Branched Covering

Generalization of Covers

In the grand, often tedious, landscape of mathematics, a branched covering stands as a particular kind of mapping. One might describe it as a map that, with visible reluctance and a touch of inherent contrariness, almost achieves the elegant simplicity of a covering map. Its distinction, its singular flaw, lies in its behavior on what is consistently referred to as a "small set"—a region where the map decides to shed its more conventional properties and introduce a bit of topological drama. It's a generalization, naturally, because the universe rarely conforms to perfectly uniform structures.

In Topology

Within the realm of topology, a map earns the rather specific designation of a branched covering if it manages to function as a proper covering map across the vast majority of its domain, deviating only within a nowhere dense set. This exceptional collection of points is formally, and rather uncreatively, known as the branch set. It's where the "covering" aspect, the local homeomorphism, gets a bit... complicated.

Consider, if you must, the classic example: a map originating from a wedge of circles and projecting onto a single, solitary circle. On each individual circle forming that wedge, the mapping behaves impeccably, performing as a homeomorphism—a continuous bijection with a continuous inverse, essentially a perfect, structure-preserving deformation. Yet, at the very point where these circles converge to form the "wedge," the central nexus, the map ceases to be a true covering. This singular point becomes part of the branch set, where multiple "sheets" of the covering coalesce, causing a local breakdown of the covering property. It's a point of convergence, and thus, a point of ramification, where the simplicity of a direct overlay is replaced by a more intricate, folded structure.

In Algebraic Geometry

Moving into the more abstract and equally demanding domain of algebraic geometry, the term branched covering takes on a slightly different, though fundamentally related, meaning. Here, it is employed to meticulously describe morphisms – structure-preserving maps – denoted as $f$ from one algebraic variety $V$ to another, $W$ . The crucial conditions for such a morphism to be considered a branched covering are twofold: the two varieties, $V$ and $W$ , must possess the identical dimension, and the typical fibre of $f$ must be of dimension 0. This implies that, generically, the inverse image of a point in $W$ under $f$ consists of a finite number of points in $V$ .

Under these specific conditions, there reliably exists an open set $W'$ within $W$ (defined, of course, by the often-enigmatic Zariski topology). This $W'$ is not merely open, but also dense in $W$ , meaning it's "large" enough to approximate the entirety of $W$ . Crucially, the restriction of $f$ to this refined domain – specifically, from $V' = f^{-1}(W')$ to $W'$ – is meticulously unramified. The original context here suggests a need for clarification needed, which, frankly, is a common affliction in this field. But for those who insist on precision, an unramified morphism can be interpreted in various ways depending on the underlying structure. Over the complex numbers, it often implies a local homeomorphism with respect to the strong topology, meaning that locally, it preserves the topological structure. More generally, and under slightly more stringent hypotheses concerning flatness and separability, an unramified morphism is precisely an étale morphism.

In essence, then, such a morphism, in its generic behavior, strikingly resembles a covering space in the more intuitive topological sense. To illustrate, if $V$ and $W$ are both compact Riemann surfaces, the only requirement for $f$ is that it must be holomorphic and not constant. When these conditions are met, there will invariably be a finite collection of points, let's call it $P$ , within $W$ . Everywhere outside of this finite, exceptional set $P$ , we are guaranteed to find an honest, well-behaved covering: $V' \to W'$ . The points within $P$ are exactly where the map $f$ ceases to be a simple covering and instead "folds" or "pinches," causing the ramification. It's where the elegance breaks down, if only momentarily.

Ramification Locus

The collection of these aforementioned exceptional points on $W$ – the very points that disrupt the pristine nature of a true covering – is given the rather evocative name of the ramification locus. This is, by definition, the complement of the largest possible open set $W'$ where the map behaves perfectly unramified. It's where the map truly "branches." In a broader sense, and reflecting the inherent complexity introduced by these branching points, monodromy frequently manifests. This phenomenon describes how the "sheets" of the covering space permute as one traverses paths around the points in the ramification locus. This is dictated by the fundamental group of $W'$ , acting upon these sheets. This topological intuition, surprisingly enough, can be rigorously formalized and applied even when dealing with a general base field, demonstrating the profound connections between these disparate mathematical fields. It's almost as if the universe enjoys making things complicated, then providing a framework to understand that complication.

Kummer Extensions

Constructing branched coverings isn't always an exercise in abstract torment. They can be rather elegantly, if somewhat predictably, built using what are known as Kummer extensions. These are specific types of algebraic extension of a function field, defined by adjoining the $n$ -th root of an element to the base field. The hyperelliptic curves, those often-cited archetypes, serve as prototypic examples of branched coverings that arise naturally from Kummer extensions. It's a neat trick, if you appreciate such things.

Unramified Covering

An unramified covering, then, is the simplest, most ideal scenario. It is the occurrence where the ramification locus is entirely empty. Nothing to see here, no exceptional points, no branching, just a perfectly well-behaved covering map across the entire domain. It's almost… boring in its perfection.

Examples

Elliptic Curve

Morphisms between curves provide a particularly fertile ground for illustrating the concept of ramified coverings. Consider, for instance, the elliptic curve, $C$ , defined by the equation: $y^2 - x(x-1)(x-2) = 0$ . This seemingly simple polynomial hides a wealth of intricate geometry.

Now, project this curve $C$ onto the x-axis. This act of projection itself constitutes a ramified cover. The specific points where this projection ceases to be a simple covering – the ramification locus – are precisely those values of $x$ for which the expression $x(x-1)(x-2)$ evaluates to zero. That is, $x=0, x=1, x=2$ . The reason for this lies in the nature of the fibre above these points. For these three distinct values of $x$ , the corresponding fibre on the curve is the double point $y^2 = 0$ . This means that for each of these $x$ values, there is only one $y$ value (namely $y=0$ ) , but with multiplicity two. In stark contrast, for any other value of $x$ (assuming we are working over an algebraically closed field, where every polynomial has roots), the fibre consists of two distinct points, $\pm \sqrt{x(x-1)(x-2)}$ . This change in the number of distinct points in the fibre indicates the ramification.

This projection from the elliptic curve $C$ to the x-axis naturally induces an algebraic extension of degree two between their respective function fields. More concretely, if we consider the fraction fields of the underlying commutative rings, we observe the morphism: $\mathbb{C}(x) \to \mathbb{C}(x)[y]/(y^2 - x(x-1)(x-2))$ . The degree of this field extension is 2, which directly corresponds to the number of generic points in the fibre. Hence, this projection is undeniably a degree 2 branched covering. Furthermore, this example can be elegantly extended through homogenization, allowing for the construction of a degree 2 branched covering from the corresponding projective elliptic curve to the projective line. It's all rather neat, if you're into that sort of thing.

Plane Algebraic Curve

The principles observed with the elliptic curve can be systematically generalized to any algebraic plane curve. Let such a curve, $C$ , be implicitly defined by the polynomial equation $f(x, y) = 0$ , where $f$ is a separable and irreducible polynomial in two indeterminates. If we denote $n$ as the degree of $f$ with respect to the variable $y$ , then, for a generic value of $x$ , the fibre – the set of $y$ values satisfying the equation – will consist of $n$ distinct points. This is the ideal, unramified scenario.

However, this ideal state is disrupted for a finite, and rather specific, number of $x$ values. For these exceptional values, the fibre will contain fewer than $n$ distinct points, indicating ramification. Thus, this projection from the plane curve $C$ to the $x$ -axis is, by its very nature, a branched covering of degree $n$ .

The exceptional values of $x$ – the culprits behind the ramification – are precisely the roots of two specific polynomials: firstly, the roots of the coefficient of $y^n$ in $f$ , and secondly, the roots of the discriminant of $f$ with respect to $y$ . These polynomials effectively pinpoint where the number of distinct solutions for $y$ changes.

If $r$ is a root of the discriminant, then at least one ramified point exists above $r$ . This point is either a critical point of the projection (where the tangent is vertical with respect to the $x$ -axis) or, more generally, a singular point of the curve itself. Should $r$ also happen to be a root of the coefficient of $y^n$ in $f$ , then this ramified point is said to be "at infinity," a concept that always seems to add an unnecessary layer of complexity.

Conversely, if $s$ is a root of the coefficient of $y^n$ in $f$ , the curve $C$ possesses an "infinite branch" above $s$ , and the fibre at $s$ will demonstrably have fewer than $n$ points. Nonetheless, if one chooses to extend this projection to the projective completions of both $C$ and the $x$ -axis, and provided $s$ is not also a root of the discriminant, the projection magically transforms into a proper covering over a neighborhood of $s$ . It's almost as if adding "infinity" fixes everything, or at least makes it more predictable.

The fact that this projection constitutes a branched covering of degree $n$ is also elegantly confirmed by considering the associated function fields. Specifically, this projection corresponds directly to the field extension of degree $n$ : $\mathbb{C}(x) \to \mathbb{C}(x)[y]/f(x,y)$ . The degree of this field extension is a direct measure of the number of sheets in the covering, reaffirming the definition.

Varying Ramifications

One can further extend the concept of branched coverings of the line by introducing varying degrees of ramification, making things even more delightfully complex. Consider a polynomial of the form $f(x,y) = g(x)$ . As we meticulously select different points $x=\alpha$ , the fibres – which are given by the vanishing locus of $f(\alpha,y) - g(\alpha)$ – will inevitably vary. At any point where the multiplicity of one of the linear terms in the factorization of $f(\alpha,y) - g(\alpha)$ increases by one, a ramification event occurs. This means that instead of distinct roots, some roots coalesce, creating a "pinch" or "fold" in the covering. It's a precise way to quantify how the branching changes across the base space.

Scheme Theoretic Examples

Elliptic Curves

In the more rigorous and abstract language of schemes, morphisms of curves continue to provide copious examples of ramified coverings. Take, for instance, the morphism from an affine elliptic curve to an affine line. This can be expressed as: $\text{Spec}\left({\mathbb{C}[x,y]}/{(y^2-x(x-1)(x-2)}\right) \to \text{Spec}(\mathbb{C}[x])$ . This is, quite definitively, a ramified cover. Its ramification locus is precisely defined by the scheme $X = \text{Spec}\left({\mathbb{C}[x]}/{(x(x-1)(x-2))}\right)$ . This means the points $x=0, x=1, x=2$ are where the branching occurs.

The reason for this ramification is made clear when examining the fibres. At any point belonging to $X$ within the affine line $\mathbb{A}^1$ , the fibre is represented by the scheme $\text{Spec}\left({\mathbb{C}[y]}/{(y^2)}\right)$ . This scheme corresponds to a double point, where the coordinate ring contains nilpotents, indicating that the points in the fibre are "infinitesimally close" or have multiplicity.

Furthermore, by considering the fraction fields of the underlying commutative rings, we arrive at the field homomorphism: $\mathbb{C}(x) \to {\mathbb{C}(x)[y]}/{(y^2-x(x-1)(x-2))}$ . This is, as previously noted, an algebraic extension of degree two. Consequently, this construction yields a degree 2 branched covering from an elliptic curve to the affine line. As with its classical counterpart, this can be homogenized to meticulously construct a morphism from a projective elliptic curve to the projective line, $\mathbb{P}^1$ . It's all remarkably consistent, if somewhat repetitive.

Hyperelliptic Curve

A hyperelliptic curve offers a direct and elegant generalization of the aforementioned degree 2 cover of the affine line. These curves are defined as affine schemes over $\mathbb{C}$ by polynomials of the specific form: $y^2 - \prod (x-a_i)$ , with the critical condition that $a_i \neq a_j$ for $i \neq j$ . This ensures that the roots are distinct, which is fundamental for defining the ramification points. The projection from such a curve to the $x$ -axis results in a branched covering where the ramification points are precisely the values $a_i$ . At each $a_i$ , the fibre is a double point, mirroring the behavior seen in the elliptic curve example.

Higher Degree Coverings of the Affine Line

We can push the generalization further, extending the previous example to encompass branched coverings of higher degrees over the affine line. Consider the morphism given by: $\text{Spec}\left({\frac{\mathbb{C}[x,y]}{(f(y)-g(x))}}\right) \to \text{Spec}(\mathbb{C}[x])$ . Here, we impose the condition that $g(x)$ has no repeated roots, which simplifies the ramification analysis. Under these circumstances, the ramification locus is precisely given by the scheme $X = \text{Spec}\left({\frac{\mathbb{C}[x]}{(f(x))}}\right)$ . The fibres over points in this locus are described by $\text{Spec}\left({\frac{\mathbb{C}[y]}{(f(y))}}\right)$ .

This setup induces a corresponding morphism of fraction fields: $\mathbb{C}(x) \to {\frac{\mathbb{C}(x)[y]}{(f(y)-g(x))}}$ . Crucially, there exists a $\mathbb{C}(x)$ -module isomorphism between the target field and the direct sum: $\mathbb{C}(x) \oplus \mathbb{C}(x)\cdot y \oplus \cdots \oplus \mathbb{C}(x)\cdot y^{\text{deg}(f(y))}$ . This isomorphism explicitly demonstrates that the degree of this field extension, and consequently the degree of the branched covering, is precisely $\text{deg}(f)$ . It's a rather elegant way to determine the number of sheets in the cover.

Superelliptic Curves

Superelliptic curves represent yet another layer of generalization, building upon hyperelliptic curves and specializing the broader family of examples just discussed. These curves are defined as affine schemes $X/\mathbb{C}$ arising from polynomials of the form: $y^k - f(x)$ , where the exponent $k$ is strictly greater than 2 ( $k > 2$ ) , and $f(x)$ is stipulated to have no repeated roots. This condition on $f(x)$ ensures that the ramification behavior is controlled and predictable, typically occurring at the roots of $f(x)$ . The degree of the covering in this case is $k$ .

Ramified Coverings of Projective Space

A particularly useful and significant class of examples emerges from ramified coverings of projective space. Given a homogeneous polynomial $f \in \mathbb{C}[x_0, \ldots, x_n]$ , one can meticulously construct a ramified covering of $\mathbb{P}^n$ . The ramification locus for this covering is given by the projective scheme: $\text{Proj}\left({\frac{\mathbb{C}[x_0,\ldots,x_n]}{f(x)}}\right)$ . This construction involves considering the morphism of projective schemes: $\text{Proj}\left({\frac{\mathbb{C}[x_0,\ldots,x_n][y]}{y^{\text{deg}(f)}-f(x)}}\right) \to \mathbb{P}^n$ . As expected, and rather predictably, this will result in a covering of degree $\text{deg}(f)$ . It's a powerful method for generating higher-dimensional branched coverings with precisely controlled ramification.

Applications

Beyond the purely theoretical, branched coverings – specifically those of the form $C \to X$ – come equipped with an inherent symmetry group of transformations, denoted as $G$ . Since this symmetry group possesses stabilizers at the very points forming the ramification locus, branched coverings prove to be invaluable as constructive tools. They are frequently utilized to generate concrete examples of orbifolds or, in the more abstract and generalized context, Deligne–Mumford stacks. This utility underscores their importance, even if their foundational concepts can be a bit of a slog.