On some Fano manifolds admitting a rational fibration

Let X be a smooth, complex Fano variety. For every prime divisor D in X, we set c(D):=dim ker(r:H^2(X,R)->H^2(D,R)), where r is the natural restriction map, and we define an invariant of X as c_X:=max{c(D)|D is a prime divisor in X}. In a previous paper we showed that c_X2, then either X is a product, or X has a flat fibration in Del Pezzo surfaces. In this paper we study the case c_X=2. We show that up to a birational modification given by a sequence of flips, X has a conic bundle structure, or an equidimensional fibration in Del Pezzo surfaces. We also show a weaker property of X when c_X=1.Comment: 31 pages. Revised version, minor changes. To appear in the Journal of the London Mathematical Societ

Quasi elementary contractions of Fano manifolds

Let X be a smooth complex Fano variety. We define and study 'quasi elementary' contractions of fiber type f: X -> Y. These have the property that rho(X) is at most rho(Y)+rho(F), where rho is the Picard number and F is a general fiber of f. In particular any elementary extremal contraction of fiber type is quasi elementary. We show that if Y has dimension at most 3 and Picard number at least 4, then Y is smooth and Fano; if moreover rho(Y) is at least 6, then X is a product. This yields sharp bounds on rho(X) when dim(X)=4 and X has a quasi elementary contraction, and other applications in higher dimensions.Comment: Final version, minor changes, to appear in Compositio Mathematic

On Fano manifolds with a birational contraction sending a divisor to a curve

Let X be a smooth Fano variety of dimension at least 4. We show that if X has an elementary birational contraction sending a divisor to a curve, then the Picard number of X is smaller or equal to 5.Comment: 24 pages, 6 figure

Rank 2 quasiparabolic vector bundles on P1 and the variety of linear subspaces contained in two odd-dimensional quadrics

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The number of vertices of a Fano polytope

Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.Comment: Final version, to appear in Annales de l'Institut Fourie

Accurate fundamental parameters for Lower Main Sequence Stars

We derive an empirical effective temperature and bolometric luminosity calibration for G and K dwarfs, by applying our own implementation of the InfraRed Flux Method to multi-band photometry. Our study is based on 104 stars for which we have excellent BVRIJHK photometry, excellent parallaxes and good metallicities. Colours computed from the most recent synthetic libraries (ATLAS9 and MARCS) are found to be in good agreement with the empirical colours in the optical bands, but some discrepancies still remain in the infrared. Synthetic and empirical bolometric corrections also show fair agreement. A careful comparison to temperatures, luminosities and angular diameters obtained with other methods in literature shows that systematic effects still exist in the calibrations at the level of a few percent. Our InfraRed Flux Method temperature scale is 100K hotter than recent analogous determinations in the literature, but is in agreement with spectroscopically calibrated temperature scales and fits well the colours of the Sun. Our angular diameters are typically 3% smaller when compared to other (indirect) determinations of angular diameter for such stars, but are consistent with the limb-darkening corrected predictions of the latest 3D model atmospheres and also with the results of asteroseismology. Very tight empirical relations are derived for bolometric luminosity, effective temperature and angular diameter from photometric indices. We find that much of the discrepancy with other temperature scales and the uncertainties in the infrared synthetic colours arise from the uncertainties in the use of Vega as the flux calibrator. Angular diameter measurements for a well chosen set of G and K dwarfs would go a long way to addressing this problem.Comment: 34 pages, 20 figures. Accepted by MNRAS. Landscape table available online at http://users.utu.fi/luccas/IRFM

Fano 4-folds with a small contraction

Let X be a smooth complex Fano 4-fold. We show that if X has a small elementary contraction, then the Picard number rho(X) of X is at most 12. This result is based on a careful study of the geometry of X, on which we give a lot of information. We also show that in the boundary case rho(X)=12 an open subset of X has a smooth fibration with fiber the projective line. Together with previous results, this implies if X is a Fano 4-fold with rho(X)>12, then every elementary contraction of X is divisorial and sends a divisor to a surface. The proof is based on birational geometry and the study of families of rational curves. More precisely the main tools are: the study of families of lines in Fano 4-folds and the construction of divisors covered by lines, a detailed study of fixed prime divisors, the properties of the faces of the effective cone, and a detailed study of rational contractions of fiber type.Comment: 45 page

Numerical invariants of Fano 4-folds

Let X be a (smooth, complex) Fano 4-fold. For any prime divisor D in X, consider the image of N_1(D) in N_1(X) under the push-forward of 1-cycles, and let c_D be its codimension in N_1(X). We define an integral invariant c_X of X as the maximal c_D, where D varies among all prime divisors in X. One easily sees that c_X is at most rho_X-1 (where rho is the Picard number), and that c_X is greater or equal than rho_X-rho_D, for any prime divisor D in X. We know from previous works that if c_X > 2, then either X is a product of Del Pezzo surfaces and rho_X is at most 18, or c_X=3 and rho_X is at most 6. In this paper we show that if c_X=2, then rho_X is at most 12.Comment: 11 page