Geometry of ℙ2 blown up at seven points

In this paper, we prove that blown up at seven general points admits a conic bundle structure over ℙ1 and it can be embedded as (2, 2) divisor in ℙ1 × ℙ2. Conversely, any smooth surface in the complete linear system |(2, 2)| of ℙ1 × ℙ2 can be obtained as an embedding of blowing up ℙ2 at seven points. We also show that smooth surface linearly equivalent to (2, 2) in ℙ1 × ℙ2 has at most four (−2) curves.

A Characterization of Some Fano 4-folds Through Conic Fibrations

We find a characterization for Fano 4-folds $X$ with Lefschetz defect $\delta _{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure $f\colon X\to Y$ with $\rho _{X}-\rho _{Y}=3$. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano $4$-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric $4$-folds with $\delta _{X}=3$.

Fiberwise bimeromorphic maps of conic bundles

Given a holomorphic conic bundle without sections, we show that the orders of finite groups acting by its fiberwise bimeromorphic transformations are bounded. This provides an analog of a similar result obtained by Bandman and Zarhin for quasi-projective conic bundles.

Stable rationality of higher dimensional conic bundles

We prove that a very general nonsingular conic bundle $X\rightarrow\mathbb{P}^{n-1}$ embedded in a projective vector bundle of rank $3$ over $\mathbb{P}^{n-1}$ is not stably rational if the anti-canonical divisor of $X$ is not ample and $n\geq 3$. Comment: Final version. To appear in Epijournal de Geometrie Algebrique

Conic bundles that are not birational to numerical Calabi--Yau pairs

Let $X$ be a general conic bundle over the projective plane with branch curve of degree at least 19. We prove that there is no normal projective variety $Y$ that is birational to $X$ and such that some multiple of its anticanonical divisor is effective. We also give such examples for 2-dimensional conic bundles defined over a number field.