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On Specializations of Curves. I
| Content Provider | Scilit |
|---|---|
| Author | Nobile, A. |
| Copyright Year | 1984 |
| Description | The following is proved: Given a family of projective reduced curves $X \to T$ ($T$ irreducible), if ${X_t}$ (the general curve) is integral and ${X_0}$ is a special curve (having irreducible components ${X_1}, \ldots ,{X_r}$), then $\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g({X_t})}$, where $g(Z) =$ geometric genus of $Z$. Conversely, if $A$ is a reduced plane projective curve, of degree $n$ with irreducible components ${X_1}, \ldots ,{X_r}$, and $g$ satisfies $\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g \leqslant \frac {1} {2}(n - 1)(n - 2)}$, then a family of plane curves $X \to T$ (with $T$ integral) exists, where for some ${t_0} \in T,{X_{{t_0}}} = Z$ and for $t$ generic, ${X_t}$ is integral and has only nodes as singularities. Results of this type appear in an old paper by G. Albanese, but the exposition is rather obscure. |
| Related Links | https://www.ams.org/tran/1984-282-02/S0002-9947-1984-0732116-X/S0002-9947-1984-0732116-X.pdf |
| Ending Page | 748 |
| Page Count | 10 |
| Starting Page | 739 |
| ISSN | 00029947 |
| e-ISSN | 10886850 |
| DOI | 10.2307/1999262 |
| Journal | Transactions of the American Mathematical Society |
| Issue Number | 2 |
| Volume Number | 282 |
| Language | English |
| Publisher | Duke University Press |
| Publisher Date | 1984-04-01 |
| Access Restriction | Open |
| Subject Keyword | Logic Special Curve X_t Irreducible T_0 Conversely Genus Old X_0 |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics |
