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Let X
m,n
be the Segre–Veronese variety ℙ
m
× ℙ
n
embedded by
the morphism given by O(1, 2). In this paper, we provide two
functions 
such that the sth secant variety of
X
m,n
has the expected dimension if 
or 
.
We also present a conjecturally complete list of defective
secant varieties of such Segre–Veronese varieties.
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