@ARTICLE{10.21494/ISTE.OP.2025.1341, 

TITLE={From a defective Segre-Veronese embedding to a non-defective one adding a factor}, 
  
AUTHOR={Edoardo Ballico, }, 
	
JOURNAL={Advances in Pure and Applied Mathematics}, 
	
VOLUME={16}, 

NUMBER={Issue 4 (September 2025)}, 
	
YEAR={2025}, 

URL={https://www.openscience.fr/From-a-defective-Segre-Veronese-embedding-to-a-non-defective-one-adding-a}, 
	
DOI={10.21494/ISTE.OP.2025.1341}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={Fix $$$x\in \mathbb{N}$$$, a multiprojective space $$$Y$$$ and a very ample line bundle $$$L$$$ on $$$Y$$$ . We say that $$$(Y,L)$$$ satisfies $$$\pm{x}\star$$$-non-defectivity if the $$$s$$$-secant variety of $$$(Y, L)$$$ has the expected dimension if either $$$(\dim Y+1)(s+x)\le h^0(L)$$$ or $$$(\dim Y +1)(s-x)\ge h^0(L)$$$. Natural examples arise when $$$L$$$ is the Segre line bundle and all factors have the same dimension (Abo - Ottaviani - Peterson). We take integers $$$r > 0$$$, $$$t\ge 2$$$ and set $$$X:= Y\times \mathbb{P}^r$$$. Let $$$L[t]$$$ be the line bundle on $$$X$$$ coming from $$$L$$$ and $$$\mathcal{O}_{\mathbb{P}^r}(t)$$$. Under certain assumptions on $$$x$$$, dim $$$Y, h^0(L)$$$, $$$r$$$ and $$$t$$$ we prove that $$$L[t]$$$ is not secant defective. Two of the main results are for $$$r\le 2$$$. In particular we extend a recent result by Ballico, Bernardi and Mańduz on the non-defectivity of Segre-Veronese embeddings of multidegree $$$(t_1,\dots ,t_k)$$$ of $$$(\mathbb{P}^2)^k$$$, $$$k\ge 3$$$, to the case in which $$$t_i=1$$$ for $$$y > 0$$$ integers $$$i:$$$ we require $$$y\ge 9$$$.}}