# When are Segre- and Veronese embeddings Gorenstein?

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 $f(1/t)=(-1)^n t^sf(t)$.

• Do you really mean Gorenstein (in the intrinsic sense) or arithmetically Gorenstein? All smooth varieties are Gorenstein. Apr 3, 2016 at 9:05
• The Veronese varieties will almost never be arithmetically Gorenstein. The affine cone over $\nu_d(\mathbb{P}^n)$ is the quotient of $\mathbb{A}^{n+1}$ by the diagonal action of the group of roots of unity, $\mu_{d}$. The determinant of that action is the character $(\bullet)^{n+1}:\mu_d\to \mu_d$. Thus, this action factors through the special linear group if and only if $d$ divides $n+1$. On the positive side, these are always arithmetically Cohen-Macaulay. That follows from the Hochster-Roberts theorem (among others). Apr 3, 2016 at 10:16

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 $m_1=\ldots =m_k$.

• I have edited my question further and can you point me to some reference please
– MIQ
Apr 4, 2016 at 5:34
• 1) In case you are not sure: saying that the graded ring you mention is Gorenstein (or Cohen-Macaulay) is the same as saying that the variety is arithmetically Gorenstein (or Cohen-Macaulay). 2) One possible reference is chapter 4 of Migliore's book Introduction to Liaison Theory and Deficiency Modules. There is also a paper by Hartshorne: Geometry of arithmetically Gorenstein curves in $\mathbb{P}^4$.
– abx
Apr 4, 2016 at 7:07
• Thanks but I meant a reference for Veronese and Segre to be Arithmetically Gorenstein when satisfying the above conditions, if one needs to cite the reference in some piece of thesis or article...
– MIQ
Apr 8, 2016 at 14:00
• I do not know a direct reference, but the result is obvious in view of the characterization of arithmetically Gorenstein by the canonical line bundle being $\mathcal{O}(\ell)$.
– abx
Apr 8, 2016 at 15:31
• This is too long for a comment, so I am editing my answer.
– abx
Apr 9, 2016 at 11:47