Secant Varieties: Why The Map Is Well-Defined

Hey guys! Ever wondered about the fascinating world of secant varieties in algebraic geometry? It's a pretty cool topic, and today we're going to dive deep into one specific aspect: why the map used to define secant varieties is actually well-defined. This might sound a bit technical, but trust me, we'll break it down in a way that's easy to understand. So, grab your favorite beverage, get comfy, and let's explore this together!

Why is the Secant Variety a Variety? The Well-Defined Map

Let's get started by understanding the core question: Why is the secant variety considered a variety in the context of algebraic geometry? This question stems from a construction involving a rational map, which we'll call φ. This map, φ: V × V × ℙ¹ ⟶ ℙⁿ, plays a crucial role in defining the secant variety. Here, V represents a projective variety embedded in the projective space ℙⁿ, and ℙ¹ is the projective line. The map φ essentially takes a pair of points from our variety V and a point from the projective line ℙ¹ and maps them to a point in ℙⁿ. The secant variety is then, roughly speaking, the closure of the image of this map. Now, the tricky part is ensuring that this map φ is well-defined. What does well-defined even mean in this context? Well, a rational map is not defined everywhere; it has points where it becomes indeterminate, often due to divisions by zero. For the secant variety construction to work, we need to make sure that these indeterminate points don't mess things up too much. Specifically, we need to show that we can resolve these indeterminacies in a way that allows us to take the closure and still get a variety. This involves a careful analysis of the map φ and its behavior, ensuring that the construction leads to a well-behaved geometric object. We achieve this by blowing up the product V × V along the diagonal. This process effectively replaces the problematic diagonal, where the two points coincide, with the projectivized normal bundle. This resolution allows us to extend the rational map φ to a regular map, which is defined everywhere. The image of this regular map then gives us the secant variety, and since the map is well-defined after the blow-up, we can be confident that the resulting secant variety is indeed a variety in the algebraic geometric sense. The map's well-defined nature ensures that the resulting secant variety has the expected geometric properties and is a valuable object of study.

The Nitty-Gritty: Defining the Rational Map φ

Alright, let's break down this rational map φ a little further, because understanding its definition is key to grasping why it's well-defined. Remember, this map is the heart of our secant variety construction. We need to define this map precisely, because its properties determine the properties of the secant variety itself. The map φ takes a triple (P, Q, [s:t]) as input, where P and Q are points on our variety V, and [s:t] represents a point on the projective line ℙ¹. Think of [s:t] as homogeneous coordinates; it's the ratio of s and t that matters, not the specific values themselves. Now, how does φ map this triple to a point in ℙⁿ? Well, if P and Q are distinct points, the map is defined quite naturally: φ(P, Q, [s:t]) = [sP + tQ]. This means we're taking a linear combination of the homogeneous coordinates of P and Q, using s and t as the coefficients. Geometrically, this corresponds to a point on the line passing through P and Q. As [s:t] varies, we trace out the entire line. But what happens when P and Q are the same point? That's where things get a bit tricky, and the map becomes a priori undefined. This is because if P = Q, then sP + tQ = (s+t)P, which is just a scalar multiple of P. We need a way to extend the definition of φ to include this case, and that's where the blow-up comes in. The blow-up allows us to separate these coinciding points and define the map in a consistent way. By carefully defining φ in this manner, we ensure that it captures the intuitive idea of taking linear combinations of points on V, even when those points coincide. This rigorous definition is essential for ensuring that the secant variety is a well-behaved object that reflects the underlying geometry of V.

The Trouble with the Diagonal: Where Things Get Indeterminate

So, why all this fuss about the diagonal? What's so problematic about the case where P = Q? Let's dive into the heart of the issue: the indeterminacy of the rational map φ. When we say a map is indeterminate at a point, it means that the map is not defined at that point in a straightforward way. In our case, the issue arises when we try to evaluate φ(P, Q, [s:t]) when P and Q are the same point. Remember, the formula φ(P, Q, [s:t]) = [sP + tQ] works perfectly well when P and Q are distinct. We're simply taking a linear combination of their homogeneous coordinates, which gives us a point on the line connecting them. But when P = Q, we get φ(P, P, [s:t]) = [sP + tP] = [(s+t)P]. This looks innocent enough, but here's the catch: we're working in projective space, where points are defined up to scalar multiplication. So, [(s+t)P] is the same point as [P], as long as s + t ≠ 0. But what happens if s + t = 0? Then we get [0P] = [0], which is not a well-defined point in projective space. This is our indeterminacy! The map φ doesn't have a clear, unique value when P = Q and s + t = 0. This indeterminacy is not just a minor technicality; it's a fundamental issue that needs to be addressed. If we simply ignored it, our secant variety construction would be ill-defined, and we wouldn't be able to guarantee that the resulting object is a variety in the algebraic geometric sense. This is why we need a more sophisticated approach, and that's where the blow-up comes to the rescue. The blow-up allows us to resolve this indeterminacy and extend the definition of φ in a consistent and meaningful way, ultimately ensuring that our secant variety is a well-behaved geometric object.

Blowing Up the Diagonal: A Clever Trick to Resolve Indeterminacy

Okay, guys, let's talk about the hero of our story: the blow-up. This is a powerful technique in algebraic geometry that allows us to resolve singularities and indeterminacies in a clever way. In our case, we're going to blow up the product V × V along the diagonal Δ, where Δ is the set of all pairs (P, P) where P is a point in V. Think of the blow-up as a surgical procedure on our space. We're zooming in on the diagonal and replacing it with something new, something that allows us to extend our map φ. Imagine the diagonal as a problematic line in a surface. The blow-up essentially