Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation. Finally, we establish the stabilization of the degree of the dual variety of a Segre product X×Qn, where X is a projective variety and Qn⊂Pn+1 is a smooth quadric hypersurface.
Asymptotics of degrees and ED degrees of Segre products / Ottaviani, G.; Sodomaco, L.; Ventura, E.. - In: ADVANCES IN APPLIED MATHEMATICS. - ISSN 0196-8858. - 130:(2021), p. 102242. [10.1016/j.aam.2021.102242]
Asymptotics of degrees and ED degrees of Segre products
Ottaviani G.;Ventura E.
2021
Abstract
Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation. Finally, we establish the stabilization of the degree of the dual variety of a Segre product X×Qn, where X is a projective variety and Qn⊂Pn+1 is a smooth quadric hypersurface.File | Dimensione | Formato | |
---|---|---|---|
Asymptotics of degrees and ED degrees of Segre products.pdf
accesso aperto
Tipologia:
2a Post-print versione editoriale / Version of Record
Licenza:
Creative commons
Dimensione
557.51 kB
Formato
Adobe PDF
|
557.51 kB | Adobe PDF | Visualizza/Apri |
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
https://hdl.handle.net/11583/2958175