A proof of the degree-genus formula

The proof of the degree-genus formula {g=\frac{(d-1)(d-2)}{2}} for complex algebraic plane curves, which works from topological genus in an intuitive case, was the one given in my introductory algebraic geometry class. My TA took it from Kirwan’s book, so it ended up being pretty confusing, since Kirwan’s book is terrible and characteristically did not really explain the gaps in rigor in the argument. (but Oishee you’re still my favorite!)

Looking around, it seems this is kind of a “folklore proof,” in that I haven’t found a fully rigorous version in any book, but everybody knows about the idea. I liked the proof a lot, so here I’ll present a short account of it.

Implicitly, this idea takes the nice correspondence of nonsingular curves with Riemann surfaces as the origin of genus of curves. Riemann surfaces, once you forget the complex structure, just become smooth orientable real surfaces, and that’s pretty much the appeal – genus for smooth orientable surfaces is very geometric (“number of handles”), and probably the first place people see “genus,” so it’s an intuitive way to correspond it to the algebraic case. Of course, in modern treatments, the first principles of genus (arithmetic and geometric) are usually taken to be cohomological, but who cares.

So let’s say we have a projective plane curve of degree {d} – hence, defined by a homogenous polynomial of degree {d}. The degenerate case is just a union of {d} lines. Generically, each pair of lines intersects at a point; taking the analytification correspondences, this is the wedge sum of {d} spheres.

The idea is that by shifting the coefficients slightly to make the curve nonsingular, the global topology stays the same besides at the singular wedge points, which open up into holes between the spheres. Then, if we take one sphere as a “central sphere,” it’s clear that each pair of the remaining {d-1} spheres creates a handle, so for the Riemann surface, we get {g=\binom{d-1}{2}}, as desired. (Actually, this first stage can be replaced by ANY exhibition of a curve with genus {\frac{(d-1)(d-2)}{2}}, so there are many ways to proceed, but this one is most in line with the spirit of the proof.)

This is the first step which is usually skipped: proving that this is indeed what happens to the topology upon perturbing the coefficients to get a nonsingular curve. To rigorize this argument, we use the Milnor fibration.

Let our curve be {C}; initially take it to be {d} hyperplanes intersecting generically, with the consequent polynomial {P(X,Y,Z)} being a product of {d} linear factors. What’s more, we can take all intersections to be away from, say {Z=0}, so we affinize in {x} and {y} to {p(x,y)}, defining an affine slice of the curve {C} where all the singularities happen. Fix one singularity; WLOG it is at the origin {(0,0)}. If we take a ball {B} of sufficiently small radius about the origin of {\mathbb{A}^2}, then for every sufficiently small disc {D_r} of radius {r} about the origin of the complex plane, the map {p:B\cap p^{-1}(D_r\setminus \{0\}) \rightarrow D_r\setminus \{0\}} is a locally trivial fibration.

I won’t go into the general theory of Milnor fibers here; consult chapter 6 of Wall’s book for a treatment. It is clear in this case that the Milnor fiber is a cyllinder (without the ends); see lemma 6.3.3. To visualize why this is true, just picture what is happening locally as the hyperbola {xy=a}.

But the Milnor fiber above {a\in \mathbb{C}} is precisely the curve defined by {p(x,y)-a} in the small ball {B}. This same logic applies to every singularity, so if we make a sufficiently small perturbation to the constant term, all the singularities indeed get resolved into connecting tubes. For sufficiently small {a}, we have to check that the global geometry away from the singularities does not change.

Notice on the full projective curve, what we have done is replace {P(X,Y,Z)} with {P(X,Y,Z)-aZ^d}. In particular, there can clearly be no new singularities on the hyperplane {Z=0}, so for the global geometry, we can continue restricting ourselves to the affine case. In this case, the Milnor map at infinity, the same as above where we take {B} to be an arbitrary large ball, is also a locally trivial fibration for a certain class of polynomials, known as tame polynomials; see Nemethi and Zaharia. This class includes all polynomials with isolated critical points, including our {p(x,y)}. Hence globally the analytic surfaces corresponding to {p(x,y)-a}, parametrized by {a}, are all topologically the same for {a} in some punctured disc in the complex plane. In particular, no new singularities can appear, and the global geometry away from the singularities remains the same. Thus we get {d} spheres pairwise joined by little tubes, and consequently the desired genus of {\binom{d-1}{2}}

Now that we have our “base case,” we work inside the parameter space: the homogenous polynomials of degree {d} in the coordinate ring {\mathbb{C}[X,Y,Z]}, which we will denote {\mathbb{C}[X,Y,Z]_d}, given the obvious topology as a {\mathbb{C}}-vector space – finite dimensional, too, since there are finitely many monomials of fixed degree. We have the subspace of polynomials yielding nonsingular curves {\mathbb{C}[X,Y,Z]_d^{\text{nonsing}}}.

First, we will show that genus is locally constant on this subspace. This is the other rigorizing section typically skipped. Let {M_1,M_2,\ldots, M_k} be the monomials of degree {d}, so that they are a basis for {\mathbb{C}[X,Y,Z]_d^{\text{nonsing}}}. Then take the projective variety {X} defined by the homogenous (in {x,y,z}) coordinate ring {\mathbb{C}[t_2,\ldots,t_k][x,y,z]/(M_1+\sum t_iM_i)} – here we are affine in {t_i}, projective in {x,y,z}. There is an obvious rational map {f:X\rightarrow \mathbb{A}^{k-1}} (affine space over {t_2,\ldots, t_k}), whose fiber over a point {(t_2,\ldots,t_k)} is precisely the degree {d} curve with those coefficients on the monomials. Every curve is equivalent to some curve in this family because the coefficient of {M_1} can always be made nonzero via a projective transformation.

Notice that it’s clear that the only singular points of {X} are singular points on the individual curves. Further, it’s not hard to see that the image of the singular curves on affine space is Zariski closed, since it ultimately consists of polynomial conditions on the coefficients, by the theory of resultants. Hence if we take the analytification of the morphism {f:X\rightarrow \mathbb{A}^{k-1}}, yielding {\text{an}(f):\text{an}(X)\rightarrow \mathbb{C}^{k-1}}, by local compactness we can find an (analytic/Euclidean) compact neighborhood of a point on {\mathbb{C}^k} whose fiber is a nonsingular curve. The preimage of this is compact, as a closed subset of {\text{an}(X)}, since every Proj construction yields a compact manifold and closed subsets of compact spaces are compact.

Then {\text{an}(f)} restricted to this preimage is a surjective map of a smooth manifold onto its image, which is clearly a submersion by checking algebraic tangent spaces. It is also a (analytic) proper map by construction. Hence by We will use Ehresmann’s theorem, it’s a locally trivial fibration, so every fiber is the same topologically, hence has the same genus. By identifying {\mathbb{P}^{k-1}} with {\mathbb{C}^\times}-scaling orbits in {\mathbb{C}[X,Y,Z]_d^{\text{nonsing}}}, we have our desired result.

Finally, it’s a snap to show that {\mathbb{C}[X,Y,Z]_d^{\text{nonsing}}} is connected. Indeed, inside the vector space {\mathbb{C}[X,Y,Z]_d}, it’s the complement of {\mathbb{C}[X,Y,Z]_d^{\text{sing}}}. A polynomial {P(X,Y,Z)} defines a singular curve if and only if there is a common zero of {P} with its partial derivatives in {X}, {Y}, and {Z} simultaneously. Since the partial derivative is just a linear transformation of the coefficients (hence of the vector space), by the theory of resultants, this corresponds to a codimension {\ge 2} surface in {\mathbb{C}[X,Y,Z]_d} considered as an affine space in the coefficients.

Thus, {\mathbb{C}[X,Y,Z]_d^{\text{nonsing}}} is indeed connected as the complement of a codimension {\ge 2}, so genus is constant. The result follows.

One reason I like this proof a lot is that it uses proto-versions of two big ideas in algebraic geometry. First, family arguments, where we turn an easy case into a hard one while keeping something invariant, hinting towards cycle classes, flat families, and deformation theory. Indeed, the second step is trivial with the theory of flat families, just a specialization of the fact that the Hilbert polynomial is constant. Second, working with a geometric space which parametrizes the objects we’re talking about (a proto-moduli space).