Here's a gnat's-eye view at the face of a nut as seen from the leeward side of the second fret. The slots for these two strings are cut so that they completely support the string. 

The sketch above relates to fretted instruments, but the principle is no different for violin family and other unfretted instruments. The .001" clearance is somewhat hyperbolic—I explain this in a minute..

Here's an idea of how it works on a bass:

Having the slot cut too high above the frets (or an unfretted board of some type) means that the act of pressing the string down to the first few frets actually stretches the string, raising the pitch and throwing the intonation off in the process. Ideally the nut slot height is identical to any other fret. Remember that strings can 'saw' themselves through a nut just through normal use. I ordinarily leave a nut slot a little higher than necessary at first, to allow for the string to cut itself a little lower. I also avoid synthetic and elephant ivory, both of which are too soft and rubbery to make good nuts for steel strings.

Here's a shimmed-up mess of a nut that has all the problems:

These slots are all too deep, but the B is still so high it doesn't play in tune, so someone shoved a piece of ebony under it to try and correct the intonation. Big "Ugh" for this one.

The sketch below illustrates how - and how not - to shape a slot for any string.

Left: a slot that's cut with a saw has a roughly flat bottom and also affords poor acoustic coupling. Saws seldom match the precise width of the string, which will roll side to side in the slot. 

Center: strings will work their way down a v-cut, often bottoming out on frets of the board, as the case may be. The signal transfer is compromised because of the limited contact. They also tend to bind and squeak.

Moreover, the slot itself needs to be shaped in a way that it not only fits the diameter of each individual string, but also such that the string has firm contact with the nut at the very front of the slot. This defines the end of the vibrating string length, and if it's not right, intonation will be impaired. 

I prefer to shape my slots in the shape of a horn's bell:

The point of this is to offer a smooth surface for the string to travel from the tuning machine to the critical point of final contact at the front of the slot, where it is held firmly to define the end of the vibrating string length.

Strings have to make a compound bend at the nut, and to make tuning easiest while ensuring complete firm contact at the front of the slot, this horn bell shape makes certain the string glides smoothly, no matter the angle of approach. Here's a treble side view:

The bell here is imaginary. The nut is in yellow, the fingerboard is dark brown. The string is the green line, and the tuning machines are off to the right somewhere. Notice that the string connects with a smooth curved surface, no corner or edge. Whether the string is coming from the top or the bottom of the string post, it will slide smoothly into the nut slot. The string is in complete contact with the front 30% of the nut. There's plenty of substance there to keep the string from sawing its way deeper into the bone. 

Here's the same slot seen looking straight down from above:

The string's other curve, from, say, the farthest peg on the bass side of the headstock, also elides with the inside of the bell-shaped slot, guided gently and firmly to the front where it's held firmly by its own tension inside the confines of a well cut slot. 

If the slot isn't properly angled back, several problems can arise. If it's too flat (some repair books actually advocate this!) the string soon wears away the front of the slot and the functional point of contact is as much a 40% of the width of the nut back from the front edge, which can cause the note to ring poorly (because it's vibrating along a surface, not held to a point) and perhaps cause intonation problems. This is bad:

If the slot is angled back, but left a straight line, it will bind on the back edge, and the front edge will wear down from playing and the string is at risk for sizzling on the first fret or on the surface of the board. 

The precise shape of the slot at the front edge is extremely important for sound quality, stability of the setup, and intonation. 

More on bridges in due time, but the principles here apply to bridge slots on the viol and violin families, archtop guitars, mandolins, and so on.
 
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