In this note, we classify smooth equivariant compactifications of G n a which are Fano manifolds with index ≥ n − 2. 1 See [FH2, Definition 2.5].

In this article, we determine all equivariant compactifications of the three-dimensional vector group G 3 a which are smooth Fano threefolds with Picard number greater or equal than two.

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 $\delta _{X}=3$.

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In this article, we study Newton-Okounkov bodies on projective vector bundles over curves. Inspired by Wolfe's estimates used to compute the volume function on these varieties, we compute all Newton-Okounkov bodies with respect to linear flags. Moreover, we characterize semi-stable vector bundles over curves via Newton-Okounkov bodies.Following the idea that numerical information encoded by the Harder-Narasimhan filtration of E should be related to asymptotic numerical invariants of P(E), we study the geometry of Newton-Okounkov bodies on P(E). 1. In other words, µmax(E) = σ1 ≥ . . . ≥ σr = µmin(E) can be viewed as the components of the vector σ = (σ1, . . . , σr) = (µ ℓ , . . . , µ ℓ r ℓ times , µ ℓ−1 , . . . , µ ℓ−1 r ℓ−1 times , . . . , µ1, . . . , µ1 r 1 times ) ∈ Q r .