Tohoku Mathematical Journal
2026
June
SECOND SERIES VOL. 78, NO. 2

Tohoku Math. J.
78 (2026), 277-290

Title DESCENDING CHAIN CONDITION FOR FINITE MORPHISMS OF ALGEBRAIC VARIETIES

Author Rajendra Vasant Gurjar and Masayoshi Miyanishi

(Received September 30, 2024, revised February 4, 2025)
Abstract. The descending chain condition ((DCC), for short) for finite surjective morphisms of algebraic varieties belonging to a category $\mathcal{C}$ asserts that for any descending chain with $X_i$ and $f_i$ being objects and morphisms in $\mathcal{C}$, \[ X_1 \st{f_1} X_2 \lto \cdots\cdots \to X_n \st{f_n} X_{n+1} \lto \cdots \] there exists an integer $N > 0$ such that $f_n$ is an isomorphism for every $n \ge N$. We discuss if the (DCC) holds for various subcategory $\mathcal{C}$ of the category of algebraic varieties, especially, the category of algebraic surfaces.

Mathematics Subject Classification. Primary 14A10; Secondary 14J29.
Key words and phrases. Descending chain condition, finite morphism, algebraic surfaces, fundamental group at infinity, 3-manifold at infinity.

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