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9 votes
6 answers
5k views

Examples of naturally occurring Quadratic forms or quadrics.

I am always fascinated when a quadratic form (or a quadric) arises naturally. I have some elementary examples, but most of all, I want to learn more examples. I hope this question isn't considered too ...
9 votes
4 answers
752 views

Is any quadric birational to a product of Brauer-Severi varieties?

Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let $$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$ be a non-singular ...
Daniel Loughran's user avatar
7 votes
0 answers
259 views

K3 surfaces with no −2 curves

I seem to remember that a K3 surface with big Picard rank always has smooth rational curves. This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
Misha Verbitsky's user avatar
5 votes
0 answers
308 views

Algorithm for solutions to quadratic forms over number fields

Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)? I am especially interested in the quaternary case. There exist some ...
Casaubon's user avatar
  • 101
3 votes
1 answer
220 views

Reference request: relationship between discriminant and smoothness of a conic over arbitrary fields

I'm looking for an (ideally modern) reference of the relationship between the discriminant and smoothness of projective conics over arbitrary fields (including those of characteristic 2). Let $k$ be a ...
stupid_question_bot's user avatar
0 votes
1 answer
125 views

Polyhedral conditions for quadratic inequalities in fixed dimension

Denote $\mathcal T$ be set of $(T_1,T_2,T_3,T_4)\in\mathbb Z^4$ that satisfy $$0<T_1,T_2,T_3,T_4$$ conditions? Define the level set $$M_{\gamma}(Q,\mathcal T)=\{(T_1,T_2,T_3,T_4)\in\mathcal T:Q(...
Turbo's user avatar
  • 13.9k
0 votes
0 answers
179 views

Points at which a polynomial becomes reducible

Let $n \geq 10$ and set $\mathbf{y} = (y_1,\ldots,y_n)$. Let $Q_1(\mathbf{y}),\ldots,Q_5(\mathbf{y})$ be non-zero quadratic forms with integer coefficients such that the cubic form $x_1Q_1(\mathbf{y})+...
admissiblecycle's user avatar