Given a Segre product $\mathbb P^m \times \mathbb P^n$, or more generally $\mathbb P^{m\_1}\times\cdots\times\mathbb P^{m\_n}$, is there a characterization in terms of $m$ and $n$, or the $m\_i$, for the Segre product to be Gorenstein?


Similar question for $\nu\_d(\mathbb P^n)$


Added: I am interested in the case when when the corresponding graded ring of Segre/Veronese variety is Gorenstein i.e. it has a self dual free resolution or equivalently its Hilbert series $f(t)$ is symmetric so that $f(1/t)=(-1)^n t^sf(t) $.



When are Segre- and Veronese embeddings Gorenstein?
I assume that you mean arithmetically Gorenstein, i.e. the cone over the variety is Gorenstein. Then:


$\bullet$ The Veronese variety $\nu\_d(\mathbb{P}^n)\subset \mathbb{P}^N$ is always arithmetically Cohen-Macaulay; the extra condition you need is $\omega \_{\mathbb{P}^n}\cong \nu\_d^\*\mathcal{O}\_{\mathbb{P}^N}(\ell)$ for some $\ell$, which means that $n+1$ is a multiple of $d$.


$\bullet$ Similarly the Segre variety $s(\mathbb{P}^{m\_1}\times \ldots \times \mathbb{P}^{m\_k})\subset \mathbb{P}^N$ is ACM; it is AG if and only if $\omega\_{\mathbb{P}^{m\_1}\times \ldots \times \mathbb{P}^{m\_k}} \cong s^\*\mathcal{O}\_{\mathbb{P}^N}(\ell)$ for some $\ell$, that is, if and only if $m\_1=\ldots =m\_k\,$.


  

**Edit**: Here are some precisions at the request of the OP. By e.g. Migliore's book *Introduction to Liaison Theory and Deficiency Modules*, pp. 9-11, a smooth projective variety $V\subset \mathbb{P}^N$ is arithmetically Gorenstein if and only if:


1) $H^i(V, \mathcal{O}\_V(k))=0$ for $0<i<\dim(V)$ and all $k$; 


2) The canonical line bundle $\omega\_V$ is induced by $\mathcal{O}\_{\mathbb{P}^N}(k)$ for some integer $k$.


Condition 1) always holds for a product of projective spaces. When $V=\mathbb{P}^n$ embedded by the Veronese embedding $\nu\_d$, one has $\omega \_V=\mathcal{O}\_{\mathbb{P}^n}(-n-1)$ and $\nu\_d^\*\mathcal{O}\_{\mathbb{P}^N}(k)=\mathcal{O}\_{\mathbb{P}^n}(kd)$, hence the condition $d\mid n+1$. If $V=\mathbb{P}^{m\_1}\times \ldots \times \mathbb{P}^{m\_k}$ embedded by the Segre embedding $s$, $\omega \_V= \mathcal{O}\_{\mathbb{P}^{m\_1}}(-m\_1-1)\boxtimes\ldots \boxtimes \mathcal{O}\_{\mathbb{P}^{m\_k}}(-m\_k-1)$ and $s^\*\mathcal{O}\_{\mathbb{P}^N}(k)=\mathcal{O}\_{\mathbb{P}^{m\_1}}(k)\boxtimes\ldots \boxtimes \mathcal{O}\_{\mathbb{P}^{m\_k}}(k)$, hence the condition $m\_1=\ldots =m\_k$.


