Enriched Gromov–Witten Invariants of Del Pezzo Surfaces

Definitions and mathematical statement

A field $k$ is perfect if every finite algebraic extension of $k$ is separable. A finite étale $k$-algebra is a finite product of finite separable field extensions of $k$. A del Pezzo surface over $k$ is a smooth projective surface $X$ whose anticanonical divisor $-K_X$ is ample; its degree is $K_X^2$. Here rationality means birationality over $k$ to $\mathbb{P}^2_k$.

The supplied Conjecture 1 has the following partially specified formulation. Let $k$ be a perfect field with $\operatorname{char}(k)\notin\{2,3\}$, and fix $(n_1,\ldots,n_r)\in{\mathbb{N}}^r$. Let $A_1,\ldots,A_r$ be the finite étale $k$-algebras specifying zero-dimensional subschemes \[ p_i\subset\mathbb{P}^2_k,\qquad p_i\cong\operatorname{Spec}(A_i),\qquad 1\leq i\leq r. \] Suppose that the surface obtained by blowing up $\mathbb{P}^2_k$ along the prescribed subschemes $p_1,\ldots,p_r$ is a rational del Pezzo surface $X$ satisfying $K_X^2\geq4$. For the intended auxiliary data $A_0,D$, the asserted identity is \[ W^{\mathrm{WD}}(A_0,A_1,\ldots,A_r)=Q_{X,D}(A_0). \] Unresolved qualifications. The record does not define either invariant, their common target, the datum $A_0$, or the divisor or curve-class datum denoted by $D$. It therefore does not specify the admissibility conditions or quantifiers for these data. The original transcription places $(A_1,\ldots,A_r)$ in an unexplained collection written $\mathrm{Et}_{n}(k)$; the relation between this notation and $(n_1,\ldots,n_r)$, including any required algebra dimensions, is not supplied. These omissions prevent a complete precise statement of the conjecture. The preceding real-specialization formula in the original transcription also contains undefined notation and damaged indices, so it cannot be reliably formulated as an additional identity from this record alone.

Short English statement

The conjecture compares two enriched curve-counting invariants for rational del Pezzo surfaces of degree at least $4$, obtained from the specified finite étale blow-up centers over a perfect field of characteristic different from $2$ and $3$. The invariant definitions and the conditions on $A_0$ and $D$ are missing, so the displayed equality remains a partially specified, statement-only draft.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006377$, exact source problem code catalogue entry, title: Enriched Gromov--Witten Invariants of Del Pezzo Surfaces. The record attributes its provenance to the Oberwolfach Reports Open Problems collection; no individual proposer is supplied.
• Supplied report citation: Algebraic K-theory (2026), Oberwolfach Reports, pp. 1949--2012, DOI 10.4171/owr/2025/36: https://doi.org/10.4171/owr/2025/36. The supplied literature-source label is "Wickelgren--Brugalle--Rau Oberwolfach report."
• Supplied literature source labeled "primary report text": https://ems.press/content/serial-article-files/52244?nt=1.
• Literature checked at: 2026-08-22. This is supplied dated curation, not independently verified. Its status label is "partially solved," and its assessment states that the conjectural identity is proved for del Pezzo surfaces of degree greater than five and in toric cases, while the full stated degree-at-least-four assertion remains open. The background explicitly requests independent verification of the time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that only formula delimiters, notation, grammar, and typography were repaired. This supplied label does not establish mathematical correctness or resolve the missing definitions.