On the extendability of projective surfaces and a genus bound for Enriques-Fano threefolds.

*(English)*Zbl 1238.14026The extendability of projective varieties is a very classical problem in algebraic geometry: a smooth projective variety \(Y\subseteq \mathbb{P}^r\) is said to be nonextendable if the only variety \(X\subseteq \mathbb{P}^{r+1}\) that has \(Y\) as an hyperplane section is a cone over \(Y\).

The classical tool used to detect nonextendable varieties is Zak’s theorem, stating that if \(Y\subseteq \mathbb{P}^r\) has codimension at least two, then it is nonextendable if \(h^0(N_{Y/\mathbb{P}^r}(-1))\leq r+1\).

The key point to apply Zak’s theorem is to calculate the cohomology of the normal bundle. In the case \(Y\) is a curve Wahl’s formula allows to do this by making use of the Gaussian map \(\Phi_{H_Y,\omega_Y}\), associated to the canonical and hyperplane bundle \(H_Y\) of \(Y\).

In this paper the authors prove an analogue of Wahl’s formula for surfaces and they apply it to the study of Enriques-Fano threefolds: three dimensional varieties that are non-trivial extensions of smooth Enriques surfaces.

G. Fano [Mem. Mat. Sci. Fis. Natur. Soc. Ital. Sci., III. Ser. 24, 41–66 (1938; Zbl 0022.07702)] claimed a classification of such threefolds, but the proof contained several gaps. However, after the works of L. Bayle [J. Reine Angew. Math. 449, 9–63 (1994; Zbl 0808.14028)] and T. Sano [J. Math. Soc. Japan 47, No. 2, 369–380 (1995; Zbl 0837.14031)] it has been conjectured to have a complete list. The authors show that in fact this is not the case by finding a completely new example of Enriques-Fano threefold of genus 9 and, on the other hand, they prove the bound \(g\leq 17\) on the genus of Enriques-Fano threefolds under no assumptions on their singularities.

The classical tool used to detect nonextendable varieties is Zak’s theorem, stating that if \(Y\subseteq \mathbb{P}^r\) has codimension at least two, then it is nonextendable if \(h^0(N_{Y/\mathbb{P}^r}(-1))\leq r+1\).

The key point to apply Zak’s theorem is to calculate the cohomology of the normal bundle. In the case \(Y\) is a curve Wahl’s formula allows to do this by making use of the Gaussian map \(\Phi_{H_Y,\omega_Y}\), associated to the canonical and hyperplane bundle \(H_Y\) of \(Y\).

In this paper the authors prove an analogue of Wahl’s formula for surfaces and they apply it to the study of Enriques-Fano threefolds: three dimensional varieties that are non-trivial extensions of smooth Enriques surfaces.

G. Fano [Mem. Mat. Sci. Fis. Natur. Soc. Ital. Sci., III. Ser. 24, 41–66 (1938; Zbl 0022.07702)] claimed a classification of such threefolds, but the proof contained several gaps. However, after the works of L. Bayle [J. Reine Angew. Math. 449, 9–63 (1994; Zbl 0808.14028)] and T. Sano [J. Math. Soc. Japan 47, No. 2, 369–380 (1995; Zbl 0837.14031)] it has been conjectured to have a complete list. The authors show that in fact this is not the case by finding a completely new example of Enriques-Fano threefold of genus 9 and, on the other hand, they prove the bound \(g\leq 17\) on the genus of Enriques-Fano threefolds under no assumptions on their singularities.

Reviewer: Salvatore Cacciola (Rome)