# Singularities of Pfaffian hypersurfaces

Let $X\subset\mathbb{P}^4$ be an hypersurface of degree six given by the Pfaffian of a $6\times 6$ matrix $M$ whose entries are quadratic forms in the homogeneous coordinates of $\mathbb{P}^4$. I am interested in the singular locus of $X$.

Is it true that $Sing(X)$ is the curve defined by the $4\times 4$ sub-Pfaffians of $M$?

In this case, is it known that $X$ has ordinary singularities along $Sing(X)$, and what is the multiplicity of $Sing(X)$ for $X$?

What about the same questions for the following $$\left( \begin{array}{cccccc} 0 & x_1^2 & x_2^2 & x_3^2 & x_4^2 & x_5^2\\ -x_1^{2} & 0 & x_1x_3 & x_2x_4 & x_3x_5 & -x_4x_5\\ -x_2^{2} & -x_1x_3 & 0 & x_1x_5 & -x_2x_5 & -x_3x_4\\ -x_3^{2} & -x_2x_4 & -x_1x_5 & 0 & -x_1x_4 & -x_2x_3\\ -x_4^{2} & -x_3x_5 & x_2x_5 & x_1x_4 & 0 & -x_1x_2\\ -x_5^{2} & x_4x_5 & x_3x_4 & x_2x_3 & x_1x_2 & 0 \end{array} \right)$$ particular Pfaffian?

• A Macaulay 2 calculation with a generic skew-symmetric 6 x 6 matrix $N$ of quadratic forms in the homogeneous coordinates of $\mathbb{P}^{4}$ shows that the singular locus of the associated sextic Pfaffian threefold is empty. May 23 '15 at 1:57
• Indeed, an easy computation shows that in the space of skew-symmetric $6\times 6$ matrices, those which are of rank $\leq 2$ form a subvariety of codimension 6. This implies that if the matrix defining $X$ is general, $X$ is smooth.
– abx
May 23 '15 at 2:30
• Thanks a lot to both of you. I added a new piece in my question asking for a particular Pfaffian as well.
– gxg
May 23 '15 at 12:23

As Sasha proved the general Pfaffian is smooth. On the other hand the special Pfaffian $X$ you wrote is an irreducible hypersurface of degree $6$ in $\mathbb{P}^4$ singular along a smooth curve $C$ of degree $20$ and genus $26$. Indeed $X$ has ordinary double points along $C$.

Your Pfaffian is indeed birational to the moduli space of $(1,11)$-polarized abelian surfaces, endowed with a symmetric theta structure and an odd theta characteristic. In this sense you can find a detailed description of this hypersurface in Lemma $2.1$ of this paper:

M. GROSS, S. POPESCU, The Moduli Space of (1,11)-Polarized Abelian Surfaces is Unirational, Compositio Mathematica 126: 1-23, 2001.

• So presumably that genus 26 curve is isomorphic with the modular curve $X(11)$, right? May 24 '15 at 21:57
• That's it. Exactly!
– F_L
May 24 '15 at 22:49

Let me add a bit more details to abx comment. Consider the projectivization $P^{14}$ of the space of all $6 \times 6$ skew-symmetric matrices. It contains the Pfaffian cubic hypersurface, which parameterizes matrices of rank 4, and its singular locus is the locus of matrices of rank 2, i.e. $Gr(2,6)$. Indeed, its codimension in the ambient space is $14-8 = 6$ and in the Pfaffian hypersurface its codimension is $5$. Now a skew-symmetric matrix with quadratic forms as entries gives a map from $P^4$ to $P^{14}$ of degree 2, and what you are interested in is the intersection of its image with the Pfaffian hypersurface. So, for generic choice of a map by Bertini Theorem the result is smooth.

• Thanks a lot. I added a new piece in my question asking for a particular Pfaffian as well.
– gxg
May 23 '15 at 12:23