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Let $X=\mathbb{P}(a_0,\ldots, a_n)$ be a well-formed weighted projective space, and let $a=\mathrm{lcm}(a_0,\ldots,a_n)$. Then $\mathcal{O}(a)$ embeds $X$ in projective space $\mathbb{P}^N$.

Question. Is $X$ cut out by quadrics in $\mathbb{P}^N$?

As far as I can tell, in general this is non-obvious for knapsack problem related reasons, though it is true for surfaces by Koelman's paper A criterion for the ideal of a projectively embedded toric surface to be generated by quadrics.

EDIT 3-3: Mateusz Michaelek has pointed out that $\mathcal{O}(a)$ does not generally embed X in projective space; for instance $\mathbb{P}(1,6,10,15)$ is not embedded by $\mathcal{O}(30)$, as the degree 60 monomial $x_0 x_1^4 x^2_2 x_3$ cannot be factored into degree 30 monomials. This example is example 2.17 of Toric geometry of path signature varieties. So in the question above, $\mathcal{O}(a)$ should be replaced with a very ample bundle $\mathcal{L}$, in which case I still don't know how to answer it.

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    $\begingroup$ I believe this is related to the conjectures of Neil White and of Herzog-Hibi. $\endgroup$ – Jason Starr Feb 24 at 15:45
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    $\begingroup$ Regarding EDIT 3-3: the fact that a degree 60 monomial is not a product of degree 30 monomials is showing that $\mathcal{O}(a)$ does not give a projectively normal embedding, that is the polytope is not normal. However the fact that it's not very ample (which is also true) is a more subtle check, right? $\endgroup$ – Evgeny Shinder Mar 22 at 16:35
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The following code in Sage gives information about the embedding of $\mathbb{P}(1,1,2,3)$ in $\mathbb{P}^{22}$ using $\mathcal{O}_{\mathbb{P}(1,1,2,3)}(6)$.

$\texttt{toric_varieties.WP(1,1,2,3).divisor([6,0,0,0]).Kodaira_map()}$

It produces the following output:

Scheme morphism:

From: 3-d toric variety covered by 4 affine patches

To: Closed subscheme of Projective Space of dimension 22 over Rational Field defined by:

-z16^2 + z0*z22, -z1*z20 + z0*z21, z21^2 - z14*z22, -z7*z16 + z0*z20, z20*z21 - z13*z22, z20^2 - z12*z22, -z1*z18 + z0*z19, z19*z21 - z11*z22, z19*z20 - z10*z22, z19^2 - z6*z22, -z1*z17 + z0*z18, z18*z21 - z10*z22, z18*z20 - z9*z22, z18*z19 - z5*z22, z18^2 - z4*z22, -z1*z16 + z0*z17, z17*z21 - z9*z22, z17*z20 - z8*z22, z17*z19 - z4*z22, z17*z18 - z3*z22, z17^2 - z2*z22, z16*z21 - z8*z22, z16*z20 - z7*z22, z16*z19 - z3*z22, z16*z18 - z2*z22, z16*z17 - z1*z22, -z7*z12 + z0*z15, z15*z19 - z14*z21, z15*z18 - z13*z21, z15*z17 - z12*z21, z15*z16 - z12*z20, -z1*z13 + z0*z14, z14*z20 - z13*z21, z14*z19 - z11*z21, z14*z18 - z10*z21, z14*z17 - z9*z21, z14*z16 - z8*z21, z14^2 - z11*z15, -z1*z12 + z0*z13, z13*z20 - z12*z21, z13*z19 - z10*z21, z13*z18 - z9*z21, z13*z17 - z8*z21, z13*z16 - z7*z21, z13*z14 - z10*z15, z13^2 - z9*z15, -z7^2 + z0*z12, z12*z19 - z9*z21, z12*z18 - z8*z21, z12*z17 - z7*z21, z12*z16 - z7*z20, z12*z14 - z9*z15, z12*z13 - z8*z15, z12^2 - z7*z15, -z1*z10 + z0*z11, z11*z20 - z10*z21, z11*z19 - z6*z21, z11*z18 - z5*z21, z11*z17 - z4*z21, z11*z16 - z3*z21, z11*z14 - z6*z15, z11*z13 - z5*z15, z11*z12 - z4*z15, z11^2 - z6*z14, -z1*z9 + z0*z10, z10*z20 - z9*z21, z10*z19 - z5*z21, z10*z18 - z4*z21, z10*z17 - z3*z21, z10*z16 - z2*z21, z10*z14 - z5*z15, z10*z13 - z4*z15, z10*z12 - z3*z15, z10*z11 - z5*z14, z10^2 - z4*z14, -z1*z8 + z0*z9, z9*z20 - z8*z21, z9*z19 - z4*z21, z9*z18 - z3*z21, z9*z17 - z2*z21, z9*z16 - z1*z21, z9*z14 - z4*z15, z9*z13 - z3*z15, z9*z12 - z2*z15, z9*z11 - z4*z14, z9*z10 - z3*z14, z9^2 - z2*z14, -z1*z7 + z0*z8, z8*z20 - z7*z21, z8*z19 - z3*z21, z8*z18 - z2*z21, z8*z17 - z1*z21, z8*z16 - z1*z20, z8*z14 - z3*z15, z8*z13 - z2*z15, z8*z12 - z1*z15, z8*z11 - z3*z14, z8*z10 - z2*z14, z8*z9 - z1*z14, z8^2 - z1*z13, z7*z19 - z2*z21, z7*z18 - z1*z21, z7*z17 - z1*z20, z7*z14 - z2*z15, z7*z13 - z1*z15, -z1*z15*z20 + z7*z12*z21, z7*z11 - z2*z14, z7*z10 - z1*z14, z7*z9 - z1*z13, z7*z8 - z1*z12, -z1*z12*z20 + z7^2*z21, -z1*z5 + z0*z6, z6*z20 - z5*z21, z6*z18 - z5*z19, z6*z17 - z4*z19, z6*z16 - z3*z19, z6*z13 - z5*z14, z6*z12 - z4*z14, z6*z10 - z5*z11, z6*z9 - z4*z11, z6*z8 - z3*z11, z6*z7 - z2*z11, -z1*z4 + z0*z5, z5*z20 - z4*z21, z5*z18 - z4*z19, z5*z17 - z3*z19, z5*z16 - z2*z19, z5*z13 - z4*z14, z5*z12 - z3*z14, z5*z10 - z4*z11, z5*z9 - z3*z11, z5*z8 - z2*z11, z5*z7 - z1*z11, z5^2 - z4*z6, -z1*z3 + z0*z4, z4*z20 - z3*z21, z4*z18 - z3*z19, z4*z17 - z2*z19, z4*z16 - z1*z19, z4*z13 - z3*z14, z4*z12 - z2*z14, z4*z10 - z3*z11, z4*z9 - z2*z11, z4*z8 - z1*z11, z4*z7 - z1*z10, z4*z5 - z3*z6, z4^2 - z2*z6, -z1*z2 + z0*z3, z3*z20 - z2*z21, z3*z18 - z2*z19, z3*z17 - z1*z19, z3*z16 - z1*z18, z3*z13 - z2*z14, z3*z12 - z1*z14, z3*z10 - z2*z11, z3*z9 - z1*z11, z3*z8 - z1*z10, z3*z7 - z1*z9, z3*z5 - z2*z6, z3*z4 - z1*z6, z3^2 - z1*z5, -z1^2 + z0*z2, z2*z20 - z1*z21, z2*z18 - z1*z19, z2*z17 - z1*z18, z2*z16 - z1*z17, z2*z13 - z1*z14, z2*z12 - z1*z13, z2*z10 - z1*z11, z2*z9 - z1*z10, z2*z8 - z1*z9, z2*z7 - z1*z8, z2*z5 - z1*z6, z2*z4 - z1*z5, z2*z3 - z1*z4, z2^2 - z1*z3

Defn: Defined on coordinates by sending [z0 : z1 : z2 : z3] to

(z0^6 : z0^5*z1 : z0^4*z1^2 : z0^3*z1^3 : z0^2*z1^4 : z0*z1^5 : z1^6 : z0^4*z2 : z0^3*z1*z2 : z0^2*z1^2*z2 : z0*z1^3*z2 : z1^4*z2 : z0^2*z2^2 : z0*z1*z2^2 : z1^2*z2^2 : z2^3 : z0^3*z3 : z0^2*z1*z3 : z0*z1^2*z3 : z1^3*z3 : z0*z2*z3 : z1*z2*z3 : z3^2)

Which you can see contains cubics such as $-z_1 z_{15} z_{20} + z_7 z_{12} z_{21}$.

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  • $\begingroup$ Thanks for computing this! I am confused though by the cubic, because that cubic is given by $$ z_{15}(z_0z_{21}-z_1z_{20})+ z_{21}(z_7z_{12}-z_0z_{15}),$$ and I think those two quadrics are part of the ideal (at least, I see them on the list). Do you have any idea what makes the cubic appear in there as well? $\endgroup$ – Geoff Mar 16 at 5:13

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