When are Segre- and Veronese embeddings Gorenstein?

Question

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) $.

Answer 1

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 $m_1=\ldots =m_k$.