English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It is known that the optimal Noether inequality $\mathrm{vol}(X) \ge\frac{4}{3}p_g(X) - \frac{10}{3}$ holds for every $3$-fold $X$ of general typewith $p_g(X) \ge 11$. In this paper, we give a complete classification of$3$-folds $X$ of general type with $p_g(X) \ge 11$ satisfying the aboveequality by giving the explicit structure of a relative canonical model of $X$.This model coincides with the canonical model of $X$ when $p_g(X) \ge 23$. Wealso establish the second and third optimal Noether inequalities for $3$-folds$X$ of general type with $p_g(X) \ge 11$. These results answer two openquestions raised by J. Chen, M. Chen and C. Jiang, and in dimension three anopen question raised by J. Chen and C. Lai. A novel phenomenon shows that thereis a one-to-one correspondence between the three Noether inequalities and threepossible residues of $p_g(X)$ modulo $3$.

Algebraic threefolds of general type with small volume