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 ? 



Proj of some graded algebra
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.


