Бидний тухай
Багш ажилтан
Бид энд Эллиптик муруйн рационал торшин цэгүүдийг хэрхэн олох талаар судалсан. Эллиптик муруйн рационал торшин цэгүүдийг одоогийн байдлаар ерөнхий тохиолдолд олоогүй ба тухайн тохиолдолд, тодорхой хэлбэртэй үед олсон байдаг. Бид $y^2=x^3+ax^2+c^2(c-a)$, $a, c\in \ZZ$ байх муруйн рационал торшин цэгүүдийг судлан тодорхой үр дүнгүүд гаргасан.
Энэ ажилд Q гэсэн 2 cusp, 1node тэй 4-р эрэмбийн муруйн бүх боломжит сул шүргэгч коникуудыг шинж чанараар нь ангилан олсон. Үүний тулд эллиптик гадаргуйн онол ашигласан болно.
Here we study weak contact conics to an irreducible quartic $\mcQ$ with 3 points as its singularities via rational elliptic surfaces. We restrict to the case when $\mcC$ is a weak contact conic to $\mcQ$ such that $\mcQ \cap \mcC$ contains at least one smooth point $z_o$ of $\mcQ$ . Even under such restriction, we still obtain non-trivial examples of Zariski pairs.
Let $\mcQ$ be an irreducible quartic with two nodes and one cusp as its singularities and let $\mcC$ be a conic such that the intersection multiplicity at each point of $\mcC \cap \mcQ$ is even and $\mcC \cap \mcQ$ contain at least one smooth point $z_o$ of $\mcQ$. In this paper, for every $\mcQ$ we find all possible conics $\mcC$ as above via studying geometry of $\mcC$ and $\mcQ$ through that of integral sections of a rational elliptic surface which canonically arises from $\mcQ$ and $z_o \in \mcC \cap \mcQ$. As an application, we construct Zariski pairs of degree 7 and degree 8, whose irreducible components consist of $\mcQ$, $\mcC$ and line passing through two of the singular points of $\mcQ$ . Let $\mcB$ be a reduced plane curve and let $\mcC$ be a smooth conic in $\PP^2$. Let $I_x(\mcB,\mcC)$ denote the intersection multiplicity at $x \in \mcB \cap \mcC$. We first define a weak contact conic as follows: \begin{defin} If $I_x(\mcB,\mcC) $ is even for $\forall x \in \mcB \cap \mcC$, then $\mcC$ is called a weak contact conic to $\mcB$. Moreover if $\forall x \in \mcB \cap \mcC$ is a smooth point of $\mcB$ and $I_x(\mcB,\mcC) $ is even, then $\mcC$ is called a contact conic to $\mcB$. \end{defin} Here we study weak contact conics to an irreducible quartic $\mcQ$ with 2 nodes and 1 cusp only as its singularities via rational elliptic surfaces. We restrict to the case when $\mcC$ is a weak contact conic to $\mcQ$ such that $\mcQ \cap \mcC$ contains at least one smooth point $z_o$ of $\mcQ$ . Even under such restriction, we still obtain non-trivial examples of Zariski pairs, which we explain later. We first consider the following problem: \begin{prbm} Find all weak contact conics that are tangent to $\mcQ$ at $z_o$.
Бид энд Эллиптик муруйн рационал торшион цэгүүдийг хэрхэн олох талаар судалсан. Эллиптик муруйн рационал торшион цэгүүдийг одоогийн байдлаар ерөнхий тохиолдолд олоогүй ба тухайн тохиолдолд, тодорхой хэлбэртэй үед олсон байдаг. Бид $y^2=x^3+ax^2+c^2(c-a)$, $a, c\in \ZZ$ байх муруйн рационал торшион цэгүүдийг судлан тодорхой үр дүнгүүд гаргасан. \begin{defin}{Эллиптик муруй} $k$ талбар дээрх $E$ гэсэн онцгой биш кубик муруй болон $O$ цэгийг хамтад нь $k$ талбар дээрх эллиптик муруй гээд $E(k)$ гэж тэмдэглэнэ. \end{defin} \begin{mprop} Эллиптик муруйн цэгүүдийн нэмэх үйлдлийг хөвч шүргэгчийн аргаар тодорхойлвол энэ үйлдлийн хувьд эллиптик муруйн цэгүүд нь бүлэг үүсгэнэ. \end{mprop} \begin{defin} Хэрэв $E$ эллиптик муруйн $P$ цэгийн хувьд $mP=O$ байдаг $m$ эерэг бүхэл тоо олддог бол $P$ цэгийг $E$ эллиптик муруйн торшион цэг гэнэ. \end{defin} $E$ эллиптик муруйн торшион цэгүүд нь бүлэг үүсгэдэг билээ. \noindent $y^2=x^3+ax^2+c^2(c-a)$, $a, c\in \ZZ$ муруйн Тейтийн нормал хэлбэр лүү хувирган дискриминантыг тооцоолсон ба тодорхой тохиолдлуудад бүх торшион цэгүүдийг олсон. Тухайлбал $a=p$, $c=2p$, $p$ нь анхны тоо үед торшион цэгүүд оршин байхгүй гэдэг нь батлагдаж байгаа юм. $y=0$ үед $y_1+y_2+a_1x_2+a_3=0$ байгаа учир $$x+c=0\quad \text{ эсвэл }x^2+(c-a)x+c(c-a)=0$$ байх бүх бүхэл шийдүүд нь хоёр эрэмбэтэй байна. Энэ үед $x=-c$ шийдтэй ба бусад хоёр шийд нь бодит тоо байх нь нөхцөл нь $x^2+(c-a)x+c(c-a)=0$ тэгшитгэл бүхэл тоон шийдтэй байх зайлшгүй бөгөөд хүрэлцээтэй нөхцөл нь $(c-a)^2-4c(c-a)=(a+3c)(a-c)=A^2$ буюу бүхэл тооны квадрат байх юм. Эндээс \begin{itemize} \item[I.] $a=d((m+n)^2+mn)$, $c=dmn$ байна. $(m, n\in \ZZ)$ \item[II.] $a=d(m^2+mn+n^2)$, $c=-dmn$ байна. $(m, n\in \ZZ)$ \item[III.] $a=d(2(m+n)+mn)$, $c=dmn$ байна. $(m, n\in \ZZ)$ \end{itemize} Өөр бусад тохиолдлуудад торшион цэгүүдийг олох судалгаа явагдаж байгаа ба программ ашиглан тодорхой торшион цэгүүдийг олоод байгаа юм. $-10\leq c\leq 10$, $-10\leq a\leq 10$ үед python 3.0 ашиглан код бичиж бодуулахад хамгийн ихдээ 6 эрэмбийн цэг гарсан. \begin{align*} 3P=0 \text{ цэг } &c=-1, a=-2 \text{ үед } P = (2, 1)\\ 4P=0 \text{ цэг } &c=2, a=-1 \text{ үед } P = (2, 4), \\ &c=8, a=-4 \text{ үед } P = (8, 32),\\ &c=9, a=1 \text{ үед } P = (6, 30)\\ 6P=0 \text{ цэг } &c=-4, a=-8 \text{ үед } P = (0, 8), \\ &c=2, a=5 \text{ үед } P = (6, 8),\\ &c=4, a=0 \text{ үед } P = (8, 24),\\ &c=9, a=0 \text{ үед } P = (12, 81) \end{align*} \begin{mthm}{Мазур} $E$ нь $\QQ$ дээр тодорхойлогдсон эллиптик муруй байг. Тэгвэл түүний торшион бүлэг $E_T(\QQ)$ нь дараах 2 төрлийн бүлгийн аль нэгтэй изоморф байна: $C_n$ ($n$ эрэмбийн цикл бүлэг), $n=1,2,\dots,10,12$, эсвэл $C_2\times C_{2n}$, $n=1,2,3,4$. \end{mthm} Эндээс цэгийн эрэмбэ нь дээрээсээ зааглагдаж байгаа учир тодорхой эрэмбэтэй боломжит бүх цэгүүдийг тооцоолсноор бидний сонирхож буй хэлбэртэй тэгшитгэлээр өгөгдөх эллиптик муруйн боломжит бүх торшион цэгийг олох боломжтой болох юм. Цаашид эдгээр үр дүнгүүдээ нэгтгэн бусад ерөнхий тохиолдолд үр дүн гаргах зорилготой байна.
{\bf Abstract.} Let $\mcQ$ be an irreducible quartic with two nodes and one cusp as its singularities and let $\mcC$ be a conic such that the intersection multiplicity at each point of $\mcC \cap \mcQ$ is even and $\mcC \cap \mcQ$ contain at least one smooth point $z_o$ of $\mcQ$. In this paper, for every $\mcQ$ we find all possible conics $\mcC$ as above via studying geometry of $\mcC$ and $\mcQ$ through that of integral sections of a rational elliptic surface which canonically arises from $\mcQ$ and $z_o \in \mcC \cap \mcQ$. As an application, we construct Zariski pairs of degree 7 and degree 8, whose irreducible components consist of $\mcQ$, $\mcC$ and line passing through two of the singular points of $\mcQ$ . Let $\mcB$ be a reduced plane curve and let $\mcC$ be a smooth conic in $\PP^2$. Let $I_x(\mcB,\mcC)$ denote the intersection multiplicity at $x \in \mcB \cap \mcC$. We first define a weak contact conic as follows: \begin{defin} If $I_x(\mcB,\mcC) $ is even for $\forall x \in \mcB \cap \mcC$, then $\mcC$ is called a weak contact conic to $\mcB$. Moreover if $\forall x \in \mcB \cap \mcC$ is a smooth point of $\mcB$ and $I_x(\mcB,\mcC) $ is even, then $\mcC$ is called a contact conic to $\mcB$. \end{defin} Here we study weak contact conics to an irreducible quartic $\mcQ$ with 2 nodes and 1 cusp only as its singularities via rational elliptic surfaces. We restrict to the case when $\mcC$ is a weak contact conic to $\mcQ$ such that $\mcQ \cap \mcC$ contains at least one smooth point $z_o$ of $\mcQ$ . Even under such restriction, we still obtain non-trivial examples of Zariski pairs, which we explain later. We first consider the following problem: \begin{prbm} Find all weak contact conics that are tangent to $\mcQ$ at $z_o$. \end{prbm}
In this article, all varieties are defined over the field of complex numbers $\CC$. Let $\varphi : S \to \PP^1$ be a rational elliptic surface with a section $O$. Here a section means an irreducible curve on $S$ intersecting a fiber at one point or a morphism $s: \PP^1 \to S$ such that $\varphi\circ s = id_{\PP^1}$ (note that these two notions can be canonically identified). It is known that, if $\varphi$ has a reducible singular fiber, $S$ coincides with a rational elliptic surface $S_{\mcQ, z_o}$ associated with a reduced plane quartic $\mcQ$, which is not $4$ distinct lines meeting at one point, in $\PP^2$ and a smooth point $z_o$ on $\mcQ$ obtained in the following way: \begin{enumerate} \item[(i)] Let $S_o$ be the minimal resolution of the double cover of $\PP^2$ branched along $\mcQ$. \item[(ii)] Choose a smooth point $z_o$ of $\mcQ$. The pencil of lines through $z_o$ gives rise to a pencil $\Lambda_{\mcQ, z_o}$ of curves of genus $1$ on $S_o$. \item[(iii)] By resolving the base points of $\Lambda_{\mcQ, z_o}$, we have a rational elliptic surface $\varphi: S_{\mcQ, z_o} \to \PP^1$. We denote the generically $2$ to $1$ morphism from $S_{\mcQ, z_o}$ to $\PP^2$ by $f_{\mcQ,z_o} : S_{\mcQ, z_o} \to \PP^2$. \end{enumerate}
In this note, all varieties are defined over the field of complex numbers C. Let d be an even positive integer and let p(t, x) ∈ C[t, x] be a polynomial of the form x 3 + a1(t)x 2 + a2(t)x + a3(t) = 0, where degt ai(t) ≤ id. Our aim of this note is to consider when p(t, x) has a decomposition of the form (∗) p(t, x) = (x − xo(t))3 + (c0(t)x + c1(t))2 , xo(t), c0(t), c1(t) ∈ C[t]. The right hand side of (∗) is called a (2, 3) torus decomposition of the affine curve given by p(t, x) = 0. We will show that the above plane curve has degenerated (2, 3) torus decompositions by using arithmetic properties of elliptic surfaces and show that a 3-cuspidal quartic has infinitely many degenerated (2, 3) torus decompositions. Let E be an elliptic curve defined over the rational function field of one variable C(t) given by E : y 2 = p(t, x), and we denote the set of C(t)-rational points and the point at infinity O by E(C(t)). It is well-known that E(C(t)) becomes an abelian group, O being the zero element. Now our first statement is as follows: Proposition 1 Assume that both of plane curves given by p(t, x) = 0 and s 3d p(1/s, x′ /sd ) = 0 have at worst simple singularities in both of (t, x) and (s, x′ ) planes. Then p(t, x) has a decomposition as in (∗) if and only if E(C(t)) has a point P of order 3. The polynomial xo(t) is given by the x-coordinate of P. As an application of Proposition 1, we have the following theorem: Theorem 1 Let Q be a quartic with 3 cusps and choose a smooth point zo on Q. There exists a unique irreducible conic C as follows: (i) C is tangent to Q at zo and passes through three cusps of Q. (ii) Let FQ, FC, and Lzo be defining equations of Q, C and the tangent line Lzo of Q at zo, respectively. Then there exists a homogeneous polynomial G of degree 3 such that (∗∗) L 2 zo FQ = F 3 C + G 2 .
Энэхүү илтгэлдээ биноидын Гильберт-Кунзийн функцийг тооцоолох нэгэн аргын талаар авч үзнэ.
Let U be a universal subclass of a universal class V of rings. We investigate connections between radicals in U and V . We dene T and Ts as follows: T = {A\in A_{ass}| every prime homomorphic image of A is not a hereditary Amitsur ring} Ts = {A\in A_{ass}| every prime homomorphic image of A has no nonzero ideal which is a hereditary Amitsur ring}. Let gamma is one of T, Ts and let A be a commutative ring with minimum condition on ideals. We give a sufficient and necessary condition for A to be gamma semi-simple.
Let $\mcQ$ be an irreducible quartic with two nodes and one cusp as its singularities and let $\mcC$ be a conic such that the intersection multiplicity at each point of $\mcC \cap \mcQ$ is even and $\mcC \cap \mcQ$ contain at least one smooth point $z_o$ of $\mcQ$. In this talk, for every $\mcQ$ we find all conics $\mcC$ as above via studying geometry of $\mcC$ and $\mcQ$ through that of integral sections of a rational elliptic surface which canonically arises from $\mcQ$ and $z_o \in \mcC \cap \mcQ.$ As an application, we construct Zariski pairs of degree 7 and 8, whose irreducible components consist of $\mcQ$, $\mcC$ and line passing through two of the singular points of $\mcQ$ .
We investigate connections in the classes of rings with chain property and the lattice of strongly hereditary radicals.
