On the automorphisms of Mukai varieties

Abstract

Mukai varieties are Fano varieties of Picard number one and coindex three. In genus seven to ten they are linear sections of some special homogeneous varieties. We describe the generic automorphism groups of these varieties. When they are expected to be trivial for dimensional reasons, we show they are indeed trivial, up to three interesting and unexpected exceptions in genera 7, 8, 9, and codimension 4, 3, 2 respectively. We conclude in particular that a generic prime Fano threefold of genus g has no automorphisms for $7\leqslant g\leqslant 10$. In the Appendix by Y. Prokhorov, the latter statement is extended to $g=12$.

Appendix A: Automorphism groups of prime Fano threefolds of genus twelve

by Yuri Prokhorov ¹

Theorem A.1

The automorphism group of a general (in the moduli sense) prime Fano threefold of genus 12 is trivial.

Proof

For a prime Fano threefold X we denote by $\mathrm {F}_1(X)$ the Hilbert scheme of lines, i.e. curves in X with Hilbert polynomial $h_1(t) = t + 1$. It is known that $\mathrm {F}_1(X)$ is of pure dimension 1 (see e.g. [15]).

Claim A.1.1

For any prime Fano threefold $X=X_{22}\subseteq \mathbf {P}^{13}$ the natural homomorphism

$$\begin{aligned} \Psi :\mathrm {Aut}(X)\rightarrow \mathrm {Aut}(\mathrm {F}_1(X)) \end{aligned}$$

is injective.

Proof

Assume that $\Psi$ is not injective. Take a non-trivial element $\varphi \in {\mathrm{Ker}}(\Psi )$. Thus $\varphi$ acts trivially on $\mathrm {F}_1(X)$. Fix a line $l\subseteq X$. Apply the double projection [12, Theorem 4.3.3, Theorem 4.3.7] from l. This is the birational map

$$\begin{aligned} \theta : X \dashrightarrow Y \subseteq {\mathbf {P}}^6 \end{aligned}$$

given by the linear system $|-K_X-2l|$ of hyperplane sections which are singular along l. Here $Y=Y_5\subseteq {\mathbf {P}}^6$ is a smooth quintic del Pezzo threefold and the $\theta$-exceptional divisor is contracted to a rational normal quintic curve $\Gamma \subseteq Y\subseteq {\mathbf {P}}^6$. The map $\theta$ induces a $\varphi$-action on $Y\subseteq {\mathbf {P}}^6$ by a projective transformation and the curve $\Gamma$ is $\varphi$-invariant. A general line $l'\subseteq X$ is mapped to a line $m'\subseteq Y$ meeting $\Gamma$ at one point. The set of lines in Y passing through any point $y\in Y$ is finite (see e.g. [15, Corollary 5.1.5]). Since $\dim \mathrm {F}_1(X)=1$, the automorphism $\varphi$ acts trivially on $\Gamma$. Thus the fixed point locus $Y^\varphi$ contains the hyperplane section $S:=Y\cap \langle \Gamma \rangle$. Recall that $H^2(Y,{\mathbf {Z}})\simeq {\text {Pic}}(Y)\simeq {\mathbf {Z}}$ and $H^3(Y,{\mathbf {Z}})=0$ (see e.g. [12, § 12.2]). Hence the induced action of $\varphi$ on $H^q(Y,{\mathbf {C}})$ is trivial for any q.

Assume that $\varphi$ is an element of finite order. Then its fixed point locus $Y^\varphi$ is smooth. Hence $Y^\varphi$ contains no one dimensional components (because $\uprho (Y)=1$) and S is a smooth del Pezzo surface. In particular, $\upchi _{\mathrm {top}}(Y^\varphi )\geqslant 7$. This contradicts the topological Lefschetz fixed point formula [10, Prop. 5.3.11]:

$$\begin{aligned} \upchi _{\mathrm {top}}(Y^\varphi )=\sum _q (-1)^q {\mathrm{Tr}} \big (\varphi ^*|_{H^q(Y,{\mathbf {C}})} \big ) =\sum _q (-1)^qh^q(Y,{\mathbf {C}}) =\upchi _{\mathrm {top}}(Y) =4. \end{aligned}$$

Therefore $\varphi$ is an element of infinite order. Any line on Y meets S hence $\varphi ^m$ acts trivially on $\mathrm {F}_1(Y)$ for some m (in fact, $m\leqslant 3$). Recall that there are exactly three lines in Y passing through a general point $y\in Y$. This implies that $\varphi ^m$ acts trivially on Y, a contradiction. $\square$

Now we use Mukai’s realization of $X=X_{22}\subseteq \mathbf {P}^{13}$ as VSP(C, 6) where C is a plane quartic [22]. Take a general quartic $C\subseteq \mathbf {P}^2$ and let $X=VSP(C,6)$. Then the curve $\mathrm {F}_1(X)$ is also a smooth plane quartic $F_C$ which is covariant of C [28, Theorem 6.1]. The curve $\mathrm {F}_1(X)=F_C$ has a natural (3, 3)-correspondence of intersecting lines which defines an even theta characteristic $\Theta$ on $F_C$. There is a map $C \longmapsto (F_C,\Theta )$ of the corresponding moduli spaces which is called Scorza map. It is birational [9, Theorem 7.8]. In particular, this implies that the curve $F_C$ is general in the moduli space of plane quartics. Since the plane quartic $F_C$ is general, we have $\mathrm {Aut}(F_C)=\{1\}$. Hence $\mathrm {Aut}(X)=\{1\}$ for $X=VSP(C,6)$ by Claim A.1.1. $\square$

Remark A.2

Note that in contrast with the cases $g\leqslant 10$ the automorphism group of a prime Fano threefold of genus $g=12$ can be infinite. We refer to [15, 16, 26] for description of infinite groups of automorphisms.