Coming Together: With Curves

So now we define a family of stable curves over a base to be a flat morphism with fibers stable curves and sections. The functor taking to the set of families of stable curves over is not representable , sadly. However, it is coarsely representable.

Secant lines

We denote it by. If two points are coming together in the moduli space, then the limit will be non-irreducible. Now, an interesting thing is to look at. This is actually a divisor, called the boundary divisor, and it behaves rather well. Now we move on to families of maps. A family of maps of -points genus quasi-stable curves to over is a family of quasi-stable curves along with a map.

The members of the family are the restriction of to the fibers. A family of maps will be a family of stable maps if the map restricted to each fiber is stable. It will just be a condition on the components: Then the map is stable if the two following conditions hold:. The last point in defining the moduli space is breaking it up into components.

Let a homology class on. Then we say that represents if. There exists a projective coarse moduli space of stable maps representing. Now we define to be the subset of maps with no automorphisms, and we say that a variety is convex if for all , we have. We also define the boundary of to be the locus of non-irreducible domains.

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Then we have the following theorem:. So let's imagine a curve that looks something like this.

So what is the rate of change of y with respect to x of this curve? Well, let's look at it at different points.

And we could at least try to approximate what it might be in any moment. So let's say that this is one point on a curve. Let's call that x1, and then this is y1. And let's say that this is another point on a curve right over here, x2. And let's call this y2. So this is a point x1, y1, this is a point x2, y2.

So we don't have the tools yet. And this is what's exciting about calculus, we will soon have the tools to figure out, what is the rate of change of y with respect to x at exactly this point?

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But we don't have that tool yet. But using just the tools from algebra, we could at least start to think about, what is the average rate of change over the interval from x1 to x2? Well, what's the average rate of change? Well, that's just how much did my y change-- so that's my change in y-- for this change in x.

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And so we would calculate it the same way. So our change in y over this interval is equal to y2 minus y1, and our change in x is going to be equal to x2 minus x1. So just like that we were able to figure out the rate of change between these two points. Or another way of thinking about it is, this is the average rate of change for the curve between x equals x1, and x is equal to x2.

This is the average rate of change of y with respect to x over this interval. But what if we have we also figured out here? Well, we figured out the slope of the line that connects these two points. And what we call a line that intersects a curve in exactly two places? Well, we figured out, we call that a secant line. So this right over here is a secant line.

So the big idea here is we're extending the idea of slope. We said, OK, we already knew how to find the slope of a line. A curve, we don't have the tools yet, but calculus is about to give it to us. But let's just use our algebraic tools. We can at least figure out the average rate of change of a curve, or a function, over an interval. That is the same exact thing as the slope of a secant line. Now just as a little bit of foreshadowing, where is this all going? How will we eventually get the tools, so that we can figure out the instantaneous rate of change, not just the average?

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Well just imagine what happens if this point right over here got closer and closer to this point. Michelle marked it as to-read May 12, Tracey marked it as to-read May 12, Sonia marked it as to-read May 13, Darlene Howard marked it as to-read May 13, Simon marked it as to-read May 13, Sarah marked it as to-read May 13, Patricia marked it as to-read May 13, Julie Alvarez marked it as to-read May 13, Lisa Ann marked it as to-read May 13, Kim Coomey marked it as to-read May 13, Jennifer beck marked it as to-read May 13, Shelly marked it as to-read May 13, Janelea added it May 13, Cindy Gates marked it as to-read May 13, Gaynor Baker marked it as to-read May 13, Melissa Herston marked it as to-read May 13, Christiane marked it as to-read May 13, Betsy Hover marked it as to-read May 14, Jenn marked it as to-read May 14, Kirsty marked it as to-read May 15, There are no discussion topics on this book yet.

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