Question

Show that if \(V \subset W \subset \mathbb{P}^{n}\) are varieties, and \(V\) is a hypersurface, then \(W=V\) or \(W=\mathbb{P}^{n}\) (see Problem 1.30).

Short answer

In conclusion, given varieties \(V\) and \(W\) with \(V \subset W \subset \mathbb{P}^{n}\) and \(V\) being a hypersurface, we demonstrated that either \(W=V\) or \(W=\mathbb{P}^{n}\). We achieved this by analyzing the degrees of the defining polynomials of \(V\) and \(W\) and considering two cases for the degree of the polynomial that relates them. Depending on the degree, we were able to deduce that either \(W=V\) or \(W=\mathbb{P}^{n}\).

Step-by-step solution

Define hypersurface

A hypersurface is a projective variety defined by a single homogeneous polynomial. Let \(F \in k[x_0, x_1, \dots, x_n]\) be the defining polynomial of \(V\). Since \(V\) is a hypersurface, \(F\) must be non-constant and have degree \(d \geq 1\).

Relate degrees of \(V\) and \(W\)

Since \(V \subset W\), any point in \(V\) also belongs to \(W\). In other words, any point \(p \in V\) satisfies the defining equation of \(W\), which means that the defining polynomial of \(V\) divides the defining polynomial of \(W\). Let \(G \in k[x_0, x_1, \dots, x_n]\) be the defining polynomial of \(W\). We can write \(G = FH\) for some polynomial \(H \in k[x_0, x_1, \dots, x_n]\) and \(\deg(G) = \deg(F) + \deg(H)\).

Consider the cases for the degree of \(H\)

Since \(\deg(G) = \deg(F) + \deg(H)\), we have two possibilities: either \(\deg(H) = 0\) (i.e., \(H\) is a constant) or \(\deg(H) \geq 1\). Case 1: \(\deg(H) = 0\). In this case, \(H\) is a nonzero constant and \(G = FH\), so \(W\) and \(V\) have the same defining polynomial, and thus \(W = V\). Case 2: \(\deg(H) \geq 1\). In this case, we will show that \(W = \mathbb{P}^{n}\).

Show that \(W=\mathbb{P}^{n}\) when \(\deg(H) \geq 1\)

Since \(\deg(H) \geq 1\), there must be at least one point \(q \in \mathbb{P}^{n}\) for which \(H(q) \neq 0\). This is because if \(H\) vanishes everywhere, it must be the zero polynomial, contradicting the fact that \(\deg(H) \geq 1\). Now consider any point \(r \in \mathbb{P}^{n}\). Since \(V\) is a hypersurface, there exists a line \(L\) in \(\mathbb{P}^{n}\) that passes through \(q\) and \(r\). Let \(p\) be any point on \(L\). Then all points of \(L\) can be written as \(p = \lambda q + \mu r\) for some \(\lambda, \mu \in k\). Since \(H\) is homogeneous, we have \(H(\lambda q + \mu r) = \lambda^{\deg(H)}H(q) + \mu^{\deg(H)}H(r)\). Since \(H(q) \neq 0\) and \(\deg(H) \geq 1\), this means that \(H(p) \neq 0\) for all points \(p \in L\). In particular, \(H(r) \neq 0\), so \(r \in W\). This argument applies to all points \(r \in \mathbb{P}^{n}\), so \(W = \mathbb{P}^{n}\) in this case.

Conclusion

We have shown that if \(V \subset W \subset \mathbb{P}^{n}\) are varieties with \(V\) being a hypersurface, then either \(W=V\) or \(W=\mathbb{P}^{n}\).

Key concepts

Hypersurface

In algebraic geometry, a hypersurface is one of the fundamental concepts. It is a special type of projective variety. To better understand this, a projective variety is essentially a set of solutions to a system of polynomial equations in projective space. A hypersurface simplifies this idea; it is defined by just a single homogeneous polynomial. Consider a polynomial \(F\) in variables \(x_0, x_1, \ldots, x_n\). If \(F\) is non-constant and homogeneous, then the hypersurface is the set of all points where this polynomial equals zero. For the hypersurface to exist in our example, it's crucial that the polynomial has a degree \(d \geq 1\). That is, it's really essential for the polynomial to have meaningful geometric properties.

So, in summary, a hypersurface is just a very neat, often complex shape in projective space, defined simply by one polynomial equation. It represents much more than just zeros of a polynomial; it's a window into multidimensional geometric structures.

Projective Variety

Varieties are central to algebraic geometry. A projective variety, in particular, is like a playground where algebra meets geometry. It lives in a projective space, which is a mathematical way to handle points 'at infinity'. Without getting too technical, a projective variety is a set of common solutions to one or more homogeneous polynomial equations, defined in projective space \(\mathbb{P}^{n}\).

To imagine this visually, think of it as a surface or perhaps a multi-dimensional shapeshift that stretches across infinite planes. Projective varieties can take many forms, ranging from simple curves to complex surfaces and beyond. The fascinating part about them is that they compactly wrap infinite points, allowing insights into geometric features that continue indefinitely.

In our example, the varieties \(V\) and \(W\) form a hierarchy within these infinite spaces, where \(V\) is a hypersurface. The beauty of projective varieties is that they allow us to explore and uncover deep geometric properties using algebraic techniques.

Polynomial Divisibility

Polynomial divisibility is a useful tool in understanding relationships between different geometric entities in algebraic geometry. When we say a polynomial \(F\) divides another polynomial \(G\), we mean there's a third polynomial \(H\) such that \(G = FH\).

In this context, divisibility becomes a method to assess how two varieties relate. If a polynomial \(F\) defines one variety \(V\), and it divides another polynomial \(G\) that defines a variety \(W\), it implies all solutions of \(V\) are also solutions of \(W\). This is a critical way to link these geometric structures through their algebraic representations.

It’s a bit like saying that all shapes defined by \(F\) fit perfectly within those defined by \(G\). This mathematical insight is crucial when determining whether two geometric shapes, or varieties like \(V\) and \(W\) in our example, are the same or nested within one another.

Degrees of Polynomials

The degree of a polynomial is a pivotal concept in polynomial algebra and geometry. It tells you about the highest power of the variable in the polynomial, encapsulating its complexity and shape. In geometric terms, the degree relates to dimension and the shape of the algebraic set defined by the polynomial.

In the given problem, understanding how degrees work helps distinguish between whether two varieties are identical or different. If a polynomial \(G\) is the product of \(F\) and \(H\), the degree of \(G\) is the sum of the degrees of \(F\) and \(H\). Thus, if \(H\) has a degree of zero, it indicates \(H\) is merely a constant and \(V\) and \(W\) are essentially the same shape or variety.

On the other hand, if \(H\) has a degree of one or more, \(W\) extends beyond \(V\), potentially encompassing the entire projective space. Degrees offer a succinct way to measure these relationships between polynomials, influencing how we interpret their geometric implications.