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Register a new userDefect formula for nodal complete intersection threefolds
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S. Cynk
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We generalize Werners defect formula for nodal hypersurfaces in $mathbb P^{4}$ to the case of a nodal complete intersection threefold.
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The aim of this paper is to classify indecomposable rank 2 arithmetically Cohen-Macaulay (ACM) bundles on compete intersection Calabi-Yau (CICY) threefolds and prove the existence of some of them. New geometric properties of the curves corresponding to rank 2 ACM bundles (by Serre correspondence) are obtained. These follow from minimal free resolutions of curves in suitably chosen fourfolds (containing Calabi-Yau threefolds as hypersurfaces). Also the existence of an indecomposable vector bundle of higher rank on a CICY threefold of type (2,4) is proved.
We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let $N geq 2$, and consider an isolated complete intersection curve singularity germ $f colon (mathbb{C}^N,0) to (mathbb{C}^{N-1},0)$. We introduce a numerical function $m mapsto operatorname{AD}_{(2)}^m(f)$ that arises as an error term when counting $m^{mathrm{th}}$-order weight-$2$ inflection points with ramification sequence $(0, dots, 0, 2)$ in a $1$-parameter family of curves acquiring the singularity $f = 0$, and we compute $operatorname{AD}_{(2)}^m(f)$ for various $(f,m)$. Particularly, for a node defined by $f colon (x,y) mapsto xy$, we prove that $operatorname{AD}_{(2)}^m(xy) = {{m+1} choose 4},$ and we deduce as a corollary that $operatorname{AD}_{(2)}^m(f) geq (operatorname{mult}_0 Delta_f) cdot {{m+1} choose 4}$ for any $f$, where $operatorname{mult}_0 Delta_f$ is the multiplicity of the discriminant $Delta_f$ at the origin in the deformation space. Furthermore, we show that the function $m mapsto operatorname{AD}_{(2)}^m(f) -(operatorname{mult}_0 Delta_f) cdot {{m+1} choose 4}$ is an analytic invariant measuring how much the singularity counts as an inflection point. We obtain similar results for weight-$2$ inflection points with ramification sequence $(0, dots, 0, 1,1)$ and for weight-$1$ inflection points, and we apply our results to solve various related enumerative problems.
We construct a Kawamata type semiorthogondal decomposition for the bounded derived category of coherent sheaves of nodal quintic del Pezzo threefolds.
The family of varieties of minimal rational tangents on a quasi-homogeneous projective manifold is isotrivial. Conversely, are projective manifolds with isotrivial varieties of minimal rational tangents quasi-homogenous? We will show that this is not true in general, even when the projective manifold has Picard number 1. In fact, an isotrivial family of varieties of minimal rational tangents needs not be locally flat in differential geometric sense. This leads to the question for which projective variety Z, the Z-isotriviality of varieties of minimal rational tangents implies local flatness. Our main result verifies this for many cases of Z among complete intersections.
In this paper we give a geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to $2n-2$, is exactly $2n-2$ if and only if the cone is a bipyramidal cone, where $n>1$ is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones.
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