The hyperelliptic curves of genus 4 form an irreducible family of dimension 7. The nonhyperelliptic ones form an irreducible family of dimension 9. The subset of those having only one \(g_3^1\) is an irreducible family of dimension 8.
Hint: Use (5.2.2) to count how many complete intersections \(Q \intersect F_3\) there are.
Solution
We work over an algebraically closed field of characteristic \(0\), as in the surrounding discussion.
1. Hyperelliptic curves of genus \(4\) are parametrized by unorderedconfigurations of \(10\) distinct points of \(\PP^1\), modulo \(\PGL_2\).
A hyperelliptic genus-\(4\) curve is a double cover \[ f:X\longrightarrow\PP^1. \] Riemann–Hurwitz gives \[ 2g-2 = 2(-2)+\deg R_f, \] so for \(g=4\) the branch divisor has degree \[ 2g+2=10. \] In characteristic \(0\) these are \(10\) distinct points.
Conversely, an unordered reduced divisor of degree \(10\) on \(\PP^1\) determines a double cover branched exactly there, unique up to isomorphism: one takes the square root of the corresponding section of \(\OO_{\PP^1}(10)\), and scalar choices differ by a square over the algebraically closed field. Changing coordinates on \(\PP^1\) changes the cover only by isomorphism. Hence the asserted parametrization.
2. The hyperelliptic genus-\(4\) locus is irreducible of dimension\[ \boxed{7}. \]
The open configuration space of \(10\) ordered distinct points of \(\PP^1\) is irreducible of dimension \(10\). Quotienting by the finite symmetric group does not change dimension or irreducibility, so unordered branch divisors also form an irreducible \(10\)-dimensional family.
The group \[ \PGL_2 \] has dimension \(3\). The stabilizer of a reduced set of at least three points is finite, since an automorphism of \(\PP^1\) fixing three points is the identity. Thus the orbit dimension is \(3\), and the family of isomorphism classes has dimension \[ 10-3=7. \] Its image is irreducible because the parameter space is irreducible.
3. A nonhyperelliptic genus-\(4\) curve has canonical model\[ \boxed{C=Q\cap F_3\subset\PP^3,} \]where \(Q\) is its unique quadric, and the cubic \(F_3\) is unique moduloaddition of \(Q\) times a linear form.
This is the canonical genus-\(4\) description recalled in (5.2.2). The canonical system embeds a nonhyperelliptic genus-\(4\) curve as a degree-\(6\) curve in \(\PP^3\). It lies on a unique quadric \(Q\) and on a cubic not divisible by \(Q\). Since \[ \deg(Q\cap F_3)=2\cdot3=6=\deg C, \] the complete intersection is exactly \(C\).
The degree-\(3\) part of the ideal of a \((2,3)\) complete intersection is \[ kF_3+Q\cdot H^0(\PP^3,\OO(1)). \] Thus replacing \(F_3\) by \[ cF_3+QL, \qquad c\in k^*, \quad L\in H^0(\PP^3,\OO(1)), \] does not change \(C\), and these are precisely the redundancies.
4. Canonically embedded smooth \((2,3)\) complete intersections form anirreducible parameter space of dimension\[ \boxed{24}. \]
Quadrics in \(\PP^3\) form \[ \PP H^0(\PP^3,\OO(2)) \cong \PP^9, \] because \[ h^0(\PP^3,\OO(2))=10. \] For a fixed nonzero quadric \(Q\), the cubic is determined by a line in \[ H^0(\PP^3,\OO(3)) \big/ QH^0(\PP^3,\OO(1)). \] The two vector-space dimensions are \[ 20 \qquad\text{and}\qquad 4, \] so this quotient has dimension \(16\) and its projectivization is \(\PP^{15}\).
As \(Q\) varies, these quotients form a rank-\(16\) vector bundle over \(\PP^9\): multiplication by the tautological quadric gives an injective bundle map \[ \OO_{\PP^9}(-1) \tensor H^0(\PP^3,\OO(1)) \longrightarrow H^0(\PP^3,\OO(3))\tensor\OO_{\PP^9}. \] Its projectivized cokernel is therefore irreducible of dimension \[ 9+15=24. \] The condition that \(Q\cap F_3\) be a smooth curve is open and nonempty, so the smooth-complete-intersection locus is still irreducible of dimension \(24\).
5. Nonhyperelliptic genus-\(4\) curves form an irreducible family ofdimension\[ \boxed{9}. \]
Two canonical models represent isomorphic curves exactly when they differ by a projective coordinate change in \(\PP^3\). The group \[ \PGL_4 \] has dimension \[ 4^2-1=15. \] By Exercise IV.5.2, the stabilizer of a smooth genus-\(4\) canonical curve is finite. Thus every orbit has dimension \(15\).
Step 4 gives an irreducible \(24\)-dimensional parameter space dominating the nonhyperelliptic genus-\(4\) family, so the latter is irreducible and has dimension \[ 24-15=9. \]
6. Let \(C=Q\cap F_3\) be a nonhyperelliptic canonical genus-\(4\) curve.Its \(g^1_3\)’s are in bijection with the rulings of \(Q\).
First, every ruling of \(Q\) gives a \(g^1_3\). Indeed, a ruling line \[ \ell\subset Q \] meets the cubic \(F_3\) in a divisor of degree \(3\) on \(C\), and the one-parameter family of ruling lines gives a base-point-free pencil of such divisors.
Conversely, let \(|D|\) be a \(g^1_3\) on \(C\). Thus \[ \deg D=3, \qquad \ell(D)=2. \] Riemann–Roch gives \[ \ell(K-D) = \ell(D)-\deg D-1+g = 2-3-1+4 =2. \] In the canonical embedding, hyperplanes containing \(D\) therefore form a pencil. Their common intersection is a line \[ \ell_D\subset\PP^3 \] containing the degree-\(3\) divisor \(D\).
The restriction of the quadratic equation of \(Q\) to \(\ell_D\) has degree \(2\), but it vanishes on the length-\(3\) divisor \(D\). Hence it vanishes identically: \[ \ell_D\subset Q. \] As \(D\) varies in its pencil, these lines vary in a ruling of \(Q\). The ruling recovers \(|D|\) by intersection with \(C\), proving the bijection.
7. The curve \(C\) has exactly two \(g^1_3\)’s when \(Q\) is smooth and exactlyone when \(Q\) has rank \(3\).
A smooth quadric surface is \[ Q\cong\PP^1\times\PP^1 \] and has exactly two rulings. Their induced pencils on \(C\) are distinct: if the two projection maps on \(C\) differed only by an automorphism of \(\PP^1\), then the image of \(C\) in \(\PP^1\times\PP^1\) would lie in the graph of that automorphism, a curve of bidegree \((1,1)\), whereas \[ C\sim(3,3). \] Thus step 6 gives exactly two \(g^1_3\)’s.
An irreducible singular quadric containing a nondegenerate smooth canonical curve has rank \(3\), hence is a quadric cone. Such a cone has exactly one ruling by lines, so step 6 gives exactly one \(g^1_3\).
8. The rank-\(3\) quadrics form an irreducible locally closed subset of\(\PP^9\) of dimension\[ \boxed{8}. \]
A quadric in \(\PP^3\) is represented by a symmetric \(4\times4\) matrix, up to scalar. Every rank-\(3\) quadric is projectively equivalent to \[ x_0^2+x_1^2+x_2^2=0. \] Hence the rank-\(3\) locus is the image of the irreducible group \(\PGL_4\) acting on this quadric, and its closure is irreducible.
That closure is exactly the locus of quadrics of rank at most \(3\): matrix rank can only drop under specialization, and every lower-rank diagonal quadric is a limit of rank-\(3\) diagonal quadrics. This locus is also the zero set of the determinant, so it is a proper hypersurface in \(\PP^9\) and therefore has dimension \(8\).
The locus of rank at most \(2\) is a proper closed subset of this hypersurface. Therefore its complement, the rank-\(3\) locus, is irreducible and still has dimension \(8\).
9. Nonhyperelliptic genus-\(4\) curves having exactly one \(g^1_3\) form anirreducible family of dimension\[ \boxed{8}. \]
By step 7, this is precisely the locus whose unique canonical quadric has rank \(3\). Restrict the parameter construction of step 4 to the irreducible \(8\)-dimensional rank-\(3\) quadric locus. The cubic fibre remains \(\PP^{15}\), so the resulting parameter space has dimension \[ 8+15=23. \] The condition that the complete intersection be smooth is open and nonempty, so the smooth locus is irreducible of the same dimension.
Again quotienting by projective coordinate changes removes \[ \dim\PGL_4=15 \] dimensions, and Exercise IV.5.2 gives finite stabilizers. Hence the family of isomorphism classes is irreducible of dimension \[ 23-15=8. \]
10. Q.E.D.
Step 2 proves the hyperelliptic assertion, step 5 proves the nonhyperelliptic assertion, and step 9 proves the assertion about curves with a unique \(g^1_3\).