 
Summary: SECANT VARIETIES OF SEGREVERONESE VARIETIES Pm × Pn
EMBEDDED BY O(1, 2)
HIROTACHI ABO AND MARIA CHIARA BRAMBILLA
Abstract. Let Xm,n be the SegreVeronese variety Pm
× Pn
embedded by the morphism given by
O(1, 2). In this paper, we provide two functions s(m, n) s(m, n) such that the sth
secant variety
of Xm,n has the expected dimension if s s(m, n) or s(m, n) s. We also present a conjecturally
complete list of defective secant varieties of such SegreVeronese varieties.
1. Introduction
Let X PN be an irreducible nonsingular variety of dimension d. Then the sth secant variety of
X, denoted s(X), is defined to be the Zariski closure of the union of the linear spans of all stuples
of points of X. The study of secant varieties has a long history. The interest in this subject goes
back to the Italian school at the turn of the 20th century. This topic has received renewed interest
over the past several decades, mainly due to its increasing importance in an ever widening collec
tion of disciplines including algebraic complexity theory [B¨urgisser et al. 1997, Landsberg 2006,
Landsberg 2008], algebraic statistics [Garcia et al. 2005, Eriksson et al. 2005, Aoki et al. 2007],
and combinatorics [Sturmfels and Sullivant 2006, Sullivant 2008].
The major questions surrounding secant varieties center around finding invariants of those ob
