Using a higher voltage means you can use a lower amperage. The lower the current flow, the less energy is lost to heat in a resistive substance (copper). Thus, this is why transmission lines are several tens of thousands to hundreds of thousands of volts (and kiloamps of current). Over a long stretch the savings are very, very pronounced.
Fixed.
Big power lines (in excess of 100 kV) carry currents in the thousands of Amps range. Now that's BULK. Think about it - if you need to carry the power from a big nuke station (2 x 10^9 Watts) at 500 kV - that's 4000 A (which would be divided into multiple phases). But still, that's over 1 kA per phase, for a single plant. If you've got a cluster of plants, or a multi-unit plant, then the currents can be a lot higher.
The current is so high, that the wires get too hot to touch (that's why birds don't sit on the power conductors on high voltage lines - they only sit on the safety ground wire). Typically power lines are designed to operate at wire temperatures of about 150-250 F. There are new high-tech carbon fiber cored power cables, which are designed for upgrading power lines, which can operate at 300 F (and they have lower resistance too) significantly increasing the power handling capability of the power line - although they don't do anything for the efficiency.