le superficie algebriche

Kodaira dimension

Include non-algebraic surfaces

Candidates

About the surface

References

Hodge diamond

$\mathrm{h}^{0,0}$
$\mathrm{h}^{1,0}$ $\mathrm{h}^{0,1}$
$\mathrm{h}^{2,0}$ $\mathrm{h}^{1,1}$ $\mathrm{h}^{0,2}$
$\mathrm{h}^{2,1}$ $\mathrm{h}^{1,2}$
$\mathrm{h}^{2,2}$

1
0 0
0 1 0
0 0
1

Numerical invariants

Betti numbers: $\mathrm{b}_0=1$; $\mathrm{b}_1={\color{undefined-gray}{?}}=\mathrm{h}^{1,0}+\mathrm{h}^{0,1}=\mathrm{h}^{2,1}+\mathrm{h}^{1,2}$; $\mathrm{b}_2={\color{undefined-gray}{?}}=\mathrm{h}^{2,0}+\mathrm{h}^{1,1}+\mathrm{h}^{0,2}$; $\mathrm{b}_3={\color{undefined-gray}{?}}=\mathrm{h}^{1,0}+\mathrm{h}^{0,1}=\mathrm{h}^{2,1}+\mathrm{h}^{1,2}$; $\mathrm{b}_4=1$
Euler characteristic: $\mathrm{e}={\color{undefined-gray}{?}}=\sum_{i=0}^4(-1)^i\mathrm{b}_i$
irregularity: $\mathrm{q}={\color{undefined-gray}{?}}=\mathrm{h}^{0,1}$
geometric genus: $\mathrm{p}_{\mathrm{g}}={\color{undefined-gray}{?}}=\mathrm{h}^{0,2}$
arithmetic genus: $\mathrm{p}_{\mathrm{a}}={\color{undefined-gray}{?}}=\mathrm{p}_{\mathrm{g}}-\mathrm{q}$
holomorphic Euler characteristic
Todd genus: $\chi={\color{undefined-gray}{?}}=\mathrm{p}_{\mathrm{g}}-\mathrm{q}+1=(\mathrm{c}_1^2+\mathrm{c}_2)/12$
signature: $\tau={\color{undefined-gray}{?}}=4\chi-\mathrm{e}=\sum_{i,j}(-1)^j\mathrm{h}^{i,j}$
Chern numbers: $\mathrm{c}_2={\color{undefined-gray}{?}}=\mathrm{e}$; $\mathrm{c}_1^2={\color{undefined-gray}{?}}=\mathrm{K}^2=12\chi-\mathrm{e}$

How to use it

To use this application, please first select a Kodaira dimension using the slider at the top right. Then select a pair of Chern numbers $(\mathrm{c}_{1}^{2},\mathrm{c}_{2})$ in the graph.

Finally you can select a surface from the list of known surfaces with these invariants (if there is a class of such surfaces). You can then see a description and the other numerical invariants of this class of surfaces.

Contribute

We can be reached at superficie.algebriche@gmail.com. The code is available on GitHub. Please report bugs; if you think of a feature, we would like to hear it as well.

If you have data about a (class of) surface(s), you can contribute via GitHub, or send it to us via email.

About

le superficie algebriche is a tool for studying numerical invariants of minimal algebraic surfaces over the complex numbers. The authors implemented it in order to better understand the Enriques–Kodaira classification, and to showcase how mathematics can be visualised on the web.