Geodesic
Surfaces of fundamental type with geometric genus 2 and $c_1^2|X|=1$
Matematičeskie zametki, Tome 16 (1974) no. 4, pp. 623-632 
A. N. Todorov. Surfaces of fundamental type with geometric genus 2 and $c_1^2|X|=1$. Matematičeskie zametki, Tome 16 (1974) no. 4, pp. 623-632. http://geodesic.mathdoc.fr/item/MZM_1974_16_4_a15/
@article{MZM_1974_16_4_a15,
     author = {A. N. Todorov},
     title = {Surfaces of fundamental type with geometric genus 2 and $c_1^2|X|=1$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {623--632},
     year = {1974},
     volume = {16},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_4_a15/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In [1] E. Bombieri showed that $|4K|$ always yields a holomorphic map for surfaces of fundamental type and that $|3K|$ does not yield a holomorphic map for such surfaces with $p_g=2$ and $c_1^2|X|=1$. In this note we prove the existence of such surfaces and give a complete description of them. We prove that Torelli's local theorem is true, i.e., that the mapping of periods from the space of moduli into the space of periods is étale; we calculate the number of moduli and we show that the space of moduli is nonsingular.