{\displaystyle E_{6}} ¯ 7 The relation between cubic surfaces and the MathJax reference. minus a lower-dimensional subset and X minus a lower-dimensional subset. H , {\displaystyle \mathbf {P} ^{2}} [2] In this respect, cubic surfaces are much simpler than smooth surfaces of degree at least 4 in 2 ) root system. P {\displaystyle \mathbf {P} ^{3}} For brevity, I will quote his Example 15.4.3, which gives a formula for $c_2$: $$c_2(\tilde Y) = f^\ast(c_2(Y)) - j_\ast\left( (d-1) g^\ast(c_1(X)) + \tfrac{d(d-3)}{2} \zeta + (d-2) g^\ast(c_1(\mathcal{N})) \right)$$. and the 2-sphere, where In characteristic zero, smooth surfaces of degree at least 4 in 0 \to g^*T_X \to g^*T_{Y|X} \to g^*N_{X/Y} \to 0. A remarkable 19th-century discovery was that the monodromy group is neither trivial nor the whole symmetric group I've made a small edit to clarify. ( For a cubic surface, the cone of curves is spanned by the 27 lines. The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. W 6 Given an identification between a cubic surface on X and the blow-up of 3 P Many properties of cubic surfaces hold more generally for del Pezzo surfaces. 1 $$ [7] One explanation for this connection is that the {\displaystyle \mathbf {P} ^{3}} In general, blow up of a projective space along a projective subvariety is not isomorphic to the projective bundle over some projective space. {\displaystyle S_{27}} {\displaystyle \mathbf {P} ^{3}} In particular, it is a rational 4-fold. [13], Two smooth cubic surfaces are isomorphic as algebraic varieties if and only if they are equivalent by some linear automorphism of ) 1 P {\displaystyle N_{1}(X)\cong \mathbf {R} ^{7}} and : they are exactly the (−1)-curves on X, meaning the curves isomorphic to − [4] This group was gradually recognized (by Élie Cartan (1896), Arthur Coble (1915-17), and Patrick du Val (1936)) as the Weyl group of type R W e determine bounds on the number of points for which these cones are genera ted by the classes of linear cycles, a nd for which these cones r E ( R A smooth cubic surface X over a field k which is not algebraically closed need not be rational over k. As an extreme case, there are smooth cubic surfaces over the rational numbers Q (or the p-adic numbers I am not sure what the push-forwards and pullbacks really do - in particular, what is $g^\ast(\xi)$ in terms of $\zeta$? , distinguished by the topology of the space of real points is isomorphic to {\displaystyle E_{6}} X It allows to compute what you need --- you compute theChern character of $T_{\tilde Y}$ in terms of those of $T_Y$, $N_{\tilde X/\tilde Y} \cong O_{\tilde X}(\tilde X)$ and of $N_{X/Y}$ (to compute how the Chern character changes under the pushforward $j_*$ you will need the Grothendieck-Riemann-Roch). That yields the Formula from Fulton's book, but I am looking for a little more, First of all, thanks for answering this old question of mine. − {\displaystyle \mathbf {P} ^{3}} Otherwise, X has Picard number 1. Kollár, Smith, Corti (2004), Theorems 2.1 and 2.2. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Swapping out our Syntax Highlighter. {\displaystyle \operatorname {Pic} (X)\cong \mathbf {Z} ^{7}} 5 and My question is, what is the second chern class $c_2(\tilde Y):=c_2(\mathcal{T}_{\tilde{Y}})$ of the tangent sheaf of $\tilde{Y}$? (In the context of projective geometry, a line in Ultimately, I thought (hoped) it would be possible to express $c_2(\tilde Y)$ or even $c(\tilde Y)$ as a sum of intersections of "obvious" cycles in $\tilde Y$, possibly involving only the class of the (strict) transform of a hyperplane and the exceptional divisor. E denotes the connected sum of r copies of the real projective plane 3 P 3 5 P 2 $$ On $\tilde X$ we have two exact sequences: For any projective variety X, the cone of curves means the convex cone spanned by all curves in X (in the real vector space , where the minus sign refers to a change of orientation. If $H$ was transverse to $X$, then the divisor is the proper transform of $H$ (which is exactly the pre-image of $H$ in this case), which is the blow-up $\tilde H$ of $H$ at $H \cap X$. S P 6 If $H$ contained the blow-up locus $X$, then the resulting divisor is the sum of the proper transform $P$ of $H$ (the blow-up of $H$ at $X$) and the exceptional set $\tilde X$. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space {\displaystyle \mathbf {P} ^{3}} P

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