# Questions tagged [hilbert-schemes]

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### Degrees of syzygies of points in $\mathbb P^2$

Let $X$ be a collection of points in $\mathbb P^2$ over the complex numbers. Let $I_X$ be the defining ideal. I am interested in knowing when: The syzygies of $I_X$ contains no linear forms. Since ...
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### Hilbert scheme of points concentrated in a given point

It is well known that if $X$ is a smooth surface, then the Hilbert scheme of points $X^{[n]}$ is also smooth. What about the subscheme $S_p$ of $X^{[n]}$ consisting of all schemes of finite length $Z$ ...
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### Proposition from Kollar's Rational Curves on Algebraic Varieties

$\DeclareMathOperator\Hom{Hom}$I have some questions on a proof of Proposition II.3.10 & notations from Rational Curves on Algebraic Varieties by Janos Kollar (page 117). We work in setting ...
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### Relations between double coinvariants and affine Springer fibers

Diagonal coinvariants have an interpretation from https://arxiv.org/abs/math/0201148 in terms of the Hilbert scheme. There are two recent papers https://arxiv.org/pdf/1801.09033.pdf and https://arxiv....
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### Tangent space to Hilbert schemes of points

Let $X$ be a smooth, projective rational surface and $Z$ be a zero-dimensional subscheme of $X$. Denote by $\mathcal{I}_Z$ the ideal sheaf of $Z$ in $X$ and $\mathcal{O}_Z$ the structure sheaf. Is it ...
322 views

### How to understand the proof of Proposition 2.1 in the paper 'Nodes and the Hodge conjecture'?

In the Proposition 2.1 of the paper 'Nodes and the Hodge conjecture', R.P.THOMAS gives a proof to descending the Hodge conjecture into showing that every (n,n)-Hodge class in a $2n$-dimensional smooth ...
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### Top intersections on the Hilbert scheme of points on a surface

The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism. ...
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### If a subgroup H of a finite group G acts freely on a variety, can the G-Hilbert scheme be computed by iteration?

Let $X$ be a smooth quasi projective variety over $\mathbb{C}$. Let $G$ be a finite abelian group acting via automorphisms on $X$. Denote by $G$-$\text{Hilb}(X)$ the subscheme of the Hilbert scheme ...
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### Hilbert scheme of a plane conic union a point

In Alex Lee's undergraduate thesis (2000), it was said that the Hilbert scheme $H_{2m+2}(\mathbb{P}^3)$ has two components $\mathcal{H}',\mathcal{H}''$, where a general point of $\mathcal{H}'$ ...
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### Transversality of quadrics containing a projective curve

Let $C$ be a curve of genus $g$ and $L$ a $g^r_d$ on it and assume that we are in the range ${r+2\choose 2}>2d-g+1$. If $C$ and $L$ are chosen to be general then by the maximal rank conjecture (...
Let $C\subset X$ be a smooth irreducible curve of genus $g$, embedded in a smooth projective 3-fold $X$. So its homology class $\beta=[C]\in H_2(X)$ is an irreducible class. I want to compare two ...