Бидний тухай
Багш ажилтан
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.
We prove in a broad combinatorial setting, namely for nitely generated semipositive cancellative reduced binoids, that the Hilbert-Kunz multiplicity is a rational number independent of the characteristic
