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I was computing some GIT quotients and came up with the following question: to compute $\mathrm{Proj}(\mathbb C [f_1,f_2,f_3,f_4,f_5,f_6]/I)$ where $f_i$'s are homogeneous polynomials of same degree and $I$ is the ideal generated by $$\{f_3f_6-f_4f_5, f_1f_5-f_3^2-f_2f_3, f_1f_6-f_3f_4-f_2f_4, f_2f_4-f_3^2-f_1f_3, f_2f_6-f_3f_5-f_1f_5\}.$$

I thought it is isomorphic to $\mathbb P^2$, but here we have only 5 relations and don't know how to proceed from here. Is this isomorphic to a known projective variety ?

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  • $\begingroup$ It is very difficult to identify a variety from equations alone. Tell us where they come from. $\endgroup$ Jun 15 '16 at 6:11
  • $\begingroup$ $f_i's$ are polynomials in Plucker coordinates. They are from $G_{2,5}$ under the Torus action. $\endgroup$
    – Mathew
    Jun 15 '16 at 6:15
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    $\begingroup$ A $4$-plane hyperplane section of the Grassmannian $G_{2,5}$ is typically an elliptic normal curve with degree $5$ and Hilbert polynomial $5t$. You could plug your equations into a computer algebra program and see if this is the correct Hilbert polynomial. $\endgroup$ Jun 15 '16 at 11:08
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    $\begingroup$ I just realized that this is a 5-plane hyperplane section, not a 4-plane hyperplane section. Typically that is a quintic del Pezzo surface with Hilbert polynomial $(5/2)t^2 + (5/2)t + 1$. So you can see if Macaulay2 returns that Hilbert polynomial for your ideal. $\endgroup$ Jun 15 '16 at 12:08
  • $\begingroup$ A good test that helps understanding the structure of the variety is the following. Let $Q_1,\dots,Q_5$ be your five quadratic polynomials in $f_i$. Let $x_1,\dots,x_5$ be five formal variables. Consider the sextic hypersurface $D := \{ \det(\sum x_i Q_i) = 0 \} \subset P^4$ and its singular locus. If you can describe these, it may help understanding the original intersection as well. $\endgroup$
    – Sasha
    Jun 15 '16 at 20:58
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The OP clarified that the surface $S\subset \mathbb{P}^n$ satisfies all of the following properties: (a) $S$ is smooth, (b) $S$ is rational so that $h^1(S,\omega_S)$ is zero, and (c) the Hilbert polynomial of $S$ equals $$ p(t) = 1+ d\frac{(t+1)t}{2},$$ for some integer $d$. Up to replacing $\mathbb{P}^n$ by the span of $S$, also assume that $S$ spans $\mathbb{P}^n$. (In the original question, $d$ equals $5$ and $n\leq 5$.)

Claim. Every surface $S$ in $\mathbb{P}^n$ that satisfies (a), (b) and (c) is abstractly a del Pezzo surface embedded in projective space by a sublinear system of the anticanonical linear system.

Proof of the Claim. By Bertini's theorems, a general hyperplane section $C$ of $S$ is a smooth, connected curve with Hilbert polynomial $p(t)-p(t-1) = dt$. Since the Hilbert polynomial of a smooth curve of degree $d$ and arithmetic genus $g$ equals $dt+1-g$, $C$ is a smooth, connected, genus $1$ curve of degree $d$. By adjunction, we have a short exact sequence of coherent sheaves on $S$, $$ 0 \to \omega_S \to \omega_S(\underline{C}) \to \omega_C \to 0.$$ Since $h^1(S,\omega_S)$ equals $0$, the associated map $H^0(S,\omega_S(\underline{C})) \to H^0(C,\omega_C)$ is surjective. Since $C$ is a smooth, connected, genus $1$ curve, $\omega_C$ has an everywhere nonzero global section. This is the image of a global section of $\omega_S(\underline{C})$ whose zero locus is disjoint from $C$. Since the zero locus of a nonzero global section of an invertible sheaf is an effective Cartier divisor, and since this effective Cartier divisor is disjoint from the ample divisor $\underline{C}$, this effective Cartier divisor is empty. Therefore $\omega_S^\vee$ is isomorphic to the invertible sheaf $\mathcal{O}_S(\underline{C}) = \mathcal{O}_{\mathbb{P}^n}(1)|_S$. Thus $S$ is embedded in projective space by a sublinear system of the anticanonical linear system. Since the anticanonical divisor class is ample, $S$ is abstractly a del Pezzo surface. QED Claim.

In the case of interest to the OP, $d$ equals $5$ and $n\leq 5$. If $n$ equals $5$, then $S$ is embedded by the complete linear system of the anticanonical divisor class. Otherwise $S$ is a linear projection of the anticanonically embedded quintic del Pezzo surface. However, it is straightforward to compute that for an anticanonical quintic del Pezzo surface $S\subset \mathbb{P}^5$, every point of $\mathbb{P}^5$ is contained in a secant line of $S$, cf. Exercise III.3.13, p. 177 of Kollár's book, "Rational Curves on Algebraic Varieties". Thus, a linear projection of an anticanonical quintic del Pezzo is singular. Since the surface $S$ is smooth, $S$ is a quintic del Pezzo surface embedded in $\mathbb{P}^5$ by the complete linear system of the anticanonical divisor class.

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