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The Hilbert scheme $S^{[n]}$ of $n$ points on a smooth surface $S$ is smooth for any $n$, but it is never Fano if $n>1$ since the canonical divisor is trivial on the fibers of the morphism to $S^{(n)} …

Quantum cohomology, and more generally Gromov Witten invariants, are invariants under deformation. Given a family $X$ of compact symplectic manifolds over a base $B$, to a path in $B$ one can associa …

Assume $V$ and $W$ are quasiprojective. Let $i:V\to X$ be a locally closed embedding with $X$ projective (for instance $X$ could be $P^n$). Consider the induced map $g:V\to X\times W$; this is also a …

If p is a nonsingular point, then we can define a tangent vector as an equivalence class of (nonsingular) curves, just as in the differentiable case. In fact, being nonsingular is equivalent to every …

The classical reference is Illusie's PhD thesis "complexe cotangent et deformations".There the result without the marked points is proven in a very, very general context using the cotangent complex; t …

What naturally associates to each ramification point (or more generally, irreducible divisor in the ramification locus) is a pair $(H,\psi)$ where $H$ is a cyclic subgroup of $G$ (with $\ge 2$ element …

In the case $g=1$ modular forms are sections of a line bundle on a (geometrically meaningful) compactification of the stack obtained by quotienting the upper half plane by $SL_2(\mathbb Z)$ (usually d …

Emerton has already answered, but let me summarize all the steps: 1) to a perfect torsion complex in the derived category associate a Cartier divisor; 2) to every sequence of morphisms $C\to X\to S$ …

If $X$ is a surface, then the Hilbert polynomial of the canonical divisor determines $d:=1-\chi(T_X)$ which is a lower bound for the dimension of any component of the moduli at the point $[X]$. Hence …

The first complete definition of GWI in algebraic geometry is more or less contemporary to the first complete definition in symplectic geometry. In algebraic geometry you can, e.g., use virtual locali …