Positively Curved Projective Varieties — Problem 10

Definitions and mathematical statement

For notation, let $k$ be an algebraically closed field and let \[ \mathbb F_n=\mathbb P_{\mathbb P^1_k}\left(\mathcal O_{\mathbb P^1_k}\oplus\mathcal O_{\mathbb P^1_k}(n)\right),\qquad n\in\mathbb Z_{\geq 0}, \] be a Hirzebruch surface. The supplied record does not specify the ground field or characteristic, or explicitly state the intended range of $n$. This notation therefore does not supply a missing universal quantifier over fields or indices; the intended setting requires confirmation.

For four pairwise distinct points $p_1,\ldots,p_4\in\mathbb F_n(k)$, write \[ \pi:X=\operatorname{Bl}_{\{p_1,\ldots,p_4\}}(\mathbb F_n)\longrightarrow\mathbb F_n \] for their blow-up, and let $T_X$ denote the tangent bundle of $X$. Under the numerical tautological interpretation, a vector bundle $E$ on a smooth projective variety is called pseudo-effective if \[ [\mathcal O_{\mathbb P(E)}(1)]\in\overline{\operatorname{Eff}}^{\,1}(\mathbb P(E)), \] where $\mathbb P(E)$ parametrizes one-dimensional quotients, $\mathcal O_{\mathbb P(E)}(1)$ is the tautological quotient line bundle, and $\overline{\operatorname{Eff}}^{\,1}$ is the closure of the cone generated by effective divisor classes in the real numerical divisor space. The supplied record does not define pseudo-effectivity for vector bundles; this interpretation must be checked against the cited source.

Problem (Matsumura, 2022, Problem 10).

1. Does the blow-up of a Hirzebruch surface at four general points have pseudo-effective tangent bundle? Under the numerical interpretation above, the question asks whether \[ [\mathcal O_{\mathbb P(T_X)}(1)]\in\overline{\operatorname{Eff}}^{\,1}(\mathbb P(T_X)). \] The source does not specify whether "general" means points in a nonempty Zariski open subset of the space of distinct ordered quadruples or very general points, nor whether the answer is sought uniformly in $n$ or separately for each $n$.
2. What can be said about pseudo-effectivity of the tangent bundle of a Hirzebruch surface blown up at special points? This is an exploratory question: the supplied record specifies neither the special configurations nor the number of points in this second question. It also does not say whether infinitely near centers are allowed. Consequently, a precise classification target cannot be recovered from this record alone.

Short English statement

Matsumura asks whether blowing up a Hirzebruch surface at four general points gives a surface with pseudo-effective tangent bundle, and what happens for blow-ups at special points. The ground field, meaning of general and special points, and precise vector-bundle positivity convention require confirmation. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath source registry, permanent ID $800775$, exact source problem code catalogue entry, title "Positively Curved Projective Varieties --- Problem 10." The supplied record attributes the problem to Matsumura, with proposed year 2022, and labels it open; that label is supplied curation, not an independently verified mathematical assessment.
• Matsumura, Open Problems on Structure of Positively Curved Projective Varieties (2022), Problem 10. Supplied source link: https://arxiv.org/abs/2211.09153. The provenance reports extraction by arxiv_latex and access on 2026-10-16. These are supplied provenance details; no independent source visit or verification is claimed.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry gives creation and update timestamps of 2026-10-16T14:00:00Z; these timestamps do not establish a literature review.