1,721,117 research outputs found
On the Terracini locus of projective varieties
Get PDFWe introduce and study properties of the Terracini locus of projective varieties X, which is the locus
of finite subsets S such that 2S fails to impose independent conditions to a linear system L. Terracini loci
are relevant in the study of interpolation problems over double points in special position, but they also enter naturally
in the study of special loci contained in secant varieties to projective varieties.
We find some criteria which exclude that a set S belongs to the Terracini locus. Furthermore, in the case
where X is a Veronese variety, we bound the dimension of the Terracini locus and we determine examples
in which the locus has codimension 1 in the symmetric product of X
Minimal Terracini loci in the plane: gaps and non-gaps
Get PDFWe study minimally Terracini finite sets of points in the projective plane and we prove that
the sequence of the cardinalities of minimally Terracini sets can have any number of gaps
for degree great enough
Minimal Terracini loci in projective spaces
Get PDFWe characterize the number of points for which there exist non-empty Terracini sets of points in Pn. Then we study minimally Terracini finite sets of points in Pn and we obtain a complete description in the case of P3, when the number of points is less than twice the degree of the linear system
On the dimension of contact loci and the identifiability of tensors
No full textLet X in P^r be an integral and non-degenerate variety. Set n:= dim (X). We prove that if the (k+n-1)-secant
variety of X has (the expected) dimension (k+n-1)(n+1)-1<r and X is not uniruled by lines, then X is not k-weakly defective
and hence the k-secant variety satisfies identifiability, i.e. a general element of it
is in the linear span of a unique S in X with card(S) =k. We apply this result to many Segre-Veronese varieties
and to the identifiability of Gaussian mixtures G{1,d}. If X is the Segre embedding of a multiprojective space we prove
identifiability for the k-secant variety (assuming that the (k+n-1)-secant variety has dimension (k+n-1)(n+1)-1<r,
this is a known result in many cases), beating several bounds on the identifiability of tensors
Twistor fibers in hypersurfaces of the flag threefold
Get PDFWe study surfaces of bidegree (1, d) contained in the flag threefold in relation to the twistor projection. In particular, we focus on the number and the arrangement of twistor fibers contained in such surfaces. First, we prove that there is no irreducible surface of bidegree (1,d) containing d+2 twistor fibers in general position. On the other hand, given any collection of (d + 1) twistor fibers satisfying a mild natural constraint, we prove the existence of a surface of bidegree (1,d) that contains them. We improve our results for d = 2 or d = 3, by removing all the generality hypotheses
Generic Power Sum Decompositions and Bounds for the Waring Rank
Get PDFA notion of open rank, related with generic power sum decompositions of forms, has recently been introduced in the literature. The main result here is that the maximum open rank for plane quartics is eight. In particular, this gives the first example of n, d, such that the maximum open rank for degree d forms that essentially depend on n variables is strictly greater than the maximum rank. On one hand, the result allows to improve the previously known bounds on open rank, but on the other hand indicates that such bounds are likely quite relaxed. Nevertheless, some of the preparatory results are of independent interest, and still may provide useful information in connection with the problem of finding the maximum rank for the set of all forms of given degree and number of variables. For instance, we get that every ternary form of degree d>=3 can be annihilated by the product of d-1 pairwise independent linear forms
Bounds on the tensor rank
No full textWe give a sufficient criterion for a lower bound of the cactus rank of a tensor.
Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant
decomposition of a tensor to be minimal and unique
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