# regular map

## English

### Noun

regular map (plural regular maps)

1. (algebraic geometry) A morphism between algebraic varieties.
• 1994, Miles Reid (translator), Igor R. Shafarevich, Basic Algebraic Geometry 1, 2nd Edition, Springer, page 29,
We now find out how a regular map acts on the ring of regular functions on a closed set. [] Moreover, regular maps can be characterised as the maps that take regular functions into regular functions.
• 2003, Igor Dolgachev, Lectures on Invariant Theory, Cambridge University Press, page xiii,
Since one expects that the canonical projection ${\displaystyle f:X\rightarrow X/G}$ is a regular map of algebraic varieties and so has closed fibres, all orbits must be closed subsets in the Zariski topology of ${\displaystyle X}$.
• 2017, José F. Fernando, José M. Gamboa, Carlos Ueno, Polynomial, regular and Nash images of Euclidean spaces, Fabrizio Broglia, Françoise Delon, Max Dickmann, Danielle Gondard-Cozette, Victoria Ann Powers (editors), Ordered Algebraic Structures and Related Topics: International Conference, American Mathematical Society, page 160,
The 1-dimensional semialgebraic set ${\displaystyle {\mathcal {S}}:={x>0,xy=1}}$ is the image of the regular map
${\displaystyle f:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2},(x,y)\mapsto \left((xy-1)^{2}+x^{2},{\frac {1}{(xy-1)^{2}+x^{2}}}\right)}$.
2. (graph theory) A symmetric tessellation of a closed surface; a decomposition of a two-dimensional manifold into topological disks such that every flag (incident vertex-edge-face triple) can be transformed into any other flag by a symmetry (i.e., an automorphism) of the decomposition.
• 1990 [McGraw-Hill], Jay Kappraff, Connections: The Geometric Bridge Between Art and Science, 2001, World Scientific, page 141,
Just as there are only five regular maps on the sphere (or plane), there are only three classes of regular maps that can be created on a torus.
• 2010, Jozef Širáñ, Yan Wang, Maps with highest level of symmetry that are even more symmetric than other such maps: Regular maps with largest exponent groups, Richard A. Brualdi, Samat Hedayat, Hadi Kharaghani, Gholamreza B. Khosrovshahi, Shahriar Shahriari (editors), (editors), Combinatorics and Graphs: The Twentieth Anniversary Conference of IPM Combinatorics, American Mathematical Society, page 98,
If n is finite, the regular map is a tessellation of the plane by congruent n-sided polygons, m of which meet at each vertex.
• 2013, Roman Nedela, Martin Škoviera, 7.6: Maps, Jonathan L. Gross, Jay Yellen, Ping Zhang (editors), Handbook of Graph Theory, 2nd Edition, CRC Press, page 845,
If ${\displaystyle M}$ is a regular map of type ${\displaystyle \{p,q\}}$, then ${\displaystyle {\mathit {Aut}}(M)\approx \Delta (p,q,2)/N}$ for some normal subgroup ${\displaystyle N\ \trianglelefteq \ \Delta (p,q,2)}$. Similar statements hold for the class of orientably regular maps and subgroups of ${\displaystyle \Delta ^{+}}$ and ${\displaystyle \Delta ^{+}(p,q,2)}$.

#### Synonyms

• (function between algebraic varieties): morphism