Rational differential forms on the variety of flexes of plane cubics

TL;DR

This paper proves that for any positive integer d, the variety of flexes of plane cubics admits no nonzero regular differential d-forms, revealing a specific geometric property of these varieties.

Contribution

It establishes the nonexistence of nonzero regular differential forms of any positive degree on the variety of flexes of plane cubics, a new result in algebraic geometry.

Findings

No nonzero regular differential d-forms exist for any positive integer d.

The result applies to all smooth irreducible projective varieties birationally equivalent to the variety of flexes.

This enhances understanding of the geometric structure of the variety of flexes of plane cubics.

Abstract

We prove that for every positive integer $d$, there are no nonzero regular differential $d$-forms on every smooth irreducible projective algebraic variety birationally isomorphic to the variety of flexes of plane cubics.

Equations

$$\pi :{\mathrm{S}}^3(V^{\ast })\setminus \{0\}\to {\mathbf{P}}^9$$

$$F:=\sum_{i_{1}+i_{2}+i_{3}=3}a_{i_{1},i_{2},i_{3}}x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}},\quad H:={\rm det}\Big{(}\frac{\partial^{2}F}{\partial x_{i}\partial x_{j}}\Big{)}.$$

$${\mathbf{P}}^2\stackrel{p_2}{\leftarrow }X\stackrel{p_9}{\to }{\mathbf{P}}^9$$

$$H^1(Y,{\mathcal{O}}_Y)=0.$$

$$x_1^3+x_2^3+x_3^3+\lambda x_1{x}_2{x}_3=0,\lambda \in \mathbf{C}.$$

$$q:=(0:-1:1).$$

$$\varphi(\lambda):=\big{(}q,\pi(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+\lambda x_{1}x_{2}x_{3})\big{)}\quad\mbox{for every $\lambda\in{\mathbb{A}}^{1}={\mathbb{C}}$}.$$

$$G\times {\mathbf{A}}^1\to X,(g,\lambda )\mapsto g(\phi (\lambda )),$$

$${\psi }^{\ast }:{\Omega }^d(Y)\to {\Omega }^d({\mathbf{P}}^9)$$

$${\Omega }^d[Y]\hookrightarrow {\Omega }^d[{\mathbf{P}}^9]$$

Full text

Below we use the standard notation from [3], [8].

Consider a three-dimensional complex vector space $V$ and fix a basis $x_{1},x_{2},x_{3}$ of the dual space $V^{*}$. In the space ${\rm S}^{3}(V^{*})$ of degree three forms on the space $V$, all monomials of the form $x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}}$, where $i_{1}+i_{2}+i_{3}=3$, constitute (for their fixed ordering) a basis. Let $\{a_{i_{1},i_{2},i_{3}}\mid i_{1}+i_{2}+i_{3}=3\}$ be the dual to it basis of the dual space $({\rm S}^{3}(V^{*}))^{*}$.

The sets of forms $\{x_{j}\}$ and $\{a_{i_{1},i_{2},i_{3}}\}$ are the projective coordinate systems on the projective spaces ${\mathbf{P}}^{2}:=PV$ and ${\mathbf{P}}^{9}:=P{\rm S}^{3}(V^{*})$ associated with $V$ and ${\rm S}^{3}(V^{*})$ respectively. Let

[TABLE]

be the natural projection. Consider the following forms on the product ${\mathbf{P}}^{2}\times{\mathbf{P}}^{9}$:

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The closed subset $X$ of ${\mathbf{P}}^{2}\times{\mathbf{P}}^{9}$, defined by the system of equations $F=H=0$ was explored in the papers [4], [6], [7]. It is irreducible, nine-dimensional and has singularities (as a matter of fact, $X$ even is non-normal) [7]. Let

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be the natural projections. For every irreducible form $f\in{\rm S}^{3}(V^{*})$ such that the equation $f=0$ defines on ${\mathbf{P}}^{2}$ a smooth cubic $C$, the fiber of the morphism $p_{9}$ over the point $\pi(f)\in{\mathbf{P}}^{9}$ is a set of nine points. The image of this set under the projection $p_{2}$ is exactly the set of all (nine) flexes (inflection points) of the cubic $C$. Therefore, the set of points of $X$ in general position is identified with the set of all pairs $(C,c)$, where $C$ is a smooth cubic on ${\mathbf{P}}^{2}$, and $c$ is its flex. For this reason, $X$ is called (see. [7]) the variety of flexes of plane cubics.

Let $Y$ be a smooth irreducible projective algebraic variety birationally isomorphic to the algebraic variety $X$. The main result (Theorem 4) of the paper [7] is the claim that the irregularity of the variety $Y$ vanishes:

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Below is proved a theorem, a special case of which is the equality (1): Theorem. Maintain the above notation. Then for every positive integer $d$, the following properties hold:

1. (i)

$H^{d}(Y,\mathcal{O}_{Y})=0$; 2. (ii)

*there are no nonzero regular differential $d$-forms on the variety *$Y$​.

Proof. First, we note that properties (i) and (ii) are equivalent. Indeed, according to the Hodge decomposition, $H^{q}(Y,\Omega_{Y}^{p})=\overline{H^{p}(Y,\Omega_{Y}^{q})}$ for all integers $p,q\geqslant 0$ (see [3]). In view of $\Omega^{0}_{Y}={\mathcal{O}}_{Y}$, for $q=d,p=0$, this gives $H^{d}(Y,{\mathcal{O}}_{Y})=\overline{H^{0}(Y,\Omega^{d}_{Y})}$, whence it follows that property (i) is equivalent to the equality $H^{0}(Y,\Omega^{d}_{Y})=0$, which, in turn, is a reformulation of property (ii).

We now prove that property (ii) holds.

The natural actions of the group ${\rm GL}(V)$ on $V$, $V^{*}$, and ${\rm S}^{3}(V^{*})$ induce its actions on ${\mathbf{P}}^{2}$, ${\mathbf{P}}^{9}$ and ${\mathbf{P}}^{2}\times{\mathbf{P}}^{9}$, whose inefficiency kernels contain the group of scalar transformations $Z:=\{\alpha\,{\rm Id}_{V}\mid\alpha\in{\mathbf{C}}^{\times}\}$. Therefore these latter actions induce the actions of the projective group $G:={\rm PGL}(V)={\rm GL}(V)/Z$ on ${\mathbf{P}}^{2}$, ${\mathbf{P}}^{9}$, and ${\mathbf{P}}^{2}\times{\mathbf{P}}^{9}$.

The variety $X$ is invariant under the specified action of the group $G$ on ${\mathbf{P}}^{2}\times{\mathbf{P}}^{9}$. The classical Hesse’s results ([5]; see also [1, pp. 291–299]) yield the following statements:

(a) For every smooth cubic on ${\mathbf{P}}^{2}$, there is a transformation from $G$, which maps this cubic to the cubic on ${\mathbf{P}}^{2}$, defined by the equation

[TABLE]

(b) For every $\lambda\in\mathbf{C}$, distinct from $-3$, $-3\varepsilon$, $-3\varepsilon^{-2}$, where $\varepsilon=e^{2\pi i/3}$, the cubic on ${\mathbf{P}}^{2}$ defined by equation (2), is smooth and has exactly nine flexes, one of which is the point

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(c) For every smooth cubic $C$ on ${\mathbf{P}}^{2}$ defined by equation (2), and every its flex $c$, there is an element $g\in G$ such that $g(C)=C$ and $g(c)=q$.

We now consider the morphism $\varphi\colon{\mathbf{A}}^{1}\to{\mathbf{P}}^{2}\times{\mathbf{P}}^{9}$ defined by the formula

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It follows from (b) that $\varphi({\mathbf{A}}^{1})$ lies in $X$, and from (a) and (c) that the morpism

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is dominant. Since the underlying variety of every connected affine algebraic group is rational (see [2, Cor. 2]), it follows from the dominance of morphism (3) that the variety $X$, and hence $Y$, is unirational. Therefore there is a dominant rational map $\psi\colon{\mathbf{P}}^{9}\dashrightarrow Y$. In view of the smoothness of ${\mathbf{P}}^{9}$ and $Y$ and the projectivity of $Y$, the induced homomorphism of the spaces of rational differential $d$-forms

[TABLE]

defines an embedding of the spaces of regular differential $d$-forms

[TABLE]

(see [8, Chap. III, §6.1, Thm. 2]). Since $\Omega^{r}[{\mathbf{P}}^{n}]=0$ for any positive $r$ and $n$ (see [3, Chap. 0, Sect. 7]), this implies that $\Omega^{d}[Y]=0$, i.e., that statement (ii) holds. $\square$

Remark. Along the way of proof, unirationality of $X$ is proved.