le superficie algebriche

$\mathrm{c}_2$

$\mathrm{c}_1^2$

Kodaira dimension

Include non-algebraic surfaces

Candidates

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: $\mathrm{b}_0=1$; $\mathrm{b}_1={\color{#dbdbdb}{?}}=\mathrm{h}^{1,0}+\mathrm{h}^{0,1}=\mathrm{h}^{2,1}+\mathrm{h}^{1,2}$; $\mathrm{b}_2={\color{#dbdbdb}{?}}=\mathrm{h}^{2,0}+\mathrm{h}^{1,1}+\mathrm{h}^{0,2}$; $\mathrm{b}_3={\color{#dbdbdb}{?}}=\mathrm{h}^{1,0}+\mathrm{h}^{0,1}=\mathrm{h}^{2,1}+\mathrm{h}^{1,2}$; $\mathrm{b}_4=1$
Euler characteristic: $\mathrm{e}={\color{#dbdbdb}{?}}=\sum_{i=0}^4(-1)^i\mathrm{b}_i$
irregularity: $\mathrm{q}={\color{#dbdbdb}{?}}=\mathrm{h}^{0,1}$
geometric genus: $\mathrm{p}_{\mathrm{g}}={\color{#dbdbdb}{?}}=\mathrm{h}^{0,2}$
arithmetic genus: $\mathrm{p}_{\mathrm{a}}={\color{#dbdbdb}{?}}=\mathrm{p}_{\mathrm{g}}-\mathrm{q}$
holomorphic Euler characteristic
Todd genus: $\chi={\color{#dbdbdb}{?}}=\mathrm{p}_{\mathrm{g}}-\mathrm{q}+1=(\mathrm{c}_1^2+\mathrm{c}_2)/12$
signature: $\tau={\color{#dbdbdb}{?}}=4\chi-\mathrm{e}=\sum_{i,j}(-1)^j\mathrm{h}^{i,j}$
Chern numbers: $\mathrm{c}_2={\color{#dbdbdb}{?}}=\mathrm{e}$; $\mathrm{c}_1^2={\color{#dbdbdb}{?}}=\mathrm{K}^2=12\chi-\mathrm{e}$
Euler characteristic of the tangent bundle: $\chi(\mathrm{T}_S)={\color{#dbdbdb}{?}}=2\mathrm{c}_1^2-10\chi=(7\mathrm{c}_1^2-5\mathrm{c}_2)/6$

About the surface

No surface selected.

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). The description, references and the other numerical invariants of this class of surfaces then appear below the graph.

You can also browse every surface in the database in the table view.

Contribute

The code is available on GitHub. Please report bugs and request features there.

If you have data about a (class of) surface(s), you can contribute via GitHub.

About

le superficie algebriche is a tool for studying numerical invariants of minimal algebraic surfaces over the complex numbers. It was made by Pieter Belmans and Johan Commelin in order to better understand the Enriques–Kodaira classification, and to showcase how mathematics can be visualised on the web.

If you use it, please cite it.

le superficie algebriche

About the surface

References

How to use it

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