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I'm looking for an exact number and a reference and I searched papers about Gromov-Witten invariants but I failed to find an exact number of the degree zero Gromov-Witten invariant of quintic threefol …

Let $X,Y,Z$ are irrreducible varieties. $f:X\to Y$ is prpoer surjective and $g:Z \to Y$ is dominant. Then, $X\times_Y Z$ is irreducible? Moreover, it will be very helpful for me if there are other c …

In many references (Toen, Higher and derived stacks: a global overview, Toen, Vezzosi, Homotopical algebraic geometry II, and so on), the definition of $n$-geometric stack appears. In the non-derived …

Assume $X$ is a subscheme of $Y$ and $X,Y$ are irreducible. Then every irreducible component of the normal cone $C_{X/Y}$ dominates $X$?

There is an analogy of Chow's lemma for a DM stack $X$ written in the Laumon's book 'Champ algebrique'. There exists a generically finite, proper surjective morphism $Y \to X$ from a quasi-projective …

More precisely, consider a Segal groupoid $X_*$ in an infinity category of derived schemes : dSch In Toen's note, 'Derived Algebraic Geometry', he defines a 1-Artin stacks as a homotopy colimits of …

Let $X$ be an Artin stack and $Z \subset X$ be a closed substack. Can we represent $Z$ as a fundamental cycle? i.e. $[Z] = \sum_i a_i [Z_i]$ where $Z_i$ are integral substacks of $X$. In other word, c …

I'm reading Andrew Kresch's paper, Cycle groups in Artin Stacks. The author defined Chow groups of Artin stacks by very technical way, instead of ordinary ways which he called 'naive chow group', quo …

I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hirzebruch(Analytic cycles on complex manifolds) There is one thing I cannot understand. The followings are in th …

Let $G$ be a simply connected, semi-simple affine algebraic group and $C$ be a smooth projective curve with $g \geq 3$. Let $\mathcal{M}_G$ be a moduli stack of principal G-bundles on the curve $C$ a …