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Surfaces with $$K^2 = 7$$ and $$p_g = 4$$
Ingrid C. Bauer, University of Gottingen, Germany
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Memoirs of the American Mathematical Society
2001; 79 pp; softcover
Volume: 152
ISBN-10: 0-8218-2689-1
ISBN-13: 978-0-8218-2689-8
List Price: US$52 Individual Members: US$31.20
Institutional Members: US\$41.60
Order Code: MEMO/152/721

The aim of this monography is the exact description of minimal smooth algebraic surfaces over the complex numbers with the invariants $$K^2 = 7$$ und $$p_g = 4$$. The interest in this fine classification of algebraic surfaces of general type goes back to F. Enriques, who dedicates a large part of his celebrated book Superficie algebriche to this problem. The cases $$p_g = 4$$, $$K^2 \leq 6$$ were treated in the past by several authors (among others M. Noether, F. Enriques, E. Horikawa) and it is worthwile to remark that already the case $$K^2 = 6$$ is rather complicated and it is up to now not possible to decide whether the moduli space of these surfaces is connected or not.

We will give a very precise description of the smooth surfaces with $$K^2 =7$$ und $$p_g =4$$ which allows us to prove that the moduli space $$\mathcal{M}_{K^2 = 7, p_g = 4}$$ has three irreducible components of respective dimensions $$36$$, $$36$$ and $$38$$.

A very careful study of the deformations of these surfaces makes it possible to show that the two components of dimension $$36$$ have nonempty intersection. Unfortunately it is not yet possible to decide whether the component of dimension $$38$$ intersects the other two or not.

Therefore the main result will be the following:

Theorem 0.1. 1) The moduli space $$\mathcal{M}_{K^2 = 7, p_g = 4}$$ has three irreducible components $$\mathcal{M}_{36}$$, $$\mathcal{M}'_{36}$$ and $$\mathcal{M}_{38}$$, where $$i$$ is the dimension of $$\mathcal{M}_i$$.

2) $$\mathcal{M}_{36} \cap \mathcal{M}'_{36}$$ is non empty. In particular, $$\mathcal{M}_{K^2 = 7, p_g = 4}$$ has at most two connected components.

3) $$\mathcal{M}'_{36} \cap \mathcal{M}_{38}$$ is empty.

Graduate students and research mathematicians interested in algebraic geometry.

• Surfaces with $$K^2=7, p_g=4$$, such that the canonical system doesn't have a fixed part
• $$\vert K\vert$$ has a (non trivial) fixed part