How Long to Read A Study of Singularities on Rational Curves Via Syzygies

How Long Does it Take to Read A Study of Singularities on Rational Curves Via Syzygies?

It takes the average reader 1 hour and 58 minutes to read A Study of Singularities on Rational Curves Via Syzygies by David A. Cox

Assuming a reading speed of 250 words per minute. Learn more

Description

Consider a rational projective curve $\mathcal{C}$ of degree $d$ over an algebraically closed field $\pmb k$. There are $n$ homogeneous forms $g_{1},\dots, g_{n}$ of degree $d$ in $B=\pmb k[x, y]$ which parameterize $\mathcal{C}$ in a birational, base point free, manner. The authors study the singularities of $\mathcal{C}$ by studying a Hilbert-Burch matrix $\varphi$ for the row vector $[g_{1},\dots, g_{n}]$. In the ``General Lemma'' the authors use the generalized row ideals of $\varphi$ to identify the singular points on $\mathcal{C}$, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch. Let $p$ be a singular point on the parameterized planar curve $\mathcal{C}$ which corresponds to a generalized zero of $\varphi$. In the `'triple Lemma'' the authors give a matrix $\varphi'$ whose maximal minors parameterize the closure, in $\mathbb{P}^{2}$, of the blow-up at $p$ of $\mathcal{C}$ in a neighborhood of $p$. The authors apply the General Lemma to $\varphi'$ in order to learn about the singularities of $\mathcal{C}$ in the first neighborhood of $p$. If $\mathcal{C}$ has even degree $d=2c$ and the multiplicity of $\mathcal{C}$ at $p$ is equal to $c$, then he applies the Triple Lemma again to learn about the singularities of $\mathcal{C}$ in the second neighborhood of $p$. Consider rational plane curves $\mathcal{C}$ of even degree $d=2c$. The authors classify curves according to the configuration of multiplicity $c$ singularities on or infinitely near $\mathcal{C}$. There are $7$ possible configurations of such singularities. They classify the Hilbert-Burch matrix which corresponds to each configuration. The study of multiplicity $c$ singularities on, or infinitely near, a fixed rational plane curve $\mathcal{C}$ of degree $2c$ is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix $\varphi$ for a parameterization of $\mathcal{C}$.

How long is A Study of Singularities on Rational Curves Via Syzygies?

A Study of Singularities on Rational Curves Via Syzygies by David A. Cox is 116 pages long, and a total of 29,696 words.

This makes it 39% the length of the average book. It also has 36% more words than the average book.

How Long Does it Take to Read A Study of Singularities on Rational Curves Via Syzygies Aloud?

The average oral reading speed is 183 words per minute. This means it takes 2 hours and 42 minutes to read A Study of Singularities on Rational Curves Via Syzygies aloud.

What Reading Level is A Study of Singularities on Rational Curves Via Syzygies?

A Study of Singularities on Rational Curves Via Syzygies is suitable for students ages 10 and up.

Note that there may be other factors that effect this rating besides length that are not factored in on this page. This may include things like complex language or sensitive topics not suitable for students of certain ages.

When deciding what to show young students always use your best judgement and consult a professional.

Where Can I Buy A Study of Singularities on Rational Curves Via Syzygies?

A Study of Singularities on Rational Curves Via Syzygies by David A. Cox is sold by several retailers and bookshops. However, Read Time works with Amazon to provide an easier way to purchase books.

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