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Register a new userOn Fano schemes of complete intersections
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We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.
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Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if $X = cap_{i=1}^r D_i subset G/P$ is a general complete intersection of $r$ ample divisors such that $K_{G/P}^* otimes mathcal{O}_{G/P}(-sum_i D_i)$ is ample, then $X$ is Fano. We first classify these Fano complete intersections which are locally rigid. It turns out that most of them are hyperplane sections. We then classify general hyperplane sections which are quasi-homogeneous.
We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.
We prove birational boundedness results on complete intersections with trivial canonical class of base point free divisors in (some version of) Fano varieties. Our results imply in particular that Batyrev-Borisov toric construction produces only a bounded set of Hodge numbers in any given dimension, even as the codimension is allowed to grow.
We prove that every smooth complete intersection X defined by s hypersurfaces of degree d_1, ... , d_s in a projective space of dimension d_1 + ... + d_s is birationally superrigid if 5s +1 is at most 2(d_1 + ... + d_s + 1)/sqrt{d_1...d_s}. In particular, X is non-rational and Bir(X)=Aut(X). We also prove birational superrigidity of singular complete intersections with similar numerical condition. These extend the results proved by Tommaso de Fernex.
We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines.
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