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The condition you are looking for is seminormality. A variety (or a reduced scheme) $Y$ is seminormal if any proper bijective morphism $f:X\to Y$, with $X$ reduced, inducing isomorphisms on residue fi …

How about: $Y$ is the cubic curve $y^2=x^2+x^3$ in $\mathbb A^2$ minus the point $(-1,0)$, $g:Y\to Z$ is the projection to the $x$-axis, and $X$ is the normalization of $Y$, which would be $\mathbb P^ …

The reason for both is that the third derivative of a (nonhomogeneous) quadratic function is zero. $\chi(O(D))$ on a surface is a quadratic function of $D$, by Riemann-Roch. Theorem of cube on an ab …

(1) A family of hypersurfaces of degree $d$ over an affine scheme $Spec\ R$ means the following: it is a closed subscheme of $\mathbb P^n_R$ given by a homogeneous polynomial $f(x_0,\dots,x_n)$ of deg …

You need to be careful about what you mean by "a general point". Usually, this means "a point in a certain Zariski open set". So in particular, this statement would say that on a surface of general ty …

As Pete already indicated, Mumford's theorem says that for any projective variety $X\subset \mathbb P^n$, its Veronese emberdding $v_d(X)\subset \mathbb P^N$ is cut out by quadrics, for $d\gg0$. So a …

1. Are there simple examples (say on a curve or surface) of line bundles that are globally generated but not ample, of ample line bundles with no sections, of ample line bundles that are globally gene …

I think EGAs and SGAs are not useful for "students of today" but they are indispensable for "researchers of today", and "tomorrow". There is just so much stuff there that is not available anywhere els …

Because the first one is the right answer in the case of affine varieties, and the second one is not. Indeed, $R/I$, $R/J$ are nilpotent-free implies $R/I\cap J$ is nilpotent-free, but not so for $R/I …

In positive characteristic, quotient singularities need not be rational. For an example, see Artin's paper "Wildly ramified Z/2-actions in dimension two". In characteristic zero, quotient singulariti …

Any smooth projective surface with nonempty $K_X$ is obtained by blowing up finitely many points on its unique minimal model. From the formula $K_X=f^*K_Y+E$ for the blowup, you see that the exception …

That is way too ambitious, I think, considering what is known about classification of algebraic varieties. Ignoring the trivial, for these purposes, case of curves, the next and best studied case is a …

Make a base change $f':Y'\to Y$ so that $f':X'=X\times_Y Y' \to Y'$ is a morphism of schemes. That should be possible if $f:X\to Y$ is representable. Then define $\deg(f):=\deg(f')$. Most, if not all …

Why aren't you happy with the moduli space of polarized K3's, i.e. pairs $(X,L)$ where $L$ is an ample line bundle? This is standard, and this at least makes sense over any field or $\mathbb Z$. And t …

The answer is always "no". By classification, a bielliptic surface over $\mathbb C$ has the form $(E\times F)/G$ where $E,F$ are elliptic curves, $G=\subset Aut(E,0)$ is an abelian group acting by com …