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(This answer is just here to explicitly spell out how Sasha's answer and the related comments work in the case of one of your examples.) If you know a set of generators $f_1,\ldots,f_m$ for the ideal …

Here is one proof that the genus is bounded (although presumably a more elegant solution exists). Write $\Sigma=d\pi^*H - \sum_{i=1}^nm_iE_i$ in the usual basis for $\operatorname{NS}(X)$ (i.e. $\Sig …

Let $\phi\colon S_1\to S_2$ be a birational morphism between singular surfaces and let $f_i\colon \widetilde{S}_i \to S_i$ be the minimal resolution of singularities, for $i=1,2$. Any birational map …

The 'permutohedral variety' (the toric variety obtained from the permutohedron) is a very attractive toric variety which can be obtained by blowing up all of the toric strata in $\mathbb P^n$ in incre …

By Proposition 3.11 (i) and (ii) of Kollár and Shepherd-Barron's paper, if $N_{n,a,d}$ is the length of the continued fraction expansion of $\frac{dn^2}{dna-1}$, then we get the formulae $N_{2,1,d}=d$ …

Just address the question as to whether one can read off the fact that $\tfrac1r(a_1,\ldots,a_n)$ is canonical from the numbers $r$ and $a_1,\ldots,a_n$: the answer is yes. It is determined by the Rei …

You have a threefold hypersurface $Y\subset \mathbb{P}^4$ of the form $V(ax_1 - bx_2)$, for some quartic polynomials $a,b\in\mathbb{C}[\mathbb{P}^4]$. You blow up the plane $Z=V(x_1,x_2)$ in the ambi …

Take two planes $D_1 = V(x),D_2=V(y)\subset \mathbb{P}^3_{(x:y:z:t)}$ and consider the line $C=V(x,z)\subset D_1$. The equation of any hypersurface $D=V(f)$ of degree $d$ which only meets $D_2$ in the …

If the two cubics $C_1=V(F_1)$ and $C_2=V(F_2)$ do not share a common component, then the ideal $I = (F_1,F_2)$ defines a $0$-dimensional subscheme $V(I)\subset \mathbb{P}^2$ length 9, which is a comp …

If you consider the pair $(Z,\Delta)$, where $\Delta=Z\setminus i(\mathbb{G}_m)$ is the divisor given as the complement of the image of the torus, then there is an numerical invariant of $(Z,\Delta)$ …

If $X=\mathbb{V}(I)$ is given by the ideal $I$, then the $m$th symbolic power $I^{[m]}$ consists of all those functions vanishing to multiplicity $m$ at the generic point of $X$. Thus a hypersurface $ …

Possibly the simplest example is to consider the blowup of a reduced and irreducible curve $C$ in a smooth 3-fold $X$ with a point $P\in C\subset X$ which is locally analytically isomorphic to $$ 0\in …

A smooth del Pezzo surface $X_d$ of degree $d\geq3$ has very ample anticanonical divisor $-K_X$. It is also the blowup $X\to \mathbb P^2$ in $k=9-d$ points $P_1,\ldots,P_k\in\mathbb{P}^2_{x,y,z}$ in g …

You can diagonalise the action on $(x_1,\ldots,x_6)$ by the change of variables $$(u_1,\ldots,u_6):=(x_1+x_6,x_2+x_5,x_3,x_4,x_2-x_5,x_1-x_6).$$ Then the invariants are $u_5,u_6$ and $A_{ij}=u_iu_j$ …

To answer your second question: it follows from the Riemann-Roch theorem. Since $\deg(L)=-1$ we have $\Gamma(X,L^m)=0$ for $m\geq0$. Therefore, by RR on $X$, $\dim \Gamma(X,L^{-m}) = 1 + \deg(L^{-m}) …