Xiaolong Liu
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I'm interested in higher dimensional algebraic geometry, birational geometry, moduli space of varieties and bundles and derived categories of sheaves.

Some Gaps and Examples in Intersection Theory by Fulton IV (The End)

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This is one of a series of blogs aiming to complete some details of the examples in this book (Intersection Theory, 2nd edition by William Fulton1) and give some comments. This blog we consider chapter 14 to chapter 17.

Previously on these:

Some Gaps and Examples in Intersection Theory by Fulton I;

Some Gaps and Examples in Intersection Theory by Fulton II;

Some Gaps and Examples in Intersection Theory by Fulton III.

Chapter 14. Degeneracy Loci and Grassmannians

Section 14.1. Localized Top Chern Class

Example 14.1.3. Let $i:Z(s)\to Z(s_2)$ and $j:Z(s_2)\to X$, then we have

\begin{align*} j_*i_*\mathbb{Z}(s)&=c_e(E)\cap [X]=c_{e_1}(E_1)c_{e_2}(E_2)\cap [X]\\ &=c_{e_1}(E_1)\cap j_*\mathbb{Z}(s_2)=j_*(c_{e_1}(j^*E_1)\cap\mathbb{Z}(s_2)), \end{align*}

as we need to show that $i_* \mathbb{Z}(s)=c_{e_1}(j^* E_1)\cap\mathbb{Z}(s_2)$, we don’t know is $j_* $ injective?

Example 14.1.5. (a) We have

\begin{align*} \deg\mathbb{Z}(s_f)&=\int_{\{t_1,...,t_n\}}\mathbb{Z}(s_f)=\int_Xc_n(f^*T_C\otimes T_X^{\vee})\\ &=(-1)^n\chi(X)+(-1)^{n-1}\int_Xc_1(f^*T_C)c_{n-1}(T_X)\\ &=(-1)^n\chi(X)+(-1)^{n-1}\int_Cc_1(T_C)\cap f_*(c_{n-1}(T_X)\cap [X]). \end{align*}

As $f_* (c_{n-1}(T_X)\cap [X])$ is $1$-cycle on $C$, it must of form $n[C]$. Pick a general fiber $X_t$ with $t\in C$, we get $n=\int_{X_t}c_{n-1}(T_X|_{X_t})=\chi(X_t)$, hence we get

\begin{align*} \deg\mathbb{Z}(s_f)&=(-1)^n\chi(X)+(-1)^{n-1}\int_Cc_1(T_C)\cap f_*(c_{n-1}(T_X)\cap [X])\\ &=(-1)^n\chi(X)+(-1)^{n-1}\chi(X_t)\int_Cc_1(T_C)\cap [C]=(-1)^n(\chi(X)-\chi(X_t)\chi(C)), \end{align*}

and well done. $\blacksquare$

Section 14.2. Gysin Formulas

Let $\underline{A}$ be a flag of vector bundles on $X$, i.e. $0\subsetneqq A_1\subsetneqq\cdots\subsetneqq A_d=A$. Now we may want to define the flag bundle $Fl(\underline{A})$ as an $X$-scheme by a representable functor, if we define

\begin{align*} Fl(\underline{A})^{???}&:(Sch/X)^{opp}\to^{???}(Sets),\\ &(f:T\to X)\mapsto^{???} \{D_1\subset\cdots\subset D_d:\dim D_i=i,D_i\subset f^*A_i\}, \end{align*}

then as the case of the Grassmannian we may lost the functorial-property, so may not that case! Note that this is NOT the triditional flag variety, so if we use the quotient definition, we can not guarantee the inclutions of $A_i$! So the only way is to consider the construction in the proof of Proposition 14.2.1.

For the triditional flag variety, we can define that as follows:

Consider the following results taken from [GT20]2 Proposition 8.10:

Lemma. Let $S$ be a scheme and let $i:\mathscr{U}\to\mathscr{E}$ be a morphism of $\mathscr{O}_S$-modules. Consider the following statements:

(a) The homomorphism $i$ is injective and $\mathscr{E}/\mathscr{U}$ is locally free;

(b) For any scheme-morphism $f:T\to S$ the map $f^* (i):f^* \mathscr{U}\to f^* \mathscr{E}$ is injective.

If $\mathscr{U}$ is any $\mathscr{O}_S$-module and $\mathscr{E}$ is quasi-coherent, then (a) implies (b); If $\mathscr{U}$ is of finite type and $\mathscr{E}$ finite locally free. Then (a) and (b) are equivalent. $\blacksquare$

Hence we can using the quotient to give a functorial-property-well difinition of triditional flag variety: Given a sequence $(d_1,…,d_m)$ of positive integers of sum $n$ and fix a vector bundle $E$ over $X$ of rank $n$. A flag of type $(E;d_1,…,d_m)$ is $0=V_0\subset V_1\subset\cdots\subset V_m=E$ of bundles in $X$ where $V_j/V_{j-1}$ is locally free of rank $d_j$. Consider the functor $\mathrm{Flag}_X(E;d_1,…,d_m):(Sch/X)^{opp}\to(Sets)$ as $(f:T\to X)\mapsto$ (type $(E;d_1,…,d_m)$ flags). Well done.

Over. $\blacksquare$

Section 14.3. Determinantal Formula

Theorem 14.3. When we consider the universal local case, the birational morphism $\eta:Z(s_{\sigma})\to\Omega(\underline{A},\sigma)$ plays an important role. To see that $\eta$ is birational, write down the elements of $\phi^{-1}(x)\cap Z(s_{\sigma})$ for some $x\in X=\mathbb{A}^{ef}$. We find that this set will contain a single element if we remove the locus where the $\dim\ker\sigma(x)|_{A_i(x)}<i$, that is, the locus where $(a_i-i)$-minors vanishing in $\Omega(\underline{A},\sigma)$. $\blacksquare$

Section 14.4. Thom-Porteous Formula

Example 14.4.5. Here we may assume that $C$ is a smooth curve over some $k$ as it is the same as Example 4.3.2/4.3.3.

As we get $0\to E_d\to E_m\to F_t$, we may then let $m\geq 2g-1$ instead of $t\geq 2g-1$ as by Riemann-Roch theorem we have \[h^0(\mathscr{O}(mP_0)\otimes\mathscr{L}_x)=m+1-g+h^0(K_C-mP_0-\mathscr{L}_x)\] and $\deg(K_C-mP_0-\mathscr{L}_x)=2g-2-m$. But as $m=t+d$, then $t\geq 2g-1$ implies $m\geq 2g-1$.

Not that the $C$ of genus $g$ is of general moduli means that the point corresponding to the curve $C$ in the moduli space of genus $g$ curves $M_g$ does not belong to denumerably many proper subvarieties. $\blacksquare$

Example 14.4.7.

Example 14.4.12. I don’t know what’s going on!

Section 14.6. Grassman Bundles

Example 14.6.2.

Example 14.6.4. Why and how do we use this?

Section 14.7. Schubert Calculus

Example 14.7.7. This is a standard example to use the Schubert calculus to deal with some simple algebraic geometry problems and we write this as a model. Note that the first step is to deduce the relations of Schubert relations as Example 14.7.2.

(a) After checking the dimension of $[W_C]$, we note that $[W_C]=ag_p+bg_e$. Then by Example 14.7.1 we get \[a=\int_{G_1(\mathbb{P}^3)}[W_C]\cap (0,3),b=\int_{G_1(\mathbb{P}^3)}[W_C]\cap (1,2).\] Then we find that using the definiton of $(a_0,…,a_d)$ we get $a$ is just the apparent number $d$ of double points of $C$. Similarly, $b$ is the number of 2 points in whole $\deg C=n$ points over a general plane (by Bertini) and we get $b=\binom{n}{2}$.

(b) After analyze the geometry-meanings, we find that this is just \[\int_{G_1(\mathbb{P}^3)}[W_C]\cap[V_{C_1}]\cap[V_{C_2}]\] and using the relations in Example 14.7.2 and the results in Example 14.7.6.(a) we get the result.

(c) Similar as (b), we find that this is just \[\int_{G_1(\mathbb{P}^3)}[W_C]\cap[W_{C’}]\] and well done. $\blacksquare$

Example 14.7.11. Need to think that why the hyperplane section of $G$ (in Plucker embedding) coorespond to the $n-d-1$-planes in $\mathbb{P}^n$?

Example 14.7.12.(d). How to compute $\int_Xc_1((A_X^{\vee})^{\otimes 2}\otimes\mathscr{O}(1)_X)^8=92$ and re-think the Example 3.2.22.

Example 14.7.13. This is a great result and for details of this we refer [3264EH]3 Proposition 6.4. Here we give a sketch.

By the property of $S$, we have that the fiber of $S^{\vee}$ at $[L]\in G$ is the space of linear forms on $L$, that is $H^0(\mathscr{O}_{P(L)}(1)$. Hence the surjection $E_G^{\vee}\to L^{\vee}$ is the restrict of linear forms to $L$. Hence so is acting $\mathrm{Sym}^m$ to deduce the $m$-degree forms case. Now a degree $m$ hyperplane $X$ defined by a zero of section $g$ of $\mathrm{Sym}^mE^{\vee}$ which deduce a degree $m$ section $g’$ of $\mathrm{Sym}^mS^{\vee}$ by restriction. Hence the zero of $g’$ is the points $[L]$ in $G$ such that $g’|_{L}=0$. As $X$ is the zero of $g$, then these $L$ is the $d$-planes in $X$! $\blacksquare$

Chapter 15. Riemann-Roch for Non-singular Varieties

Section 15.1. Preliminaries

Example 15.1.1. (a) Here we show that $K_0(\mathbb{P}^m)$ generated by $[\mathscr{O}(n)]$ for $0\leq n\leq m$. Pick any coherent sheaf $\mathscr{F}$ and then $\mathscr{F}\otimes\mathscr{O}(N)$ generated by global sections for some large $N$. Hence we get surjection $\bigoplus_{j=1}^r\mathscr{O}(-N)\to\mathscr{F}$ as $X$ is quasicompact. Repeat this, and at some point we will get $0$. Hence $K_0(\mathbb{P}^m)$ generated by $[\mathscr{O}(n)]$ for all $n$.

Next, we consider $\mathscr{O}(m+1)$. Let $V=\bigoplus_j^{m+1}\mathscr{O}(-1)$ and we claim that we have exact sequence \[0\to\bigwedge^{m+1}V\to\bigwedge^{m}V\to\cdots\to\bigwedge^{1}V\to\bigwedge^{0}V\to 0.\] this can be check over level of graded modules. Let $S=k[x_0,…,x_m]$ and the map $V=\bigoplus_j^{m+1}S(-1)\to S$ is multiplication by $(x_0,…,x_m)$ which induce the maps $\bigwedge^{j+1}V\to\bigwedge^{j}V$. We can check it is exact and we omit it here. Now after taking dual, this exact sequence give us \[0\to\mathscr{O}\to\bigoplus^{m+1}\mathscr{O}(1)\to\cdots\to\bigoplus^{\binom{m+1}{j}}\mathscr{O}(j)\to\cdots\to\bigoplus^{m+1}\mathscr{O}(m)\to\mathscr{O}(m+1)\to0\] is exact. Repeat this and we get $K_0(\mathbb{P}^m)$ generated by $[\mathscr{O}(n)]$ for $0\leq n\leq m$.

(b) We may use the Exercise II.6.10.(c) in [Har77]4. $\blacksquare$

Example 15.1.4. In order to compute $\int_{\mathbb{P}^m}\frac{e^{nx}x^{m+1}}{(1-e^{-x})^{m+1}}$, we just need to compute that after expanding to the power series of $x$, the coefficient of $x^m$ is that value. This is just the residue of $\frac{e^{nx}}{(1-e^{-x})^{m+1}}$ around $x=0$.

Now let $y=1-e^{-x}$, then \[\int_{C}\frac{e^{nx}}{(1-e^{-x})^{m+1}}dx=-\int_{C’}\frac{1}{y^{m+1}(1-y)^n}d\log (1-y)=\int_{C’}\frac{1}{y^{m+1}(1-y)^{n+1}}dx.\] Now $\frac{1}{y^{m+1}(1-y)^{n+1}}=\sum_{i=0}^{\infty}\binom{n+i}{i}y^{i-m-1}$. Hence well done. $\blacksquare$

Example 15.1.4. Here thay claim that $F_kK_0X$ generated by $[\mathscr{O}_V]$ for $\dim V\leq k$. WLOG we let $k=\dim X$ and $F_kK_0X=K_0X$. Now let $K\subset K_0X$ be a subgroup generated by $[\mathscr{O}_V]$ for $\dim V\leq\dim X$ and we claim that $K=K_0X$. First, for any subscheme $Z\subset X$ we have $0\to\mathscr{I}_Z\to\mathscr{O}_X\to\mathscr{O}_Z\to0$, then any ideal sheaf living in $K$. Next we need the following “Devissage” results:

Lemma. Let $X$ be a Noetherian scheme. Let $\mathscr{F}$ be a coherent sheaf on $X$. There exists a filtration \[0\subset\mathscr{F}_0\subset\cdots\subset\mathscr{F}_m=\mathscr{F}\] by coherent subsheaves such that for each $j=1,…,m$ there exist an integral closed subscheme $Z_j\subset X$ and a nonzero coherent sheaf of ideals $\mathscr{I}_j$ such that $\mathscr{F}_j/\mathscr{F}_{j-1}\cong (Z_j\to X)_* \mathscr{I}_j$.

Proof. This is a little bit complicated and we refer Tag 01YF. $\blacksquare$

Now using this lemma inductively, we can get that $K=K_0X$. Actually the next argument in the book giving the similar things. $\blacksquare$

Section 15.2. Grothendieck-Riemann-Roch Theorem

Example 15.2.10. Now as we have $0\to\mathscr{O}_{P(E)}\to p^* E\otimes\mathscr{O}(1)\to T_{P(E)/X}\to0$, we have

\begin{align*} \chi(P(E),\mathscr{O}_{P(E)}&=\int_{P(E)}\mathrm{td}(T_{P(E)})\\ &=\int_{P(E)}\mathrm{td}(p^*T_X)\mathrm{td}(p^* E\otimes\mathscr{O}(1))\\ &=\int_{X}p_*(\mathrm{td}(p^*T_X)\mathrm{td}(p^* E\otimes\mathscr{O}(1))\cap[P(E)])\\ &=\int_{X}\mathrm{td}(T_X)p_*(\mathrm{td}(p^* E\otimes\mathscr{O}(1))\cap[P(E)])=? \end{align*}

Example 15.2.17. How can we let $X$ to be the smooth projective variety?

Chapter 16. Correspondences

Section 16.1. Algebra of Correspondences

Example 16.1.4.(c). As $P$ is rational, if we let $q=P^{XY}_X$, we have \[\int_{X\times Y}\alpha\cdot[P\times Y]=\int_{X}q_* (\alpha\cdot q^* [P])=\int_{X}q_* \alpha\cap [P]=d_1(\alpha)\] and well done. $\blacksquare$

Chapter 17. Bivariant Intersection Theory

  1. [FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩

  2. [GT20] Ulrich Görtz, Torsten Wedhorn. Algebraic Geometry I: Schemes, 2nd. Springer Spektrum Wiesbaden. 2020. ↩

  3. [3264EH] David Eisenbud, Joe Harris. 3264 and All That, A Second Course in Algebraic Geometry. 2016. ↩

  4. [Har77] Robin Hartshorne. Algebraic Geometry. Springer, 1977. ↩

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