Assume \(\operatorname{char}k\ne2\). Let \(a_1, \ldots, a_r\), \(r \geqslant 5\), be distinct elements of \(k\), and let \(C\) be the curve in \(\PP^2\) given by the (affine) equation \(y^2=\prod_{i=1}^r\left(x-a_i\right)\). Show that the point \(P\) at infinity on the \(y\)-axis is a singular point. Compute \(\delta_P\) and \(g(\tilde{Y})\), where \(\tilde{Y}\) is the normalization of \(Y\). Show in this way that one obtains hyperelliptic curves of every genus \(g \geqslant 2\).
(Erratum)
The characteristic hypothesis is necessary for this model. In characteristic \(2\), the extension defined by \[ y^2=\prod_i(x-a_i) \] is purely inseparable over \(k(x)\), so it is not the separable degree-two hyperelliptic cover used in the conclusion. The proof below treats the intended case \(\operatorname{char}k\ne2\).
Solution
Put \[ f(x)=\prod_{i=1}^r(x-a_i). \] The projective closure \(Y\subseteq\PP^2\) has homogeneous equation \[ F(x,y,z) = y^2z^{r-2} - \prod_{i=1}^r(x-a_i z) =0. \]
1. The only point of \(Y\) on the line at infinity \(z=0\) is\[ P=[0:1:0]. \]
Setting \(z=0\) in the homogeneous equation gives \[ -x^r=0. \] Thus \(x=0\), and the unique projective point with \(z=x=0\) is \[ [0:1:0]. \]
2. The point \(P\) is singular on \(Y\).
Since \(r\ge5\), each first partial derivative of \[ F=y^2z^{r-2}-\prod_i(x-a_i z) \] vanishes at \([0:1:0]\).
Indeed, \[ F_y=2yz^{r-2}, \] which vanishes at \(P\), while every monomial of \(F_x\) has total degree \(r-1\) in \(x,z\) and hence vanishes at \(x=z=0\). Likewise the term \[ (r-2)y^2z^{r-3} \] in \(F_z\) vanishes because \(r-3\ge2\), and all remaining terms again contain a positive power of \(x\) or \(z\). Thus the Jacobian criterion makes \(P\) singular.
3. The affine part of \(Y\) is nonsingular.
On \(z=1\) the equation is \[ y^2-f(x)=0. \] A singular affine point would satisfy \[ 2y=0, \qquad f'(x)=0, \qquad y^2=f(x). \] Since \(\operatorname{char}k\ne2\), the first equation gives \(y=0\), so \(f(x)=0\). The roots \(a_i\) are distinct, hence \(f\) is squarefree and \[ f'(a_i)\ne0. \] This is impossible. Therefore \(P\) is the only singular point of the projective plane curve.
4. In the affine chart \(y=1\) at \(P\), the local equation has multiplicity\[ \boxed{r-2}. \]
With local coordinates \((x,z)\) at \(P\), the equation becomes \[ z^{r-2} - \prod_{i=1}^r(x-a_i z) =0. \] The second term is homogeneous of total degree \(r\), whereas the first has degree \(r-2\). Hence the lowest nonzero homogeneous term is \(z^{r-2}\), so the multiplicity at the origin is \(r-2\).
5. Blow up \(P\) and use the chart\[ x=u, \qquad z=uv. \]The strict transform has local equation\[ v^{r-2}=u^2\prod_{i=1}^r(1-a_i v). \]
Substitution in the equation of step 4 gives \[ u^{r-2}v^{r-2} - u^r\prod_i(1-a_i v) =0. \] Removing the exceptional factor \(u^{r-2}\) gives the strict-transform equation \[ v^{r-2} - u^2\prod_i(1-a_i v) =0. \] It meets the exceptional divisor \(u=0\) only at \(v=0\), corresponding to the unique tangent direction \(z=0\) at \(P\).
6. The unique infinitely near singularity after the first blowup isanalytically of type\[ U^2=v^{r-2}, \]and has delta invariant\[ \boxed{\left\lfloor\frac{r-2}{2}\right\rfloor}. \]
The factor \[ h(v)=\prod_i(1-a_i v) \] is a unit in \(k[[v]]\) with \(h(0)=1\). Since \(2\) is invertible in \(k\), this unit has a square root in \(k[[v]]\). Replacing \[ U=u\sqrt{h(v)} \] turns the completed local equation of step 5 into \[ U^2=v^{r-2}. \] This is the \(A_{r-3}\) plane-curve singularity. The standard formula on D-CRVPLSING gives \[ \delta(A_n)=\left\lfloor\frac{n+1}{2}\right\rfloor, \] so here \[ \delta=\left\lfloor\frac{r-2}{2}\right\rfloor. \]
7. The delta invariant of the original point at infinity is\[ \boxed{ \delta_P = \binom{r-2}{2} + \left\lfloor\frac{r-2}{2}\right\rfloor.} \]
For a plane curve singularity, the genus-drop formula under successive point blowups is \[ \delta_P = \sum_Q\binom{m_Q}{2}, \] where \(Q\) runs through the singular point and its infinitely near singular points with multiplicities \(m_Q\) D-CRVPLSING.
4. After that firstblowup, the residual singularity is the \(A_{r-3}\) singularity of step 6,whose complete remaining contribution to delta is\(\lfloor(r-2)/2\rfloor\). Hence the displayed formula.
8. Equivalently,\[ \delta_P = \begin{cases} \dfrac{(r-2)^2}{2},&r\text{ even},\\[6pt] \dfrac{(r-1)(r-3)}{2},&r\text{ odd}. \end{cases} \]
If \(r=2m\), then \(r-2=2m-2\) is even and step 7 gives \[ \binom{2m-2}{2}+m-1 = 2(m-1)^2 = \frac{(r-2)^2}{2}. \] If \(r=2m+1\), then \(r-2=2m-1\) is odd and step 7 gives \[ \binom{2m-1}{2}+m-1 = 2m(m-1) = \frac{(r-1)(r-3)}2. \]
9. The normalization \(\widetilde Y\) has genus\[ \boxed{ g(\widetilde Y) = \left\lfloor\frac{r-1}{2}\right\rfloor.} \]
The plane curve \(Y\) has degree \(r\), hence arithmetic genus \[ p_a(Y)=\frac{(r-1)(r-2)}2. \] By step 3, \(P\) is its only singular point. Therefore \[ g(\widetilde Y) = p_a(Y)-\delta_P. \] Using step 8 gives \[ g(\widetilde Y) = \begin{cases} (r-2)/2,&r\text{ even},\\[4pt] (r-1)/2,&r\text{ odd}, \end{cases} \] which is exactly \(\lfloor(r-1)/2\rfloor\).
10. The rational function \(x\) defines a finite morphism\[ \widetilde Y\longrightarrow\PP^1 \]of degree \(2\).
The function field of \(Y\), and hence of its normalization, is \[ k(\widetilde Y)=k(x,y), \qquad y^2=f(x). \] Because \(f\) is squarefree, it is not a square in \(k(x)\). Since \(\operatorname{char}k\ne2\), the polynomial \[ T^2-f(x) \] is separable and irreducible over \(k(x)\). Thus \[ [k(\widetilde Y):k(x)]=2. \] The corresponding nonconstant morphism to the nonsingular projective curve \(\PP^1\) is finite and has degree \(2\).
11. Hyperelliptic curves occur in every genus \(g\ge2\).
Given \(g\ge2\), choose either \[ r=2g+1 \qquad\text{or}\qquad r=2g+2 \] distinct scalars \(a_i\in k\). Step 9 gives \[ g(\widetilde Y)=g, \] and step 10 gives a degree-two morphism \[ \widetilde Y\to\PP^1. \] Thus \(\widetilde Y\) is a hyperelliptic curve of genus \(g\).
12. Q.E.D.
Steps <1>1–<1>8 resolve the unique point at infinity and compute its delta invariant, step 9 computes the genus of the normalization, and steps <1>10–<1>11 give hyperelliptic curves of every genus at least two.