo, I can't see anything adverse with the plant. It looks healthy to me. Can you post another photo - maybe a bit farther away? Have you seen any webs or any type of crawling insects on your plant? Can you take one of the tightly curled leaves, cut it off and unroll the leaf to see if there is any type of insect in there? If you find insects you can buy some Insecticidal soap spray and use it for your plant. I use one I buy at Home Depot that is specifically for house plants and it works well for different types of insects.
Hopefully someone will come along soon and maybe have some ideas and suggestions.
I don't have this plant, but in the past I've grown others that had curled leaves at first which opened up eventually--sometimes that would take quite a while. plantladylin's suggestion of unrolling a leaf to see if you see signs of anything inside it, but if you don't then I imagine it's just taking the new leaves a while to unfold.
As far as the breaks, it really doesn't look disease related to me, my suspicion would either be that the stems are a little weak and the weight of the leaves breaks some of them, or if they're a bit on the weak side it would be very easy to accidentally break them like that when you're watering. Or if you've got kids, pets, etc who might bump up against the plant and break it without realizing it that would be a possibility too.