There is no fuse in the positive battery cable to the starter; because the starter draws far too much current to go through a fuse. When you say "line to the starter" do you mean the heavy battery cable to the starter; or do you mean the small wire from the igntion switch to the starter? The wire from the ignition switch to the starter will not have power coming through it unless the ignition switch is turned to the "start" position. There also can be 2 other switches in that wire. That's why I would need to know additional details about the car in order to answer this question.
This is information that anybody will need to know whenever you write in about a car; so please always include it in your question: What year, make, and model is your car? What size engine does it have? Does it have a manual or automatic transmission?
When the battery repeatedly goes dead; it is usually caused by a shorted diode in the alternator; which drains the battery when the motor is shut off. This would also be what caused the battery to go dead in the first place.
Once the battery becomes discharged; there will be no power coming from it to the starter (or to anywhere else). There can only be power from the battery after it has been charged (and before it goes dead; if a bad alternator is draining it.)
Please do not buy an alternator from a discount parts store; or you'll run a huge risk of it being defective right out of the box. Having a lifetime warranty does not provide any assurance that it will be reliable. Alternators from those stores are shipped to Mexico; and worked on by people who do not have proper parts or test equipment. That is why stores like NAPA AUTO PARTS, who serve professional mechanics, sell alternators which are properly remanufactured, and cost more; but will last MANY TIMES LONGER.
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