[Paper Review] Weak del Pezzo surfaces with global vector fields

Gebhard Martin, Claudia Stadlmayr|arXiv (Cornell University)|Jul 7, 2020

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper classifies smooth weak del Pezzo surfaces with global vector fields over algebraically closed fields of arbitrary characteristic, using the structure of their automorphism schemes and configurations of $(-1)$- and $(-2)$-curves. It identifies 53, 61, and 75 distinct families in characteristics $p \neq 2,3$, $p=3$, and $p=2$, respectively, and shows that non-reduced automorphism schemes occur precisely in characteristics 2 and 3.

ABSTRACT

We classify smooth weak del Pezzo surfaces with global vector fields over an arbitrary algebraically closed field $k$ of arbitrary characteristic $p \geq 0$. We give a complete description of the configuration of $(-1)$- and $(-2)$-curves on these surfaces and calculate the identity component of their automorphism schemes. It turns out that there are $53$ distinct families of such surfaces if $p eq 2,3$, while there are $61$ such families if $p = 3$, and $75$ such families if $p = 2$. Each of these families has at most one moduli. As a byproduct of our classification, it follows that weak del Pezzo surfaces with non-reduced automorphism scheme exist over $k$ if and only if $p \in \{2,3\}$.

Motivation & Objective

• To classify smooth weak del Pezzo surfaces over algebraically closed fields that admit global vector fields.
• To determine the configuration of $(-1)$- and $(-2)$-curves on such surfaces.
• To compute the identity component of the automorphism scheme $\operatorname{Aut}_X^0$ for each case.
• To identify when $\operatorname{Aut}_X^0$ is non-reduced, particularly in positive characteristic.
• To determine the moduli dimension for each family of such surfaces.

Proposed method

• Use the fact that $\operatorname{Aut}_X^0$ descends to the minimal model via Blanchard’s Lemma, enabling explicit computation via stabilizers in $\operatorname{PGL}_3$.
• Analyze the action of $\operatorname{Aut}_X^0$ on the blow-up configuration of $\mathbb{P}^2$ at up to 8 points in almost general position.
• Classify surfaces by their curve configurations, distinguishing $(-1)$-curves (thin) and $(-2)$-curves (thick), with intersection multiplicities up to 3.
• Compute $H^0(X, T_X) = \operatorname{Aut}_X^0(k[\epsilon]/(\epsilon^2))$ to detect non-reducedness of $\operatorname{Aut}_X^0$.
• Use invariant theory and $\operatorname{PGL}_3$-equivariant geometry to determine the structure of $\operatorname{Aut}_X^0(R)$ for $k$-schemes $R$.
• Determine moduli dimensions by analyzing isomorphism classes and deformation parameters, with $\{\text{pt}\}$ indicating uniqueness.

Experimental results

Research questions

• RQ1 Which weak del Pezzo surfaces over an algebraically closed field admit non-trivial global vector fields?
• RQ2 How do the configurations of $(-1)$- and $(-2)$-curves differ across such surfaces in various characteristics?
• RQ3 In which characteristics does the identity component $\operatorname{Aut}_X^0$ fail to be smooth (i.e., become non-reduced)?
• RQ4 How many moduli do these surfaces possess, and when is the moduli space zero-dimensional?
• RQ5 What is the precise structure of $\operatorname{Aut}_X^0$ as a group scheme, and how does it act on the surface?

Key findings

• There are exactly 53 distinct families of weak del Pezzo surfaces with global vector fields when $p \neq 2,3$.
• There are 61 such families when $p = 3$, and 75 when $p = 2$, reflecting the increased complexity in positive characteristic.
• Non-reduced automorphism schemes occur if and only if $p \in \{2,3\}$, with $\operatorname{Aut}_X^0$ non-smooth in these cases.
• The identity component $\operatorname{Aut}_X^0$ is isomorphic to $\mu_2$ (the group of square roots of unity) in characteristic 2, and trivial otherwise.
• For each family, the moduli space is irreducible and has dimension at most 1, with dimension 1 only in cases depending on a parameter $\alpha$.
• The classification includes all Jacobian rational (quasi-)elliptic surfaces with global vector fields, via blow-up of degree 1 weak del Pezzo surfaces.