Return to Answer

1. To simplify notation, denote here by $\omega^{(n)}$ the basis $\{\omega^{(n)}_k\}$ with $k=i_1,\ldots,i_{N_n}\in\{1,\ldots,M_{n}\}$. $$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}} {\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$ are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight $$ d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ . $$ Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$, a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case of genus 4 (presumably here should also appear some interesting Number Theoretical structures).

2. For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in\{1,\ldots,M_n\}$ one has $$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x) =0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space.

3. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$

4. For $g=4$ the discriminant of the quadrics is proportional to the square root of $\chi_{68}$, the $g=4$ Thetanullwerte $$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$ with $d$ a constant. A key step here is the following lemma. Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$ or a non-trigonal surface of $g=5$. Then, the canonical model of $C$ is contained in a quadric of rank $3$ if and only if $\prod_{\delta\hbox{ even}}\theta[\delta]=0$. Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight $34$) only when restricted to ${\cal I}_4$.

5. The $g=4$, $n=2$ Mumford form is $$\mu_{4,2}=\pm{1\over c S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge \widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over (\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ , $$ with $c$ a constant.

1. To simplify notation, denote here by $\omega^{(n)}$ the basis $\{\omega^{(n)}_k\}$ with $k=i_1,\ldots,i_{N_n}\in\{1,\ldots,M_{n}\}$. $$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}} {\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$ are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight $$ d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ . $$ Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$, a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case of genus 4 (presumably here should also appear some interesting Number Theoretical structures).

2. For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in\{1,\ldots,M_n\}$ one has $$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x) =0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space.

3. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$

4. For $g=4$ the discriminant of the quadrics is proportional to the square root of $\chi_{68}$, the $g=4$ Thetanullwerte $$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$ with $d$ a constant. A key step here is the following lemma. Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$ or a non-trigonal surface of $g=5$. Then, the canonical model of $C$ is contained in a quadric of rank $3$ if and only if $\prod_{\delta\hbox{ even}}\theta[\delta]=0$.

5. The $g=4$, $n=2$ Mumford form is $$\mu_{4,2}=\pm{1\over c S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge \widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over (\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ , $$ with $c$ a constant.

Note that $S_{4ij}(Z)$ trasforms with affine terms proportional to $F_4$, so that it is a vector-valued modular form only when $F_4=0$, that is in the Jacobian. This motivates the name vector-valued Teichmueller modular forms. Note that also the square root of $\chi_{68}(\tau)$ exists only in ${\cal I}_4$. It follows that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight $34$) only when restricted to ${\cal I}_4$.

This clearly shows that the vector-valued Teichmueller modular forms $[i_{N_n+1},\ldots,i_{M_n}|\tau]$ are deeply related to the geometry of the Jacobian and to the Schottky's problem. The structure of the vector-valued Teichmueller modular at any $g$, generated by Mumford's forms, and their properties, such as the one of generating canonical curves, that is Petri's relations, strongly support Mumford's suggestion that Petri's relations are fundamental and should have basic applications: see pg.241, D. Mumford, The Red Book of Varieties and Schemes, Springer Lecture Notes in Math. 1358, 1999. Actually, it seems Mumford was right, one has just to use his forms.

A suggestion for the literature: the papers by John Fay are excellent and not very well-known as they should. The one in the Memoirs of the AMS is a masterpiece.

1. To simplify notation, denote here by $\omega^{(n)}$ the basis $\{\omega^{(n)}_k\}$ with $k=i_1,\ldots,i_{N_n}\in\{1,\ldots,M_{n}\}$. $$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}} {\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$ are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight $$ d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ . $$ Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$, a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$ (presumably here should also appear some interesting Number Theoretical structures).

2. For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in\{1,\ldots,M_n\}$ one has $$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x) =0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space.

3. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$

4. For $g=4$ the discriminant of the quadrics is proportional to the square root of $\chi_{68}$, the $g=4$ Thetanullwerte $$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$ with $d$ a constant. A key step here is the following lemma. Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$ or a non-trigonal surface of $g=5$. Then, the canonical model of $C$ is contained in a quadric of rank $3$ if and only if $\prod_{\delta\hbox{ even}}\theta[\delta]=0$. Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight $34$) only when restricted to ${\cal I}_4$.

5. The $g=4$, $n=2$ Mumford form is $$\mu_{4,2}=\pm{1\over c S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge \widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over (\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ , $$ with $c$ a constant.

1. To simplify notation, denote here by $\omega^{(n)}$ the basis $\{\omega^{(n)}_k\}$ with $k=i_1,\ldots,i_{N_n}\in\{1,\ldots,M_{n}\}$. $$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}} {\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$ are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight $$ d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ . $$ Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$, a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case of genus 4 (presumably here should also appear some interesting Number Theoretical structures).

2. For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in\{1,\ldots,M_n\}$ one has $$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x) =0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space.

3. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$

4. For $g=4$ the discriminant of the quadrics is proportional to the square root of $\chi_{68}$, the $g=4$ Thetanullwerte $$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$ with $d$ a constant. A key step here is the following lemma. Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$ or a non-trigonal surface of $g=5$. Then, the canonical model of $C$ is contained in a quadric of rank $3$ if and only if $\prod_{\delta\hbox{ even}}\theta[\delta]=0$. Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight $34$) only when restricted to ${\cal I}_4$.

5. The $g=4$, $n=2$ Mumford form is $$\mu_{4,2}=\pm{1\over c S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge \widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over (\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ , $$ with $c$ a constant.

Let ${\frak H}_g:=\{Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0\}$, be the Siegel upper half-space. Let $\{\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g\}$ be a symplectic basis of $H_1(C,{\Bbb Z})$. Denote by $\{\omega_i\}_{1\le i\le g}$ the basis of $H^0(K_C)$ satisfying the standard normalization condition $\oint_{\alpha_i}\omega_j=\delta_{ij}$, and by $\tau_{ij}:=\oint_{\beta_i}\omega_j$ the Riemann period matrix, $i,j=1,\ldots,g$. Denote by ${\cal I}_g$ the closure of the locus of Riemann period matrices in ${\frak H}_g$ and by ${\cal M}_g$ the moduli space of curves of genus $g$. Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $n=2$, $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.

Let $\{\phi^n_i\}_{1\le i\le N_n}$ be a basis of $H^0(K_C^n)$, $n\geq2$. The Mumford form is, up to a universal constant $$ \mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over \kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over (\omega_1\wedge\cdots\wedge\omega_g)^{c_n}} \ . $$ where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose the natural basis ${\rm Sym}^2 H^0(K_C)$ for $H^0(K_C^2)$, and for $g=2$ gets $${\kappa[\omega]^{9}\over \kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$ whereas for $g=3$ $${\kappa[\omega]^{9}\over \kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$ For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise.

1. To simplify notation, denote here by $\omega^{(n)}$ the basis $\{\omega^{(n)}_k\}$ with $k=i_1,\ldots,i_{N_n}\in\{1,\ldots,M_{n}\}$. $$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}} {\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$ are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight $$ d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ . $$ Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$. This may be seen as a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$ (presumably here should also appear some interesting Number Theoretical structures).

2. For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in\{1,\ldots,M_n\}$ one has $$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x) =0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space.

3. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $S_{4ij}(\tau)$, where $$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$

4. For $g=4$ the discriminant of the quadrics is proportional to the square root of $\chi_{68}$, the $g=4$ Thetanullwerte $$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$ with $d$ a constant. A key step here is the following lemma. Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$ or a non-trigonal surface of $g=5$. Then, the canonical model of $C$ is contained in a quadric of rank $3$ if and only if $\prod_{\delta\hbox{ even}}\theta[\delta]=0$. Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight $34$) only when restricted to ${\cal I}_4$.

5. The $g=4$, $n=2$ Mumford form is $$\mu_{4,2}=\pm{1\over c S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge \widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over (\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ , $$ with $c$ a constant.

Let ${\frak H}_g:=\{Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0\}$, be the Siegel upper half-space. Let $\{\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g\}$ be a symplectic basis of $H_1(C,{\Bbb Z})$. Denote by $\omega_1,\ldots,\omega_g$ the basis of $H^0(K_C)$ satisfying the standard normalization condition $\oint_{\alpha_i}\omega_j=\delta_{ij}$, and by $\tau_{ij}:=\oint_{\beta_i}\omega_j$ the Riemann period matrix, $i,j=1,\ldots,g$. Denote by ${\cal I}_g$ the closure of the locus of Riemann period matrices in ${\frak H}_g$ and by ${\cal M}_g$ the moduli space of curves of genus $g$. Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $n=2$, $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.

Let $\{\phi^n_i\}_{1\le i\le N_n}$ be a basis of $H^0(K_C^n)$, $n\geq2$. The Mumford form is, up to a universal constant $$ \mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over \kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over (\omega_1\wedge\cdots\wedge\omega_g)^{c_n}} \ , $$ where $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose the natural basis ${\rm Sym}^2 H^0(K_C)$ for $H^0(K_C^2)$, and for $g=2$ gets $${\kappa[\omega]^{9}\over \kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$ whereas for $g=3$ $${\kappa[\omega]^{9}\over \kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$ For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise.

1. To simplify notation, denote here by $\omega^{(n)}$ the basis $\{\omega^{(n)}_k\}$ with $k=i_1,\ldots,i_{N_n}\in\{1,\ldots,M_{n}\}$. $$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}} {\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$ are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight $$ d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ . $$ Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$, a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$ (presumably here should also appear some interesting Number Theoretical structures).

2. For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in\{1,\ldots,M_n\}$ one has $$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x) =0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space.

3. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$

4. For $g=4$ the discriminant of the quadrics is proportional to the square root of $\chi_{68}$, the $g=4$ Thetanullwerte $$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$ with $d$ a constant. A key step here is the following lemma. Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$ or a non-trigonal surface of $g=5$. Then, the canonical model of $C$ is contained in a quadric of rank $3$ if and only if $\prod_{\delta\hbox{ even}}\theta[\delta]=0$. Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight $34$) only when restricted to ${\cal I}_4$.

5. The $g=4$, $n=2$ Mumford form is $$\mu_{4,2}=\pm{1\over c S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge \widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over (\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ , $$ with $c$ a constant.